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A₄×#41 #5942

order	192
automorphism	Aut(G) = C₂×S₄² (order 1152) · Inn(G) = C₂²×A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₂²×#41 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	C₂²×A₄
commutator, socle, cosocle	C₂³	C₂³×C₃
Frattini	C₂	C₂³×A₄
Fitting	C₂²×#41	C₃
hypercenter	#41	A₄
hypocenter	C₂²	C₃×#41
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₄ ◁ #41

215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 12 points:

(2,3,4)(5,11,8,7)(6,12,9,10);(1,4)(2,3)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#73150, A₄×#104, #85511, …
maximal normal subgroup of	#73150, A₄×#104, #85511, …
maximal quotient of	A₄×#117, C₄×D₄×A₄, C₄×Q₈×A₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1501);