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C₄×A₄⋊C₈ #73043

order	384
isoclinism	isoclinic class of S₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

C₄×A₄⋊C₈

Splittings

A₄ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	S₄
commutator, hypocenter	A₄	C₄×C₈
Frattini	C₂×C₄	C₂×S₄
Fitting	C₂²×C₄²	D₃
socle	C₂⁴	C₂×Dic₃
prime radical	C₂²×A₄	C₂×C₄
cosocle	C₂×C₄×A₄	C₂²
LCS	G ▷ A₄
UCS	C₁ ◁ C₄²

189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×A₄⋊C₈, #11710984, #11710985, …
maximal normal subgroup of	C₈×A₄⋊C₈, #11710984, #11710985, …
maximal quotient of	C₈×A₄⋊C₈, #11710984, C₄×SL(2,𝔽₃)⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5619);