C₇⋊S₄ #4276

order	168
isoclinism	isoclinic class of C₇⋊S₄
automorphism	Aut(G) = S₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₇⋊S₄
unambiguous semidirect product	A₄⋊D₇

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×A₄	C₂
Fitting, socle	C₂²×C₇	D₃
LCS	G ▷ C₇×A₄
UCS	C₁

22 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:

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

Polycyclic presentation:

2,3,7,2^2|b^a=b^2,c^a=c^6,d^a=e,e^a=d,d^b=e,e^b=de

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 46);