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Q₈×Dic₃ #945

order	96
automorphism	Aut(G) = C₂²×D₃×#1020 (order 2304) · Inn(G) = C₂²×D₃ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×Dic₃

Splittings

C₃ ⋊ C₄×Q₈

Subgroups

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

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:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#4788, #4789, #4790, …
maximal normal subgroup of	#4788, #4789, #4790, …
maximal quotient of	Dic₃×C₄⋊C₄, #4980, #4984, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 152);