#72228145

order	57624
isoclinism	isoclinic class of #72228145
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

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

63 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:

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

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