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A₄×M₃₂ #73274

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×M₃₂
unambiguous semidirect product	(C₂²×M₃₂)⋊C₃

Splittings

C₂²×M₃₂ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂²×A₄
commutator, socle	C₂³	C₂×C₃×C₈
Fitting	C₂²×M₃₂	C₃
prime radical	C₂²×A₄	C₈
cosocle	C₂²×C₈	C₂²×C₃
hypercenter	M₃₂	A₄
hypocenter	C₂²	C₃×M₃₂
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₈ ◁ M₃₂

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×#347, A₄×#349, A₄×#350, …
maximal normal subgroup of	A₄×#347, A₄×#349, A₄×#350, …
maximal quotient of	A₄×#346, A₄×C₂²⋊C₁₆, A₄×C₄⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5850);