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C₃×#104 #847

order	96
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂³×D₄ (order 64) · Inn(G) = D₄ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

Subgroups

	subgroup	G/H
center	C₃×C₄	D₄
commutator	C₄	C₂×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂²×C₃
socle	C₂×C₃	C₂²⋊C₄
prime radical	C₃×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ G

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#104, #4821, #4822, …
maximal subgroup of	D₃×#104, #4821, #4822, …
maximal normal subgroup of	D₃×#104, #4821, #4822, …
maximal quotient of	C₃×D₄⋊C₈, C₃×Q₈⋊C₈, C₃×#327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 54);