Template:D5 honeycombs

This honeycomb is one of 20 uniform honeycombs constructed by the ${\tilde {D}}_{5}$ 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]		${\tilde {D}}_{5}$
<[31,1,3,31,1]> ↔ [31,1,3,3,4]	↔	${\tilde {D}}_{5}$×21 = ${\tilde {B}}_{5}$	, , , , , ,
[[31,1,3,31,1]]		${\tilde {D}}_{5}$×22	,
<2[31,1,3,31,1]> ↔ [4,3,3,3,4]	↔	${\tilde {D}}_{5}$×41 = ${\tilde {C}}_{5}$	, , , , ,
[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]]	↔	${\tilde {D}}_{5}$×8 = ${\tilde {C}}_{5}$×2	, ,