C₂²×C₃⋊D₃ #652

order	72
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = S₄×AGL(2,𝔽₃) (order 10368) · Inn(G) = C₃⋊D₃ · Out(G) = S₄²
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

C₂²×C₃⋊D₃

Fitting, socle	C₂²×C₃² ⋊ C₂
derived, Sylow	C₃² ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²

96 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 49);