#72258131

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×C₄⁴ ⋊ A₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Frattini	C₂⁶	#11757162
Fitting, cosocle	C₂²×C₄⁴	A₅
socle	C₂⁴	#72209602
LCS	G
UCS	C₁

49633 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 49633 conjugacy classes of subgroups — too many to draw. Only its 8 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) = 60
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 60 points:

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

Conjugacy classes & character table

88 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((1,56,45,25,20)(2,50,36,18,24)(3,44,12,40,48)(4,57,49,10,38)(5,54,43,21,34,14,58,41,26,31)(6,46,53,7,9)(8,17,29,39,22,30,35,13,55,42)(11,37,28,47,33)(15,27,16,60,51)(19,23,32,59,52), (1,2,5)(3,8,7)(4,6,13)(9,22,39)(10,21,15)(11,26,18)(12,16,14)(17,33,40)(19,25,30)(20,36,46)(23,42,55)(24,43,58)(27,41,54)(28,32,29)(31,45,47)(34,48,57)(35,49,56)(37,53,50)(38,52,60)(44,51,59));