\documentclass[12pt]{article}

%% Math packages
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\usepackage{fancyhdr,fancybox,amsmath,amsthm,amssymb}
% Removes the "Too many math alphabets used in version normal" error.
\newcommand\hmmax{0}
\newcommand\bmmax{0}
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\usepackage{cancel}
\usepackage{mathdots}
\usepackage{venndiagram}
\usepackage{mathrsfs}          % Math script font

% Graphics
%%
\graphicspath{{./}{figs/}}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{matrix}
\usepackage{tikz-3dplot}
\usepackage{tkz-graph}
\usepackage{tikz-cd}

%% Colors (most are in colors.tex file)
%%
\usepackage{xcolor}
\usepackage{color}
\usepackage{visualalgebra}  %% Put this *after* the TikZ packages

%% Page layout packages
%%
\usepackage{url}
\usepackage{multicol}
\usepackage[numbers,square,sort&compress]{natbib}

%% For HW assignments
\usepackage{fancyhdr,fancybox}
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\headsep=0.2in
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\footskip=.5in
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%% Font and formatting packages
%%
\usepackage[english]{babel}    % Removing this causes compiler error
\usepackage{alltt}             % Like verbatim, but excludes \ and { }
%\usepackage{enumerate}         % [shortlabels] option??
\usepackage{comment}
\usepackage{soul}              % strikeout text
\usepackage{bm}                % Bold math
\usepackage[T1]{fontenc}
\usepackage{relsize}

\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due March 28 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------

\item Consider the right action of $G=D_6=\<r,f\>$ on following set
  of $31$ ``binary hexagons,'' where $r$ rotates each one $60^\circ$
  counterclockwise, and $f$ flips each one horizontally (i.e.,
  across a vertical axis).
  \[
  \begin{tikzpicture}[scale=.65]
    \begin{scope}[shift={(-3.75,0)}]
      \node at (0,0) {$S=\Bigg\{$};
    \end{scope}
    %%
    \begin{scope}[shift={(-1.5,0)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(1.5,0)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(4.5,0)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(7.5,0)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(10.5,0)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(13.5,0)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(16.5,0)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(0,-3)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(3,-3)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(6,-3)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(9,-3)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(12,-3)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(15,-3)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(1.5,-6)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(4.5,-6)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(7.5,-6)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(10.5,-6)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(13.5,-6)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(16.5,-6)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(0,-9)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(3,-9)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
     \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(6,-9)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(9,-9)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(12,-9)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(15,-9)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(1.5,-12)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(4.5,-12)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(7.5,-12)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(10.5,-12)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(13.5,-12)}]
      \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    %%
    \begin{scope}[shift={(16.5,-12)}]
      \path[fill=actPurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=actPurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
    \end{scope}
    %%
    \begin{scope}[shift={(18.25,-12)}]
      \node at (0,0) {$\Bigg\}$};
    \end{scope}
    %%
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Draw the action graph.
  \item Construct the ``\emph{fixed point table}'', which has a
    checkmark in row $g$ and column $s$ if $s\Dot\phi(g)=s$.
  \item Next to each $s\in S$ on your action graph, write
    $\stab(s)$, the stabilizer subgroup, using its generators. Which
    subgroups of $D_6$ don't appear?
  \item The \emph{fixator} of each $g\in D_6$, denoted $\fix(g)$, can be read
    off of the the fixed point table. What is the average size $|\fix(g)|$? 
  \item Find $\Ker(\phi)$ and $\Fix(\phi)$.
  \end{enumerate}

\item Suppose that $G$ acts on $S$ via the homomorphism $\phi\colon
  G\to\Perm(S)$.
  \begin{enumerate}
  \item Show that $\stab(s)$ is a subgroup for all $s\in S$. Use the
    notational conventions that we have been using in lecture.
  \item Show that the stablizers of any two elements in the same orbit
    are conjugate---specifically, that
    $\stab(s\Dot\phi(g))=g^{-1}\stab(s)g$ for all $g\in G$ and $s\in
    S$. This relationship is summarized by the following commutative
    diagram.
    \[
    \begin{tikzcd}[column sep=5em,row sep=3.0em]
      s \arrow[r,mapsto,"\phi(x)"] \arrow[d,mapsto,swap,"\phi(g)"]
      & s \arrow[d,mapsto,"\phi(g)"] \\   
      s' \arrow[r,mapsto,dashed,"\phi(g^{-1}xg)"] & s'
    \end{tikzcd}
    \]
  \end{enumerate}


  %%-------------------------------------------------------------------
  
  
\item Suppose a group $G$ of order $55$ acts on a set $S$ of size
  $14$. Let $s\in S$ be an arbitrary element.
  \begin{enumerate}
  \item What are the possible sizes of the orbit of $s$?
  \item What are the possible sizes of the stabilizer of $s$?
  \item Show that this action must have a fixed point.
  \item What is the fewest number of fixed points that this action can
    have? Justify your answer. \bigskip
  \end{enumerate}

  
  %%-------------------------------------------------------------------
  
  
\item Let $G=D_6=\<r,f\>$ act on its set $S=\{H\leq D_6\}$ of
  subgroups by conjugation, i.e.,
  \[
  \phi\colon G\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize \emph{the
      permutation that sends each $H\mapsto g^{-1}Hg$.}}
  \]
  A Cayley graph, cycle graph, and subgroup lattice for $D_6$
  are shown below.
  
  \[
  \hspace*{-8mm}
  \begin{tikzpicture}
    \newcommand\Hf{6.495} % height of D_6 
    \newcommand\He{5.196} % height of C_6  
    \newcommand\Hd{3.897} % height of V_4 
    \newcommand\Hc{2.598} % height of C_3  
    \newcommand\Hb{1.299} % height of C_2 
    \newcommand\Ha{0} % height of C_1   
    \tikzstyle{R6-out} = [draw, very thick, eRed,-stealth,bend right=18]
    \tikzstyle{R6-in} = [draw, very thick, eRed,-stealth,bend left=12]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt,
      minimum size=6.5mm] 
    %%
    %% ------------- Cycle graph of D_6 ---------------------------
    %%
    \begin{scope}[shift={(0,7.5)},scale=.68]
      \tikzstyle{v-r} = [circle, draw, fill=vRed,inner sep=0pt,
        minimum size=6mm]
      \tikzstyle{v-g} = [circle, draw, fill=vGreen,inner sep=0pt,
        minimum size=6mm]
      \tikzstyle{v-b} = [circle, draw, fill=vBlue,inner sep=0pt,
        minimum size=6mm]
      \tikzstyle{v-p} = [circle, draw, fill=vPurple,inner sep=0pt, 
        minimum size=6mm]
      \tikzstyle{v-o} = [circle, draw, fill=vOrange,inner sep=0pt, 
        minimum size=6mm]
      \tikzstyle{v-gr} = [circle, draw, fill=midgray,inner sep=0pt, 
        minimum size=6mm]
      \tikzstyle{cy2} = [draw,thick]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (90:2) [v-gr] {$1$};
      \node (r) at (150:2) [v-r] {$r$};
      \node (r2) at (210:2) [v-o] {$r^2$};
      \node (r3) at (270:2) [v-p] {$r^3$};
      \node (r4) at (330:2) [v-o] {$r^4$};
      \node (r5) at (30:2) [v-r] {$r^5$};
      \node (f) at (-3,4) [v-b] {$f$};
      \node (r2f) at (-1.8,4) [v-b] {$r^2\!f$};
      \node (r4f) at (-.6,4) [v-b] {$r^4\!f$};
      \node (rf) at (.6,4) [v-g] {$rf$};
      \node (r3f) at (1.8,4) [v-g] {$r^3\!f$};
      \node (r5f) at (3,4) [v-g] {$r^5\!f$};
      \draw [cy2] (1) to [bend right=15] (r);
      \draw [cy2] (r) to [bend right=15] (r2);
      \draw [cy2] (r2) to [bend right=15] (r3);
      \draw [cy2] (r3) to [bend right=15] (r4);
      \draw [cy2] (r4) to [bend right=15] (r5);
      \draw [cy2] (r5) to [bend right=15] (1);
      \draw [cy2] (1) to (f);
      \draw [cy2] (1) to (rf);
      \draw [cy2] (1) to (r2f);
      \draw [cy2] (1) to (r3f);
      \draw [cy2] (1) to (r4f);
      \draw [cy2] (1) to (r5f);
    \end{scope}
    %%
    %% ------------- Cayley graph of D_6 ---------------------------
    %%    
    \begin{scope}[shift={(0,2.5)},scale=1.45]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (60:2) [v] {$r$};
      \node (r2) at (120:2) [v] {$r^2$};
      \node (r3) at (180:2) [v] {$r^3$};
      \node (r4) at (240:2) [v] {$r^4$};
      \node (r5) at (300:2) [v] {$r^5$};
      \node (f) at (0:1) [v] {$f$};
      \node (rf) at (60:1) [v] {$rf$};
      \node (r2f) at (120:1) [v] {$r^2\!f$};
      \node (r3f) at (180:1) [v] {$r^3\!f$};
      \node (r4f) at (240:1) [v] {$r^4\!f$};
      \node (r5f) at (300:1) [v] {$r^5\!f$};
      \path[bb] (1) to (f);
      \path[bb] (r) to (rf);
      \path[bb] (r2) to (r2f);
      \path[bb] (r3) to (r3f);
      \path[bb] (r4) to (r4f);
      \path[bb] (r5) to (r5f);
      \path[R6-out] (1) to (r);
      \path[R6-out] (r) to (r2);
      \path[R6-out] (r2) to (r3);
      \path[R6-out] (r3) to (r4);
      \path[R6-out] (r4) to (r5);
      \path[R6-out] (r5) to (1);
      \path[R6-in] (f) to (r5f);
      \path[R6-in] (r5f) to (r4f);
      \path[R6-in] (r4f) to (r3f);
      \path[R6-in] (r3f) to (r2f);
      \path[R6-in] (r2f) to (rf);
      \path[R6-in] (rf) to (f);
    \end{scope}
    %%
    %% ------------- Subgroup lattice of D_6 ------------------------
    %%
    \begin{scope}[shift={(7.25,0)},scale=1.6]
      \tikzstyle{every node}=[font=\small]
      \node(D6) at (0,\Hf) {$D_6$};
      \node(r2-f) at (-1,\He) {$\<r^2,f\>$};
      \node(r2-rf) at (0,\He) {$\<r^2,rf\>$}; 
      \node(r) at (-2,\He) {$\<r\>$}; 
      \node(r3-f) at (2.4,\Hd) {\footnotesize $\<r^3\!,\!f\>$};
      \node(r3-rf) at (3.1,\Hd) {\footnotesize $\<r^3\!,rf\>$};
      \node(r3-r2f) at (3.8,\Hd) {\footnotesize $\<r^3\!,\!r^2\!f\>$};
      \node(r2) at (-2,\Hc) {$\<r^2\>$};
      \node(r3) at (0,\Hb) {$\<r^3\>$};
      \node(f) at (.6,\Hb) {$\<f\>$};
      \node(r4f) at (1.2,\Hb) {$\<r^4\!f\>$};      
      \node(r2f) at (1.8,\Hb) {$\<r^2\!f\>$};      
      \node(r3f) at (2.4,\Hb) {$\<r^3\!f\>$};
      \node(rf) at (3.1,\Hb) {$\<rf\>$};
      \node(r5f) at (3.8,\Hb) {$\<r^5\!f\>$};
      \node (1) at (0,\Ha) {$\<1\>$};
      \draw[f] (D6) -- (r2-f);
      \draw[f] (D6) -- (r);
      \draw[f] (D6) to (r2-rf);
      \draw[f] (D6) to (r3-f); 
      \draw[f] (D6) -- (r3-rf);
      \draw[f] (D6) to (r3-r2f); 
      \draw[f] (r2-f) to (f); 
      \draw[f] (r2-f) -- (r2f);
      \draw[f] (r2-f) to (r4f);
      \draw[f] (r2-f) to (r2);
      \draw[f] (r) -- (r2);
      \draw[f] (r) -- (r3);
      \draw[f] (r2-rf) -- (r2);
      \draw[f] (r2-rf) to (rf);
      \draw[f] (r2-rf) to (r3f);
      \draw[f] (r2-rf) to (r5f);
      \draw[f] (r3-f) -- (f);
      \draw[f] (r3-f) to (r3);
      \draw[f] (r3-f) to (r3f); 
      \draw[f] (r3-rf) to (rf);
      \draw[f] (r3-rf) to (r3);
      \draw[f] (r3-rf) to (r4f); 
      \draw[f] (r3-r2f) to (r2f);
      \draw[f] (r3-r2f) to (r3);
      \draw[f] (r3-r2f) to (r5f);
      \draw[f] (r2) -- (1);
      \draw[f] (f) -- (1);
      \draw[f] (rf) to (1);
      \draw[f] (r2f) to (1);
      \draw[f] (r3f) to (1);
      \draw[f] (r4f) to (1);
      \draw[f] (r5f) to (1); 
      \draw[f] (r3) -- (1);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Construct the action graph, and superimpose it on the
    subgroup lattice.
  \item Construct the fixed point table.
  \item Find $\stab(H)$ for each subgroup $H\leq D_6$, and $\fix(g)$
    for each $g\in D_6$.
  \item Find $\Ker(\phi)$ and $\Fix(\phi)$.
  \item Interpret $\orb(H)$, $\stab(H)$, $[G:\stab(H)]$, $\Fix(\phi)$,
    $\Ker(\phi)$, $\fix(g)$, and the average size of $|\fix(g)|$ in terms of
    familiar algebraic objects. 
  \end{enumerate}
  

  %%-------------------------------------------------------------------
  \newpage
  %%-------------------------------------------------------------------

  
\item There is a \emph{Galois correspondence} between conjugacy
  classes of subgroups of $G$ and transitive actions of $G$, defined
  by collapsing the right cosets of $H$. An example of this
  correspondence for $D_6$ is shown below.
  %%
  %% Subgroup diagram of D_6 w/ cycle graphs, and the G-set poset of D_6
  \[
  \begin{tikzpicture}[scale=1]
    \hspace*{-12mm}
    \tikzstyle{cy2-r} = [draw,thick,bend right=12]
    \tikzstyle{cy2} = [draw,thick]
    \tikzstyle{lat} = [draw,very thick,lightgray]
    %% 
    \newcommand\Hf{12.5} % height of C_12 
    \newcommand\He{9} % height of C_6  
    \newcommand\Hd{7} % height of C_4 
    \newcommand\Hc{5} % height of C_3  
    \newcommand\Hb{3} % height of C_2 
    \newcommand\Ha{-.5} % height of C_1   
    %%
    \tikzstyle{v-r} = [circle, draw, fill=vRed,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-g} = [circle, draw, fill=vGreen,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-b} = [circle, draw, fill=vBlue,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-p} = [circle, draw, fill=vPurple,inner sep=0pt, 
      minimum size=2mm]
    \tikzstyle{v-o} = [circle, draw, fill=vOrange,inner sep=0pt, 
      minimum size=2mm]
    \tikzstyle{v-gr} = [circle, draw, fill=midgray,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-w} = [circle, draw, fill=white,inner sep=0pt,
      minimum size=2mm]
    %%--------------------------------------------------------------------
    %% Subgroup diagram with nodes decorated w/ cycle graphs
    \begin{scope} %%------------------------------------------------------
      \begin{scope}[shift={(0,\Hf)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-r] {};
        \node (r2) at (210:2) [v-o] {};
        \node (r3) at (270:2) [v-p] {};
        \node (r4) at (330:2) [v-o] {};
        \node (r5) at (30:2) [v-r] {};
        \node (f) at (-2.5,3.5) [v-b] {};
        \node (r2f) at (-1.5,3.5) [v-b] {};
        \node (r4f) at (-.5,3.5) [v-b] {};
        \node (rf) at (.5,3.5) [v-g] {};
        \node (r3f) at (1.5,3.5) [v-g] {};
        \node (r5f) at (2.5,3.5) [v-g] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $D_6$};
        \node (D6-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-4.2,\He)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-r] {};
        \node (r2) at (210:2) [v-o] {};
        \node (r3) at (270:2) [v-p] {};
        \node (r4) at (330:2) [v-o] {};
        \node (r5) at (30:2) [v-r] {};
        \node (f) at (-2.5,3.5) [v-w] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-w] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<r\>$};
        \node (r-top) at (0,4) {};
        \node (r-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-2.1,\He)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-o] {};
        \node (r3) at (270:2) [v-w] {};
        \node (r4) at (330:2) [v-o] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-b] {};
        \node (r2f) at (-1.5,3.5) [v-b] {};
        \node (r4f) at (-.5,3.5) [v-b] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-w] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<r^2,f\>$};
        \node (r2-f-top) at (0,4) {};
        \node (r2-f-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(0,\He)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-o] {};
        \node (r3) at (270:2) [v-w] {};
        \node (r4) at (330:2) [v-o] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-w] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-g] {};
        \node (r3f) at (1.5,3.5) [v-g] {};
        \node (r5f) at (2.5,3.5) [v-g] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<r^2,rf\>$};
        \node (r2-rf-top) at (0,4) {};
        \node (r2-rf-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(2.1,\Hd)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-w] {};
        \node (r3) at (270:2) [v-p] {};
        \node (r4) at (330:2) [v-w] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-b] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-g] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<r^3,f\>$};
        \node [midgray] at (-2.25,-2) {\tiny $\times 3$};
        \node (r3-f-top) at (0,4) {};
        \node (r3-f-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-4.2,\Hc)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-o] {};
        \node (r3) at (270:2) [v-w] {};
        \node (r4) at (330:2) [v-o] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-w] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-w] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<r^2\>$};
        \node (r2-top) at (0,4) {};
        \node (r2-bot) at (0,-2.5) {};    
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-2.1,\Hb)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-w] {};
        \node (r3) at (270:2) [v-p] {};
        \node (r4) at (330:2) [v-w] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-w] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-w] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<r^3\>$};
        \node (r3-top) at (0,4) {};
        \node (r3-bot) at (0,-2.5) {};   
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(0,\Hb)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-w] {};
        \node (r3) at (270:2) [v-w] {};
        \node (r4) at (330:2) [v-w] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-b] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-w] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<f\>$};
        \node [midgray] at (-2.25,-2) {\tiny $\times 3$};
        \node (f-top) at (0,4) {};
        \node (f-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(2.1,\Hb)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-w] {};
        \node (r3) at (270:2) [v-w] {};
        \node (r4) at (330:2) [v-w] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-w] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-g] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<rf\>$};
        \node [midgray] at (-2.25,-2) {\tiny $\times 3$};
        \node (rf-top) at (0,4) {};
        \node (rf-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-2.1,\Ha)},scale=.3]
        \draw[very thick,rounded corners,lightgray] (-3,-2.5) rectangle (3,4);
        \node (1) at (90:2) [v-gr] {};
        \node (r) at (150:2) [v-w] {};
        \node (r2) at (210:2) [v-w] {};
        \node (r3) at (270:2) [v-w] {};
        \node (r4) at (330:2) [v-w] {};
        \node (r5) at (30:2) [v-w] {};
        \node (f) at (-2.5,3.5) [v-w] {};
        \node (r2f) at (-1.5,3.5) [v-w] {};
        \node (r4f) at (-.5,3.5) [v-w] {};
        \node (rf) at (.5,3.5) [v-w] {};
        \node (r3f) at (1.5,3.5) [v-w] {};
        \node (r5f) at (2.5,3.5) [v-w] {};
        \draw [cy2-r] (1) to [bend right] (r);
        \draw [cy2-r] (r) to [bend right] (r2);
        \draw [cy2-r] (r2) to [bend right] (r3);
        \draw [cy2-r] (r3) to [bend right] (r4);
        \draw [cy2-r] (r4) to [bend right] (r5);
        \draw [cy2-r] (r5) to [bend right] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \draw [cy2] (1) to (r4f);
        \draw [cy2] (1) to (r5f);
        \node at (0,0) {\footnotesize $\<1\>$};
        \node (1-top) at (0,4) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \draw [lat] (D6-bot) to (r-top);
      \draw [lat] (D6-bot) to (r2-f-top);
      \draw [lat] (D6-bot) to (r2-rf-top);
      \draw [lat] (D6-bot) to (r3-f-top);
      \draw [lat] (r-bot) to (r2-top);
      \draw [lat] (r-bot) to (r3-top);    
      \draw [lat] (r2-f-bot) to (f-top);
      \draw [lat] (r2-rf-bot) to (rf-top);
      \draw [lat] (r2-f-bot) to (r2-top);
      \draw [lat] (r2-rf-bot) to (r2-top);
      \draw [lat] (r3-f-bot) to (r3-top);
      \draw [lat] (r3-f-bot) to (f-top);
      \draw [lat] (r3-f-bot) to (rf-top);
      \draw [lat] (r2-bot) to (1-top);
      \draw [lat] (r3-bot) to (1-top);
      \draw [lat] (f-bot) to (1-top);
      \draw [lat] (rf-bot) to (1-top);
    \end{scope}
    %%=========================================================================
    %% G-set poset 
    %%=========================================================================
    \begin{scope}[shift={(9.25,0)}]
      \tikzstyle{v-gr} = [circle, draw, fill=lightgray,inner sep=0pt,
        minimum size=1.75mm]
      \tikzstyle{r-out} = [draw, eRed,-stealth,bend right=15]
      \tikzstyle{r-in} = [draw, eRed,-stealth,bend left=12]
      \tikzstyle{every node}=[font=\footnotesize]
      \tikzstyle{r-thin} = [draw,eRed,-stealth,thick]
      \tikzstyle{rr-thin} = [draw,eRed,thick]
      \tikzstyle{bb-thin} = [draw,eBlue,thick]
      \tikzstyle{b-thin} = [draw,eBlue,-stealth,thick]
      \tikzset{every loop/.style={min distance=10mm,looseness=15}}
      %%
      \begin{scope}[shift={(0,\Hf)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:0) [v-gr] {};
          \path (1) edge [b-thin,in=150,out=210,loop,-stealth] (1);
          \path (1) edge [r-thin,in=-30,out=30,loop,-stealth] (1);
        \end{scope}
        \node (D6-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-4.2,\He)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:1.65) [v-gr] {};
          \node (r3) at (180:1.65) [v-gr] {};
          \path (1) edge [r-thin,in=60,out=120,loop,-stealth] (1);
          \path (r3) edge [r-thin,in=60,out=120,loop,-stealth] (r3);
          \draw [bb-thin] (1) to (r3);
        \end{scope}
        %%      
        \node (r-top) at (0,4) {};
        \node (r-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-2.1,\He)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:1.65) [v-gr] {};
          \node (r3) at (180:1.65) [v-gr] {};
          \path (1) edge [b-thin,in=60,out=120,loop,-stealth] (1);
          \path (r3) edge [b-thin,in=60,out=120,loop,-stealth] (r3);
          \draw [rr-thin] (1) to (r3);
        \end{scope}
        \node (r2-f-top) at (0,4) {};
        \node (r2-f-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(0,\He)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:1.65) [v-gr] {};
          \node (r3) at (180:1.65) [v-gr] {};
          \draw [bb-thin] (1) to[bend left=20] (r3);
          \draw [rr-thin] (1) to[bend right=20] (r3);
        \end{scope}
        \node (r2-rf-top) at (0,4) {};
        \node (r2-rf-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(2.1,\Hd)},scale=.3]
        \begin{scope}[shift={(-.1,.6)},scale=1.3]
          \node (1) at (0:1.75) [v-gr] {};
          \node (r2) at (120:1.75) [v-gr] {};
          \node (r4) at (240:1.75) [v-gr] {};
          \draw [r-thin] (1) to[bend right=35] (r2);
          \draw [r-thin] (r2) to[bend right=35] (r4);
          \draw [r-thin] (r4) to[bend right=35] (1);    
          \draw [bb] (r2) to (r4);
          \path (1) edge [b-thin,in=-30,out=30,loop,-stealth] (1);
        \end{scope}
        %%
        \node (r3-f-top) at (0,4) {};
        \node (r3-f-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-4.2,\Hc)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:1.65) [v-gr] {};
          \node (r) at (90:1.65) [v-gr] {};
          \node (r2) at (180:1.65) [v-gr] {};
          \node (r3) at (270:1.65) [v-gr] {};
          \draw[rr-thin] (1) to[bend right=20] (r);
          \draw[bb-thin] (r) to[bend right=20] (r2);
          \draw[rr-thin] (r2) to[bend right=20] (r3);
          \draw[bb-thin] (r3) to[bend right=20] (1);
        \end{scope}
        \node (r2-top) at (0,4) {};
        \node (r2-bot) at (0,-2.5) {};    
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-2.1,\Hb)},scale=.3]
        \begin{scope}[shift={(-.5,.6)},scale=1.3]
          \tikzstyle{r-out} = [draw, thin, eRed,-stealth,bend right=15]
          \tikzstyle{r-in} = [draw, thin, eRed,-stealth,bend left=12]
          \node (1) at (0:2) [v-gr] {};
          \node (r2) at (120:2) [v-gr] {};
          \node (r4) at (240:2) [v-gr] {};
          \node (f) at (0:1) [v-gr] {};
          \node (r2f) at (120:1) [v-gr] {};
          \node (r4f) at (240:1) [v-gr] {};
          \draw[r-thin] (1) to[bend right=35] (r2);
          \draw[r-thin] (r2) to[bend right=35] (r4);
          \draw[r-thin] (r4) to[bend right=35] (1);
          \draw[r-thin] (f) to[bend left=25] (r4f);
          \draw[r-thin] (r4f) to[bend left=25] (r2f);
          \draw[r-thin] (r2f) to[bend left=25] (f);
          \draw [bb-thin] (1) to (f);
          \draw [bb-thin] (r2) to (r2f);
          \draw [bb-thin] (r4) to (r4f);      
        \end{scope}
        %%
        \node (r3-top) at (0,4) {};
        \node (r3-bot) at (0,-2.5) {};   
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(0,\Hb)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:1.65) [v-gr] {};
          \node (r) at (60:1.65) [v-gr] {};
          \node (r2) at (120:1.65) [v-gr] {};
          \node (r3) at (180:1.65) [v-gr] {};
          \node (r4) at (240:1.65) [v-gr] {};
          \node (r5) at (300:1.75) [v-gr] {};
          \path[r-out] (1) to (r);
          \path[r-out] (r) to (r2);
          \path[r-out] (r2) to (r3);
          \path[r-out] (r3) to (r4);
          \path[r-out] (r4) to (r5);
          \path[r-out] (r5) to (1);
          \path[bb-thin] (r) to (r5); \path[bb-thin] (r2) to (r4);
          \path (1) edge [b-thin,in=-30,out=30,loop,-stealth] (1);
          \path (r3) edge [b-thin,in=150,out=210,loop,-stealth] (r3);
        \end{scope}
        %%
        \node (f-top) at (0,4) {};
        \node (f-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(2.1,\Hb)},scale=.3]
        \begin{scope}[shift={(0,.6)},scale=1.3]
          \node (1) at (0:1.65) [v-gr] {};
          \node (r) at (60:1.65) [v-gr] {};
          \node (r2) at (120:1.65) [v-gr] {};
          \node (r3) at (180:1.65) [v-gr] {};
          \node (r4) at (240:1.65) [v-gr] {};
          \node (r5) at (300:1.75) [v-gr] {};
          %%
          \draw[bb-thin] (1) to (r);
          \draw[bb-thin] (r2) to (r5);
          \draw[bb-thin] (r3) to (r4);
          \draw[r-out] (1) to[bend right=18] (r);
          \draw[r-out] (r) to (r2);
          \draw[r-out] (r2) to (r3);
          \draw[r-out] (r3) to[bend right=18] (r4);
          \draw[r-out] (r4) to (r5);
          \draw[r-out] (r5) to (1);      
        \end{scope}
        %%
        \node (rf-top) at (0,4) {};
        \node (rf-bot) at (0,-2.5) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \begin{scope}[shift={(-2.1,\Ha)},scale=.4]
        \tikzstyle{r-thin} = [draw,eRed,-stealth]
        \tikzstyle{rr-thin} = [draw,eRed,]
        \tikzstyle{bb-thin} = [draw,eBlue]
        \tikzstyle{b-thin} = [draw,eBlue,-stealth]
        \node (1) at (0:2) [v-gr] {};
        \node (r) at (60:2) [v-gr] {};
        \node (r2) at (120:2) [v-gr] {};
        \node (r3) at (180:2) [v-gr] {};
        \node (r4) at (240:2) [v-gr] {};
        \node (r5) at (300:2) [v-gr] {};
        \node (f) at (0:1) [v-gr] {};
        \node (rf) at (60:1) [v-gr] {};
        \node (r2f) at (120:1) [v-gr] {};
        \node (r3f) at (180:1) [v-gr] {};
        \node (r4f) at (240:1) [v-gr] {};
        \node (r5f) at (300:1) [v-gr] {};
        \path[bb] (1) to (f);
        \path[bb] (r) to (rf);
        \path[bb] (r2) to (r2f);
        \path[bb] (r3) to (r3f);
        \path[bb] (r4) to (r4f);
        \path[bb] (r5) to (r5f);
        \path[r-out] (1) to (r);
        \path[r-out] (r) to (r2);
        \path[r-out] (r2) to (r3);
        \path[r-out] (r3) to (r4);
        \path[r-out] (r4) to (r5);
        \path[r-out] (r5) to (1);
        \path[r-in] (f) to (r5f);
        \path[r-in] (r5f) to (r4f);
        \path[r-in] (r4f) to (r3f);
        \path[r-in] (r3f) to (r2f);
        \path[r-in] (r2f) to (rf);
        \path[r-in] (rf) to (f);
        \node (1-top) at (0,2.75) {};
      \end{scope}
      %%---------------------------------------------------------------------
      \draw [lat] (D6-bot) to (r-top);
      \draw [lat] (D6-bot) to (r2-f-top);
      \draw [lat] (D6-bot) to (r2-rf-top);
      \draw [lat] (D6-bot) to (r3-f-top);
      \draw [lat] (r-bot) to (r2-top);
      \draw [lat] (r-bot) to (r3-top);    
      \draw [lat] (r2-f-bot) to (f-top);
      \draw [lat] (r2-rf-bot) to (rf-top);
      \draw [lat] (r2-f-bot) to (r2-top);
      \draw [lat] (r2-rf-bot) to (r2-top);
      \draw [lat] (r3-f-bot) to (r3-top);
      \draw [lat] (r3-f-bot) to (f-top);
      \draw [lat] (r3-f-bot) to (rf-top);
      \draw [lat] (r2-bot) to (1-top);
      \draw [lat] (r3-bot) to (1-top);
      \draw [lat] (f-bot) to (1-top);
      \draw [lat] (rf-bot) to (1-top);
    \end{scope}
  \end{tikzpicture}
  \]
  Carry out this construction for $D_6=\<s,t\>$, where where $s=f$ and
  $t=fr$. As was done above, label each conjugacy class with a
  subgroup that contains it, but use this new generating set.

  %%---------------------------------------------------------------
  
\end{enumerate}

\end{document}    
    
  
 
