C₃×C₇×ΓL(1,𝔽₈) #89208

order	441
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₂×C₃×D₃×AGL(1,𝔽₇) (order 1512) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂²×C₃×D₃
homology	Abelianization (H₁) = C₃²×C₇ · Schur multiplier (H₂) = C₃

Aliases

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

	subgroup	G/H
center, hypercenter	C₃×C₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₇

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(441, 10);