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C₃²×SL(2,ℤ/4) #88968

order	432
isoclinism	isoclinic class of S₄
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃²×SL(2,ℤ/4)
unambiguous semidirect product	A₄⋊(C₃²×C₄)

Splittings

A₄ ⋊ C₃²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²	S₄
commutator, hypocenter	A₄	C₃²×C₄
Frattini	C₂	C₃²×S₄
Fitting, socle	C₂³×C₃²	D₃
prime radical	C₂×C₃²×A₄	C₂
cosocle	C₂×A₄	C₂×C₃²
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₃²

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#64204, C₃×(C₃⋊S₄)⋊C₄, C₃×D₃×SL(2,ℤ/4), …
maximal normal subgroup of	C₃×#64204, C₃×(C₃⋊S₄)⋊C₄, C₃×D₃×SL(2,ℤ/4), …
maximal quotient of	C₃²×A₄⋊C₈, C₃²×#859, C₃²×GU(2,𝔽₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 615);