D₇×C₂₅ #66836

order	350
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₄×C₅×AGL(1,𝔽₇) (order 840) · Inn(G) = D₇ · Out(G) = C₃×C₄×C₅
homology	Abelianization (H₁) = C₂×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₇×C₂₅
unambiguous semidirect product	C₇⋊(C₂×C₂₅)

Fitting, Hall	C₇×C₂₅ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₂₅

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 175

A minimal faithful permutation representation — 2 generators on 32 points:

(2,3)(4,5)(6,7)(8,9,11,14,18,10,12,15,19,23,13,16,20,24,27,17,21,25,28,30,22,26,29,31,32);(1,2,4,6,7,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(350, 2);