Template:D4 honeycombs

There are ten uniform honeycombs constructed by the ${\tilde {D}}_{4}$ Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)^*] (index 24), [3,3,4,3^*] (index 6), [1⁺,4,3,3,4,1⁺] (index 4), [3^1,1,3,4,1⁺] (index 2) are all isomorphic to [3^1,1,1,1].

The ten permutations are listed with its highest extended symmetry relation:

D4 honeycombs
Extended symmetry	Extended diagram	Extended group	Honeycombs
[31,1,1,1]		${\tilde {D}}_{4}$	(none)
<[31,1,1,1]> ↔ [31,1,3,4]	↔	${\tilde {D}}_{4}$×2 = ${\tilde {B}}_{4}$	(none)
<2[1,131,1]> ↔ [4,3,3,4]	↔	${\tilde {D}}_{4}$×4 = ${\tilde {C}}_{4}$	1, 2
[3[3,31,1,1]] ↔ [3,3,4,3]	↔	${\tilde {D}}_{4}$×6 = ${\tilde {F}}_{4}$	3, 4, 5, 6
[4[1,131,1]] ↔ [[4,3,3,4]]	↔	${\tilde {D}}_{4}$×8 = ${\tilde {C}}_{4}$×2	7, 8, 9
[(3,3)[31,1,1,1]] ↔ [3,4,3,3]	↔	${\tilde {D}}_{4}$×24 = ${\tilde {F}}_{4}$
[(3,3)[31,1,1,1]]+ ↔ [3+,4,3,3]	↔	½${\tilde {D}}_{4}$×24 = ½${\tilde {F}}_{4}$	10