C₂×C₇×ΓL(1,𝔽₈) #64426

order	294
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₇) (order 252) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₇×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₇)⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₂×C₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇²	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₇

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(294, 15);