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A₄×C₂²⋊C₄ #5435

order	192
automorphism	Aut(G) = S₄×#120 (order 768) · Inn(G) = C₂²×A₄ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	A₄×C₂²⋊C₄
unambiguous semidirect product	(C₂²×C₂²⋊C₄)⋊C₃

Splittings

C₂²×C₂²⋊C₄ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×A₄
commutator	C₂³	C₂×C₃×C₄
Fitting	C₂²×C₂²⋊C₄	C₃
socle, cosocle	C₂⁴	C₂²×C₃
prime radical	C₂³×A₄	C₂
hypercenter	C₂²⋊C₄	A₄
hypocenter	C₂²	C₃×C₂²⋊C₄
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#73049, A₄×#99, #85292, …
maximal normal subgroup of	#73049, A₄×#99, #85292, …
maximal quotient of	A₄×#95, C₂²⋊C₄×SL(2,𝔽₃), #2341⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 994);