Frobenius 21 and other small groups
by Ed Pegg Jr
F21=PermutationGroup[{Cycles[{{1,2,3,4,5,6,7},{8,9,10,11,12,13,14},{15,16,17,18,19,20,21}}],Cycles[{{1,7,6,5,4,3,2},{8,14,13,12,11,10,9},{15,21,20,19,18,17,16}}],Cycles[{{1,8,15},{2,12,17},{3,9,19},{4,13,21},{5,10,16},{6,14,18},{7,11,20}}],Cycles[{{1,15,8},{2,17,12},{3,19,9},{4,21,13},{5,16,10},{6,18,14},{7,20,11}}]}];
z=Graph3D[CayleyGraph[F21]]
Graph[CayleyGraph[F21]]
IsomorphicGraphQ[GraphData[{"UnitDistance",{21,2}}],Graph[UndirectedEdge@@@Union[Sort/@List@@@EdgeList[z]]]]
True
GraphData[{"UnitDistance",{21,2}}]
Order 16
Order 16
CayleyGraph/@{PermutationGroup[{Cycles[{{1,9,2,10},{3,11,4,12},{5,13,6,14},{7,15,8,16}}],Cycles[{{1,11,8,14},{2,12,7,13},{3,9,6,16},{4,10,5,15}}]}],PermutationGroup[{Cycles[{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16}}],Cycles[{{1,5,9,13},{2,8,10,16},{3,7,11,15},{4,6,12,14}}]}],PermutationGroup[{Cycles[{{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16}}],Cycles[{{1,10,7,16,5,14,3,12},{2,15,8,13,6,11,4,9}}]}],PermutationGroup[{Cycles[{{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16}}],Cycles[{{1,10,5,14},{2,13,6,9},{3,16,7,12},{4,11,8,15}}]}],PermutationGroup[{Cycles[{{1,3},{2,4},{5,7},{6,8},{9,12},{10,11},{13,16},{14,15}}],Cycles[{{1,7},{2,8},{3,6},{4,5},{9,14},{10,13},{11,15},{12,16}}],Cycles[{{1,9},{2,10},{3,11},{4,12},{5,15},{6,16},{7,13},{8,14}}]}]}
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Order 18
Order 18
CayleyGraph@PermutationGroup[{Cycles[{{1,2},{3,5},{4,6},{7,10},{8,11},{9,12},{13,15},{14,16},{17,18}}],Cycles[{{1,5},{2,7},{3,10},{4,11},{6,13},{8,15},{9,16},{12,17},{14,18}}],Cycles[{{1,6},{2,9},{3,11},{4,12},{5,14},{7,15},{8,16},{10,17},{13,18}}]}]
Order 20
Order 20
CayleyGraph@PermutationGroup[{Cycles[{{1,2,3,5},{4,6,7,9},{8,10,11,13},{12,14,15,17},{16,18,19,20}}],Cycles[{{1,5,3,2},{4,9,7,6},{8,13,11,10},{12,17,15,14},{16,20,19,18}}],Cycles[{{1,7},{2,13},{3,16},{4,11},{5,14},{6,17},{8,15},{9,18},{10,20},{12,19}}]}]
Order 21
Order 21
Order 24
Order 24
Order 27
Order 27