C₂×#72209597 #72227680

order	5376
isoclinism	isoclinic class of #72209597
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

non-solvable

Aliases

direct product

C₂×#72209597

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#13051525
commutator, hypocenter	#72209597	C₂
Frattini, cosocle	C₂⁴	C₂×GL(3,𝔽₂)
Fitting, socle	C₂⁵	GL(3,𝔽₂)
LCS	G ▷ #72209597
UCS	C₁ ◁ C₂²

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

normal subgroups only

This group has 381 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) = 30
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 30 points:

(1,13,3,14)(4,11,8,10,6,12,7,9)(15,21,25,28,17,26,29,23)(16,22,24,27,19,18,20,30);(1,14,8,4,3,13,7,6)(2,10)(5,9)(15,18,21,25,17,22,26,29)(16,24,27,28,19,20,30,23)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227690, #72227745, #72227825, …

GAP

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

gap> G := Group((1,13,3,14)(4,11,8,10,6,12,7,9)(15,21,25,28,17,26,29,23)(16,22,24,27,19,18,20,30), (1,14,8,4,3,13,7,6)(2,10)(5,9)(15,18,21,25,17,22,26,29)(16,24,27,28,19,20,30,23));