C₃²×Q₈ #641

order	72
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = S₄×GL(2,𝔽₃) (order 1152) · Inn(G) = C₂² · Out(G) = D₃×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×Q₈

	subgroup	G/H
center, socle, prime radical	C₂×C₃²	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 38);