C₅²×D₇ #66839

order	350
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×D₇
unambiguous semidirect product	C₇⋊(C₂×C₅²)

Fitting, socle, Hall	C₅²×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₅²

	subgroup	G/H
center, hypercenter	C₅²	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₅²
Frattini	C₁	G
Fitting, socle	C₅²×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₅²

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 175

A minimal faithful permutation representation — 2 generators on 17 points:

(6,7,8,9,10)(12,13)(14,15)(16,17);(1,2,3,4,5)(11,12,14,16,17,15,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(350, 5);