C₃²×SL(2,𝔽₃) #6297

order	216
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = S₄×AGL(2,𝔽₃) (order 10368) · Inn(G) = A₄ · Out(G) = C₂×AGL(2,𝔽₃)
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃²×SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊C₃³
unambiguous semidirect product	(C₃×Q₈)⋊C₃²

Fitting	C₃²×Q₈ ⋊ C₃
derived, Sylow	Q₈ ⋊ C₃³

	subgroup	G/H
center, socle, hypercenter	C₂×C₃²	A₄
commutator, cosocle, hypocenter	Q₈	C₃³
Frattini	C₂	C₃²×A₄
Fitting	C₃²×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃²

90 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 134);