C₂×D₇⋊A₄ #66749

order	336
isoclinism	isoclinic class of D₇⋊A₄
automorphism	Aut(G) = C₂×A₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₇⋊A₄ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×D₇⋊A₄
unambiguous semidirect product	D₇⋊(C₂×A₄)
unambiguous semidirect product	(C₂×D₇)⋊A₄
unambiguous semidirect product	(C₂²⋊ΓL(1,𝔽₈))⋊C₂²
unambiguous semidirect product	(C₂×C₂²⋊ΓL(1,𝔽₈))⋊C₂

	subgroup	G/H
center, hypercenter	C₂	D₇⋊A₄
commutator, cosocle, hypocenter	C₂²×C₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂

78 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 218);