This honeycomb is one of 20 uniform honeycombs constructed by the Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

D5 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,3,31,1]
<[31,1,3,31,1]>
↔ [31,1,3,3,4]

×21 = , , ,

, , ,

[[31,1,3,31,1]] ×22 ,
<2[31,1,3,31,1]>
↔ [4,3,3,3,4]

×41 = , , , , ,
[<2[31,1,3,31,1]>]
↔ [[4,3,3,3,4]]

×8 = ×2 , ,

References

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