Dic₇₀ #63319

order	280
isoclinism	isoclinic class of Dic₇₀
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₇₀

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₇	C₂²
Fitting	C₄×C₅×C₇	C₂
hypercenter	C₄	D₃₅
hypocenter	C₅×C₇	Q₈
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 280

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13)(14,15,16,17,18,19,20);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,20)(16,19)(17,18)

Polycyclic presentation:

2^3,5,7|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 24);