RubiksCubeGroup - Maple Help

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GroupTheory

 RubiksCubeGroup
 construct a permutation group isomorphic to the group of Rubik's Cube

 Calling Sequence RubiksCubeGroup()

Description

 • The group of Rubik's Cube (TM) is a group of cube transformations of the popular Rubik's Cube (TM) puzzle game.
 • The RubiksCubeGroup() command returns a permutation group isomorphic to the Rubik's Cube (TM) group.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $R≔\mathrm{RubiksCubeGroup}\left(\right)$
 ${R}{≔}⟨\left({6}{,}{25}{,}{43}{,}{16}\right)\left({7}{,}{28}{,}{42}{,}{13}\right)\left({8}{,}{30}{,}{41}{,}{11}\right)\left({17}{,}{19}{,}{24}{,}{22}\right)\left({18}{,}{21}{,}{23}{,}{20}\right){,}\left({1}{,}{14}{,}{48}{,}{27}\right)\left({2}{,}{12}{,}{47}{,}{29}\right)\left({3}{,}{9}{,}{46}{,}{32}\right)\left({33}{,}{35}{,}{40}{,}{38}\right)\left({34}{,}{37}{,}{39}{,}{36}\right){,}\left({1}{,}{17}{,}{41}{,}{40}\right)\left({4}{,}{20}{,}{44}{,}{37}\right)\left({6}{,}{22}{,}{46}{,}{35}\right)\left({9}{,}{11}{,}{16}{,}{14}\right)\left({10}{,}{13}{,}{15}{,}{12}\right){,}\left({3}{,}{38}{,}{43}{,}{19}\right)\left({5}{,}{36}{,}{45}{,}{21}\right)\left({8}{,}{33}{,}{48}{,}{24}\right)\left({25}{,}{27}{,}{32}{,}{30}\right)\left({26}{,}{29}{,}{31}{,}{28}\right){,}\left({1}{,}{3}{,}{8}{,}{6}\right)\left({2}{,}{5}{,}{7}{,}{4}\right)\left({9}{,}{33}{,}{25}{,}{17}\right)\left({10}{,}{34}{,}{26}{,}{18}\right)\left({11}{,}{35}{,}{27}{,}{19}\right){,}\left({14}{,}{22}{,}{30}{,}{38}\right)\left({15}{,}{23}{,}{31}{,}{39}\right)\left({16}{,}{24}{,}{32}{,}{40}\right)\left({41}{,}{43}{,}{48}{,}{46}\right)\left({42}{,}{45}{,}{47}{,}{44}\right)⟩$ (1)
 > $\mathrm{GroupOrder}\left(R\right)$
 ${43252003274489856000}$ (2)
 > $\mathrm{IsSimple}\left(R\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{Degree}\left(R\right)$
 ${48}$ (4)
 > $\mathrm{IsTransitive}\left(R\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{numelems}\left(\mathrm{Orbits}\left(R\right)\right)$
 ${2}$ (6)
 > $\mathrm{map}\left(\mathrm{numelems},\mathrm{Orbits}\left(R\right)\right)$
 $\left[{24}{,}{24}\right]$ (7)
 > $\mathrm{Labels}\left(R\right)$
 $\left[{\mathrm{front}}{,}{\mathrm{back}}{,}{\mathrm{left}}{,}{\mathrm{right}}{,}{\mathrm{up}}{,}{\mathrm{down}}\right]$ (8)
 > $\mathrm{zip}\left(\mathrm{assign},\mathrm{Labels}\left(R\right),\mathrm{Generators}\left(R\right)\right):$

Let us construct the so-called "slice squared" subgroup, whose generators rotate a central "slice" by a half-turn.

 > $\mathrm{SS}≔\mathrm{Subgroup}\left(\left\{{\mathrm{front}}^{2}·{\mathrm{back}}^{2},{\mathrm{right}}^{2}·{\mathrm{left}}^{2},{\mathrm{up}}^{2}·{\mathrm{down}}^{2}\right\},R\right)$
 ${\mathrm{SS}}{≔}⟨\left({1}{,}{8}\right)\left({2}{,}{7}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right)\left({9}{,}{25}\right)\left({10}{,}{26}\right)\left({11}{,}{27}\right)\left({14}{,}{30}\right)\left({15}{,}{31}\right)\left({16}{,}{32}\right)\left({17}{,}{33}\right)\left({18}{,}{34}\right)\left({19}{,}{35}\right)\left({22}{,}{38}\right)\left({23}{,}{39}\right)\left({24}{,}{40}\right)\left({41}{,}{48}\right)\left({42}{,}{47}\right)\left({43}{,}{46}\right)\left({44}{,}{45}\right){,}\left({1}{,}{48}\right)\left({2}{,}{47}\right)\left({3}{,}{46}\right)\left({6}{,}{43}\right)\left({7}{,}{42}\right)\left({8}{,}{41}\right)\left({9}{,}{32}\right)\left({11}{,}{30}\right)\left({12}{,}{29}\right)\left({13}{,}{28}\right)\left({14}{,}{27}\right)\left({16}{,}{25}\right)\left({17}{,}{24}\right)\left({18}{,}{23}\right)\left({19}{,}{22}\right)\left({20}{,}{21}\right)\left({33}{,}{40}\right)\left({34}{,}{39}\right)\left({35}{,}{38}\right)\left({36}{,}{37}\right){,}\left({1}{,}{41}\right)\left({3}{,}{43}\right)\left({4}{,}{44}\right)\left({5}{,}{45}\right)\left({6}{,}{46}\right)\left({8}{,}{48}\right)\left({9}{,}{16}\right)\left({10}{,}{15}\right)\left({11}{,}{14}\right)\left({12}{,}{13}\right)\left({17}{,}{40}\right)\left({19}{,}{38}\right)\left({20}{,}{37}\right)\left({21}{,}{36}\right)\left({22}{,}{35}\right)\left({24}{,}{33}\right)\left({25}{,}{32}\right)\left({26}{,}{31}\right)\left({27}{,}{30}\right)\left({28}{,}{29}\right)⟩$ (9)

These generators commute with one another, so they generate a commutative subgroup of $R$.

 > $\mathrm{IsAbelian}\left(\mathrm{SS}\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{GroupOrder}\left(\mathrm{SS}\right)$
 ${8}$ (11)

Compatibility

 • The GroupTheory[RubiksCubeGroup] command was introduced in Maple 17.