Finite Group Databasebeta

← Prev (#840) Next (#842) →

C₃×C₂²⋊C₈ #841

order	96
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×#120 (order 128) · Inn(G) = C₂² · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

C₃×C₂²⋊C₈

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²
commutator	C₂	C₂×C₃×C₈
Frattini, cosocle	C₂×C₄	C₂²×C₃
socle	C₂²×C₃	C₂×C₄
prime radical	C₂³×C₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#4467, #4468, #4469, …
maximal subgroup of	#4467, #4468, #4469, …
maximal normal subgroup of	#4467, #4468, #4469, …
maximal quotient of	C₃×#323, C₃×#324, C₃×D₄⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 48);