order	216
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = S₄×#272 (order 1296) · Inn(G) = A₄ · Out(G) = C₂×#272
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×Q₈⋊C₉
unambiguous semidirect product	(C₃×Q₈)⋊C₉

Splittings

Q₈ ⋊ C₃×C₉

	subgroup	G/H
center, socle, prime radical, hypercenter	C₂×C₃²	A₄
commutator, hypocenter	Q₈	C₃×C₉
Frattini	C₂×C₃	C₃×A₄
Fitting	C₃²×Q₈	C₃
cosocle	C₃×Q₈	C₃²
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃²

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 40);