order	72
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = D₃×S₄ (order 144) · Inn(G) = A₄ · Out(G) = C₂×D₃
holomorph	G ⋊ Aut(G) = #72220377
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊C₃²
unambiguous semidirect product	(C₃×Q₈)⋊C₃

Splittings

Fitting	C₃×Q₈ ⋊ C₃
derived, Sylow	Q₈ ⋊ C₃²

	subgroup	G/H
center, socle, hypercenter	C₂×C₃	A₄
commutator, cosocle, hypocenter	Q₈	C₃²
Frattini	C₂	C₃×A₄
Fitting	C₃×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 25);