A₄×#85440 #72269841

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

A₄×#85440

Splittings

Fitting

C₂⁵×C₈ ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center	C₄	A₄×C₄⋊S₄
commutator, cosocle	C₂²×C₄×A₄	C₂³×C₃
Frattini	C₄	C₂²×A₄×S₄
Fitting	C₂⁵×C₈	C₃×D₃
socle	C₂⁵	C₂×C₃×D₁₂
hypercenter	C₂×C₈	A₄×S₄
hypocenter	C₂²×A₄	C₃×#135
LCS	G ▷ C₂²×C₄×A₄ ▷ C₂³×A₄ ▷ C₂²×A₄
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

4616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 144

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

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

Conjugacy classes & character table

176 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((2,3,4)(5,8,6)(9,11,18,24,13,20,12,19)(10,15,21,23,14,22,16,17), (1,4)(2,3)(10,14)(15,22)(16,21)(17,23), (7,8)(9,10)(11,17)(12,21)(13,14)(15,19)(16,18)(20,23)(22,24));

A4×#85440 #72269841