D₅×D₇ #3649

order	140
isoclinism	isoclinic class of D₅×D₇
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₇) (order 840) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₅×D₇
unambiguous semidirect product	(C₇×D₅)⋊C₂
unambiguous semidirect product	(C₅×D₇)⋊C₂
unambiguous semidirect product	D₃₅⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇	C₂²
LCS	G ▷ C₅×C₇
UCS	C₁

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

(2,3)(4,5)(6,7)(8,9,11,12,10);(1,2,4,6,7,5,3)(9,10)(11,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(140, 7);