Dic₃₅ #3645

order	140
isoclinism	isoclinic class of D₃₅
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(1,𝔽₇) (order 1680) · Inn(G) = D₃₅ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₅
unambiguous semidirect product	C₇⋊Dic₅

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₅
commutator, hypocenter	C₅×C₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₇	C₂
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 140

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(140, 3);