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Frobenius 21 group

Posted 4 years ago

FiniteGroupData is missing a few group generations, so I've filled in some holes. One of the groups is Frobenius 21.

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}}]}]; 

Then I looked at the Cayley graph.

z =Graph3D[CayleyGraph[F21]]

Frobenius 21

I decided to check if it was in GraphData. Turns out it is.

IsomorphicGraphQ[GraphData[{"UnitDistance",{21,2}}], Graph[UndirectedEdge@@@Union[Sort/@List@@@EdgeList[z]]]] 

Which graph is this? Turns out this is my graph, a unit distance graph with chromatic number 4 and no 4-cycles.

GraphData[{"UnitDistance", {21, 2}}]

twisted 7

When I found the graph I had no idea it would pop up somewhere else.

In the notebook I also have some missing groups for orders 16, 18, 20, 21, 24 and 27. Does anyone want the missing groups for order 32?

One of the 24 groups gives the Nauru graph.

enter image description here

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POSTED BY: Ed Pegg
2 Replies
Posted 2 years ago

Any suggestion for how to generate the Cycles? I read that the SemidirectProduct was involved, but that does not evaluate in FiniteGroupData and the DirectProduct generates a different thing altogether.

FiniteGroupData[{"DirectProduct", {{"CyclicGroup", 
     21}, {"CyclicGroup", 3}}}, "CayleyGraph"] // Graph3D
POSTED BY: David Barnes

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