C₂×C₄×ΓL(1,𝔽₈) #4249

order	168
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = D₄×AGL(1,𝔽₇) (order 336) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×D₄
holomorph	G ⋊ Aut(G) = B(4,𝔽₂)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₄×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₄×C₇)⋊C₃

Fitting, Hall	C₂×C₄×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₄

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 19);