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A₄×SD₁₆ #5456

order	192
automorphism	Aut(G) = C₂×D₄×S₄ (order 384) · Inn(G) = D₄×A₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×SD₁₆
unambiguous semidirect product	(C₂²×SD₁₆)⋊C₃

Splittings

C₂²×SD₁₆ ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	D₄×A₄
commutator, cosocle	C₂²×C₄	C₂²×C₃
Frattini	C₄	C₂²×A₄
Fitting	C₂²×SD₁₆	C₃
socle	C₂³	C₃×D₄
prime radical	D₄×A₄	C₂
hypercenter	SD₁₆	A₄
hypocenter	C₂²	C₃×SD₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

83 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:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×S₄, #85460, #85461, …
maximal normal subgroup of	SD₁₆×S₄, #85460, #85461, …
maximal quotient of	A₄×#102, A₄×#103, A₄×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1015);