C₃²×D₄ #640

order	72
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = D₄×GL(2,𝔽₃) (order 384) · Inn(G) = C₂² · Out(G) = C₂×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃²×D₄
unambiguous semidirect product	C₄⋊(C₂×C₃²)

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

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2)(3,4)(5,6,7);(2,4)(8,9,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 37);