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C₂×C₄⋊Dic₃ #925

order	96
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = D₃×#347042 (order 3072) · Inn(G) = C₂×D₃ · Out(G) = #32898
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₄⋊Dic₃

Splittings

C₃ ⋊ C₂×C₄⋊C₄

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₃
commutator	C₂×C₃	C₂²×C₄
Frattini	C₂²	C₂²×D₃
Fitting	C₂²×C₃×C₄	C₂
socle, prime radical	C₂³×C₃	C₂²
cosocle	C₂²×C₃	C₂³
hypercenter	C₂²×C₄	D₃
hypocenter	C₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#4529, #4551, #4554, …
maximal normal subgroup of	#4529, #4551, #4554, …
maximal quotient of	C₂×C₄⋊(C₃⋊C₈), #4924, C₄×C₄⋊Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 132);