C₅×C₇⋊D₇ #92439

order	490
isoclinism	isoclinic class of C₇⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₅×C₇⋊D₇
unambiguous semidirect product	C₇⋊(C₅×D₇)

Fitting, socle, Hall	C₅×C₇² ⋊ C₂
derived, Sylow	C₇² ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₅	C₇⋊D₇
commutator, cosocle, hypocenter	C₇²	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²	C₂
LCS	G ▷ C₇²
UCS	C₁ ◁ C₅

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 245

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,3,4,5)(7,8)(9,10)(11,12)(14,15)(16,17)(18,19);(13,14,16,18,19,17,15);(6,7,9,11,12,10,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(490, 6);