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C₄²×A₄ #5434

order	192
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = S₄×GL(2,ℤ/4) (order 2304) · Inn(G) = A₄ · Out(G) = C₂×GL(2,ℤ/4)
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

C₄²×A₄

Splittings

Fitting, Sylow	C₂²×C₄² ⋊ C₃
derived	C₂² ⋊ C₃×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	A₄
commutator, hypocenter	C₂²	C₃×C₄²
Frattini	C₂²	C₂²×A₄
Fitting	C₂²×C₄²	C₃
socle, cosocle	C₂⁴	C₂²×C₃
prime radical	C₂²×A₄	C₂²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₄²

133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 48

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

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup of	S₄×C₄²⋊C₃, A₄⋊(#127⋊C₃), A₄⋊(#126⋊C₃)
maximal subgroup of	C₄×A₄⋊C₈, #73044, C₄⋊(A₄⋊C₈), …
maximal normal subgroup of	C₄×A₄⋊C₈, #73044, C₄⋊(A₄⋊C₈), …
maximal quotient of	A₄×#95, C₄²×SL(2,𝔽₃), #73167, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 993);