#126⋊C₃² #72258871

order	9216
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

#126⋊C₃²

Splittings

Fitting, Sylow	#126² ⋊ 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	#126²	C₃²
LCS	G ▷ C₄⁴
UCS	C₁

1810 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 1810 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) = 32
transitive degree	μ_t(G) = 96

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

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

Conjugacy classes & character table

100 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((2,13,9,6,7,3)(4,12,5)(8,14,10,15,16,11)(17,31,30,32,28,24)(18,22,25,20,23,26)(19,21,27), (1,16,12,7)(2,11,15,10)(3,8,9,6)(4,14,5,13)(18,28,27,24,25,19)(20,26,29,30,31,21)(22,32,23));