This honeycomb is one of seven unique uniform honeycombs ^[1] constructed by the ${\displaystyle {\tilde {A}}_{4}}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A4 honeycombs
Pentagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
a1	[3[5]]		${\displaystyle {\tilde {A}}_{4}}$	(None)
i2	[[3[5]]]		${\displaystyle {\tilde {A}}_{4}}$×2	1,  2,  3,   4,  5,  6
r10	[5[3[5]]]		${\displaystyle {\tilde {A}}_{4}}$×10	7

References