C₃²×ΓL(1,𝔽₈) #4435

order	189
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(2,𝔽₃) (order 18144) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×AGL(2,𝔽₃)
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃²×ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊C₃³
unambiguous semidirect product	(C₃×C₇)⋊C₃²

Fitting, socle	C₃²×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃³

	subgroup	G/H
center, hypercenter	C₃²	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃³
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃²

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(189, 12);