C₂×#72228154 #72239019

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#72228154

Splittings

Fitting	C₂×#153709 ⋊ D₃
derived	#72222569 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72209990
commutator, cosocle, hypocenter	#72222569	C₂²
Frattini	C₂⁵	C₂×#1020
Fitting	C₂×#153709	D₃
socle	C₂⁶	#1020
LCS	G ▷ #72222569
UCS	C₁ ◁ C₂²

11925 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 11925 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 32
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,8,6,4,7,3,5)(9,11,12,17)(10,20,15,18,13,19,16,14)(22,31)(26,32)(27,29)(28,30);(2,6,3)(5,8,7)(9,10,18)(11,16,20)(12,13,14)(15,19,17)(21,22,28,25,26,27)(23,29,31,24,30,32)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72239028

GAP

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

gap> G := Group((1,2,8,6,4,7,3,5)(9,11,12,17)(10,20,15,18,13,19,16,14)(22,31)(26,32)(27,29)(28,30), (2,6,3)(5,8,7)(9,10,18)(11,16,20)(12,13,14)(15,19,17)(21,22,28,25,26,27)(23,29,31,24,30,32));