#108×C₂²⋊A₄ #72253109

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#108×C₂²⋊A₄

Splittings

Fitting, Sylow

C₂⁴×#108 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	D₄×C₂²⋊A₄
commutator	C₂⁴×C₄	C₂×C₃×C₄
Frattini	C₂×C₄	C₂²×C₂²⋊A₄
Fitting	C₂⁴×#108	C₃
socle	C₂⁵	C₃×C₄⋊C₄
prime radical	C₂²×C₂²⋊A₄	Q₈
cosocle	C₂⁵×C₄	C₂²×C₃
hypercenter	#108	C₂²⋊A₄
hypocenter	C₂⁴	C₃×#108
LCS	G ▷ C₂⁴×C₄ ▷ C₂⁵ ▷ C₂⁴
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

3925 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 3925 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 96

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄²×#108
maximal normal subgroup of	A₄²×#108

GAP

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

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