#72267359

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×#585 ⋊ A₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	#72209715
commutator, hypocenter	G	C₁
Fitting, cosocle	C₂⁴×#585	A₅
socle	C₂⁵	#14515300
LCS	G
UCS	C₁ ◁ C₄

At least 88813 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has at least 88813 conjugacy classes of subgroups — too many to draw. Only its 9 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) = 74
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 74 points:

(1,2)(4,6)(7,9)(8,10)(11,13,19,34)(12,16,14,22)(15,25,47,61)(17,29,53,27)(18,32,59,69)(20,36,58,54)(21,39,52,63)(23,40,46,37)(24,45,62,72)(26,49,48,35)(28,50,66,73)(30,56,44,51)(31,57,38,55)(33,43,60,70)(41,64,68,74)(42,65,67,71);(1,6,10,5,9)(2,7,8,4,3)(11,65,62,27,55,24,58,17,56,12,53,28,44,13,23,30,63,19,45,68)(14,70,73,72,69,46,18,52,29,35,41,26,42,51,25,57,47,36,34,60)(15,40,37,21,39,43,74,64,67,71,59,31,38,54,20,48,16,22,50,66)(32,49,61,33)

Conjugacy classes & character table

76 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((1,2)(4,6)(7,9)(8,10)(11,13,19,34)(12,16,14,22)(15,25,47,61)(17,29,53,27)(18,32,59,69)(20,36,58,54)(21,39,52,63)(23,40,46,37)(24,45,62,72)(26,49,48,35)(28,50,66,73)(30,56,44,51)(31,57,38,55)(33,43,60,70)(41,64,68,74)(42,65,67,71), (1,6,10,5,9)(2,7,8,4,3)(11,65,62,27,55,24,58,17,56,12,53,28,44,13,23,30,63,19,45,68)(14,70,73,72,69,46,18,52,29,35,41,26,42,51,25,57,47,36,34,60)(15,40,37,21,39,43,74,64,67,71,59,31,38,54,20,48,16,22,50,66)(32,49,61,33));