#72227550

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived

#72209597 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72227549
commutator, hypocenter	#72209597	C₂
Frattini, socle, cosocle	C₂⁴	C₂×GL(3,𝔽₂)
Fitting	C₂⁵	GL(3,𝔽₂)
LCS	G ▷ #72209597
UCS	C₁ ◁ C₂

169 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 169 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) = 44
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 44 points:

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

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227554, #72227563

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,14,13,12,10,11,2)(4,16,9,7,15,5,6)(17,35,44,23,36,28,33)(18,39,43,34,32,19,29,22,31,37,26,24,25,38)(20,42,40,30,27,21,41), (1,14,6,13,4,16,2,9)(3,5,15,7,8,11,10,12)(17,37,32,26,20,43,24,34)(18,40,36,23,22,44,27,30)(19,28)(21,25)(29,35,33,31)(38,42,41,39));