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C₄×Dic₆ #868

order	96
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = C₂²×D₃×#120 (order 768) · Inn(G) = C₂×D₃ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₄×Dic₆

Splittings

C₃ ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₃
commutator	C₂×C₃	C₂²×C₄
Frattini	C₂²	C₂²×D₃
Fitting	C₃×C₄²	C₂
socle, prime radical, cosocle	C₂²×C₃	C₂³
hypercenter	C₄²	D₃
hypocenter	C₃	C₄×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#4456, #4484, C₈×Dic₆, …
maximal normal subgroup of	#4456, #4484, C₈×Dic₆, …
maximal quotient of	#4646, #4647, #4654, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 75);