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C₃×#95 #838

order	96
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#87524 (order 768) · Inn(G) = C₂² · Out(G) = C₂×GL(2,ℤ/4)
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²
commutator	C₂	C₃×C₄²
Frattini, cosocle	C₂³	C₂²×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊(#392⋊C₃), C₃⋊(#401⋊C₃), C₃⋊(#376⋊C₃), …
Fitting subgroup of	#4451, #4452, #4646, …
maximal subgroup of	#4451, #4452, C₃×#327, …
maximal normal subgroup of	#4451, #4452, C₃×#327, …
maximal quotient of	C₃×#336, C₃×H(3,ℤ/4), C₃×#338, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 45);