Dihedral Groups

Defintition

The dihedral group $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } D_{2n}$ is the group of isometries on a polygon of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ vertices. In other words, $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } D_{2n}$ is the group of order $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 2n$ consisting of the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ rotational and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ reflection(al) symmetries on the regular $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$-gon.

Example

We can consider the dihedral group $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } D_6$ of symmetries of a regular triangle

Let $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } R$ denote a counter-clockwise rotation of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \frac 1 3$ of a circle, and let $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S$ denote a reflection accross the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } y$-axis. Then: $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } D_6 = \{1, R, R^2, S, SR, SR^2\}$$ (Remark: $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } D_6 \cong \mathbf 2 \times \mathbf 3$ both as sets and linear spaces)