S₇ #72210209

order	5040
automorphism	Aut(G) = S₇ (order 5040) · Inn(G) = G · Out(G) = C₁
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

S₇

A₇ ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	A₇	C₂
LCS	G ▷ A₇
UCS	C₁

96 conjugacy classes of subgroups.

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 7

A minimal faithful permutation representation — 2 generators on 7 points:

(1,2,3,4,5,6,7);(1,2)

maximal subgroup of	S₈
automorphism group of	A₇, S₇

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,4,5,6,7), (1,2));