C₂×C₃×Q₁₆ #974

order	96
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×#23882 (order 512) · Inn(G) = D₄ · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

C₂×C₃×Q₁₆

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ G

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2)(3,4,5);(1,2)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15);(1,2)(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 181);