C₃×#135 #975

order	96
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂²×AGL(1,ℤ/8) (order 128) · Inn(G) = D₄ · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center	C₃×C₄	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃
socle	C₂×C₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ G

62 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2)(3,10)(4,7)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18,19);(1,3)(2,6)(4,11)(5,13)(7,10)(8,16)(9,15)(12,14);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 182);