#127⋊C₃² #72228275

order	9216
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₄²

Aliases

direct product

#127⋊C₃²

Splittings

Fitting, Sylow	#127² ⋊ C₃²
derived	C₄⁴ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄⁴	C₂²×C₃²
Frattini, socle	C₂⁴	C₂²×A₄²
Fitting	#127²	C₃²
LCS	G ▷ C₄⁴
UCS	C₁

7318 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 7318 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 24
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258960
maximal normal subgroup of	#72258960

GAP

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

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