cycles = {{{}}, {{1, 2, 3}}, {{1, 3, 2}}, {{1, 2}, {4, 5}}, {{1, 3}, {4, 5}},
{{2, 3}, {4, 5}}, {{1, 3}}, {{2, 3}}, {{1, 2}}, {{4, 5}},
{{1, 2, 3}, {4, 5}}, {{1, 3, 2}, {4, 5}}};
pg = PermutationGroup[Cycles /@ cycles];
GroupOrder[pg]
12
GroupMultiplicationTable[pg] // MatrixForm // TeXForm
$\left(
\begin{array}{cccccccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
2 & 1 & 4 & 3 & 6 & 5 & 8 & 7 & 10 & 9 & 12 & 11 \\
3 & 4 & 1 & 2 & 7 & 8 & 5 & 6 & 11 & 12 & 9 & 10 \\
4 & 3 & 2 & 1 & 8 & 7 & 6 & 5 & 12 & 11 & 10 & 9 \\
5 & 6 & 9 & 10 & 1 & 2 & 11 & 12 & 3 & 4 & 7 & 8 \\
6 & 5 & 10 & 9 & 2 & 1 & 12 & 11 & 4 & 3 & 8 & 7 \\
7 & 8 & 11 & 12 & 3 & 4 & 9 & 10 & 1 & 2 & 5 & 6 \\
8 & 7 & 12 & 11 & 4 & 3 & 10 & 9 & 2 & 1 & 6 & 5 \\
9 & 10 & 5 & 6 & 11 & 12 & 1 & 2 & 7 & 8 & 3 & 4 \\
10 & 9 & 6 & 5 & 12 & 11 & 2 & 1 & 8 & 7 & 4 & 3 \\
11 & 12 & 7 & 8 & 9 & 10 & 3 & 4 & 5 & 6 & 1 & 2 \\
12 & 11 & 8 & 7 & 10 & 9 & 4 & 3 & 6 & 5 & 2 & 1 \\
\end{array}
\right)$
PermutationCycles /@ GroupMultiplicationTable[pg]
{Cycles[{}],
Cycles[{{1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}, {11, 12}}],
Cycles[{{1, 3}, {2, 4}, {5, 7}, {6, 8}, {9, 11}, {10, 12}}],
Cycles[{{1, 4}, {2, 3}, {5, 8}, {6, 7}, {9, 12}, {10, 11}}],
Cycles[{{1, 5}, {2, 6}, {3, 9}, {4, 10}, {7, 11}, {8, 12}}],
Cycles[{{1, 6}, {2, 5}, {3, 10}, {4, 9}, {7, 12}, {8, 11}}],
Cycles[{{1, 7, 9}, {2, 8, 10}, {3, 11, 5}, {4, 12, 6}}],
Cycles[{{1, 8, 9, 2, 7, 10}, {3, 12, 5, 4, 11, 6}}],
Cycles[{{1, 9, 7}, {2, 10, 8}, {3, 5, 11}, {4, 6, 12}}],
Cycles[{{1, 10, 7, 2, 9, 8}, {3, 6, 11, 4, 5, 12}}],
Cycles[{{1, 11}, {2, 12}, {3, 7}, {4, 8}, {5, 9}, {6, 10}}],
Cycles[{{1, 12}, {2, 11}, {3, 8}, {4, 7}, {5, 10}, {6, 9}}]}
PermutationProduct[Cycles[{{1, 2}, {4, 5}}], Cycles[{{1, 2, 3}, {4, 5}}]]
Cycles[{{1, 3}}]
PermutationGroup[]
andCycles[]
to specify the group? It seems to me that you might have just made a syntax error somewhere. $\endgroup$Cycles[]
toPermutationGroup[]
, e.g.PermutationGroup[{Cycles[{}], Cycles[{{1, 3, 2}}], Cycles[{{1, 2}, {4, 5}}]}]
$\endgroup$