#72228144

order	57624
isoclinism	isoclinic class of #72228144
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle

C₇³ ⋊ GL(3,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Fitting, socle, cosocle	C₇³	GL(3,𝔽₂)
LCS	G
UCS	C₁

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 56 points:

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

Conjugacy classes & character table

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