#72267635

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	#72209715
commutator, hypocenter	G	C₁
Fitting, cosocle	C₂⁵×2₋¹⁺⁴	A₅
socle	C₂⁶	#11757163
LCS	G
UCS	C₁ ◁ C₂²

At least 29986 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has at least 29986 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 72 points:

(1,8,2)(3,6,12)(4,10,5)(9,16,15)(11,13,14)(17,22,38)(18,27,36)(19,26,21)(20,28,35)(23,33,25)(24,34,37)(29,32,30)(31,39,40)(41,51,70,65,49,53)(42,61)(43,50,69,64,48,45)(44,62,47,56,72,68)(46,66)(52,63,55,60,71,67)(54,57)(58,59);(1,8,14,3)(2,11,9,15)(4,13,10,6)(5,16,7,12)(17,19,23,30)(18,34,24,27)(20,39,31,28)(21,36,32,37)(22,29,33,26)(25,35,38,40)(41,48,61,53,43,49,66,45)(42,54,64,67,46,59,65,68)(44,55,63,58,52,47,62,57)(50,71,70,60,51,72,69,56)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,8,2)(3,6,12)(4,10,5)(9,16,15)(11,13,14)(17,22,38)(18,27,36)(19,26,21)(20,28,35)(23,33,25)(24,34,37)(29,32,30)(31,39,40)(41,51,70,65,49,53)(42,61)(43,50,69,64,48,45)(44,62,47,56,72,68)(46,66)(52,63,55,60,71,67)(54,57)(58,59), (1,8,14,3)(2,11,9,15)(4,13,10,6)(5,16,7,12)(17,19,23,30)(18,34,24,27)(20,39,31,28)(21,36,32,37)(22,29,33,26)(25,35,38,40)(41,48,61,53,43,49,66,45)(42,54,64,67,46,59,65,68)(44,55,63,58,52,47,62,57)(50,71,70,60,51,72,69,56));