#72244262

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂×C₄⁴

Aliases

No aliases for this group.

Splittings

Fitting

H(5,𝔽₄) ⋊ A₅

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	#72209703
commutator, hypocenter	G	C₁
Fitting, cosocle	H(5,𝔽₄)	A₅
LCS	G
UCS	C₁ ◁ C₂²

26466 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 26466 conjugacy classes of subgroups — too many to draw. Only its 12 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) = 64
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 64 points:

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

Conjugacy classes & character table

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