order	168
isoclinism	isoclinic class of D₄×ΓL(1,𝔽₈)
automorphism	Aut(G) = D₄×AGL(1,𝔽₇) (order 336) · Inn(G) = C₂²×ΓL(1,𝔽₈) · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #453×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×ΓL(1,𝔽₈)
unambiguous semidirect product	C₄⋊(C₂×ΓL(1,𝔽₈))
unambiguous semidirect product	(C₇×D₄)⋊C₃

Splittings

Fitting, Hall	C₇×D₄ ⋊ C₃
Sylow	C₇ ⋊ C₃×D₄

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 20);