D₃₅ #601

order	70
isoclinism	isoclinic class of D₃₅
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₇) (order 840) · Inn(G) = G · Out(G) = C₃×C₄
holomorph	G ⋊ Aut(G) = #72225770
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₃₅
unambiguous semidirect product	C₅⋊D₇
unambiguous semidirect product	C₇⋊D₅

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

8 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:

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

Polycyclic presentation:

2,5,7|b^a=b^4,c^a=c^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(70, 3);