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A₄×#108 #73247

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×#108
unambiguous semidirect product	(C₂²×#108)⋊C₃

Splittings

C₂²×#108 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	D₄×A₄
commutator	C₂²×C₄	C₂×C₃×C₄
Frattini	C₂×C₄	C₂²×A₄
Fitting	C₂²×#108	C₃
socle	C₂³	C₃×C₄⋊C₄
prime radical	C₂²×A₄	Q₈
cosocle	C₂³×C₄	C₂²×C₃
hypercenter	#108	A₄
hypocenter	C₂²	C₃×#108
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

130 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)(5,15,9,7,10,16,8,17)(6,14,18,11,13,20,12,19);(1,4)(2,3)(5,12,8,6,10,18,9,13)(7,19,15,14,17,11,16,20)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×#359, A₄×#361, A₄×#362, …
maximal normal subgroup of	A₄×#359, A₄×#361, A₄×#362, …
maximal quotient of	A₄×#334, A₄×#335, #17835⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5823);