D₃₁₅ #10597859

order	630
isoclinism	isoclinic class of D₃₁₅
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₃₁₅
unambiguous semidirect product	C₅⋊D₆₃
unambiguous semidirect product	C₇⋊D₄₅
unambiguous semidirect product	C₉⋊D₃₅

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉	C₂
Frattini	C₃	D₁₀₅
socle	C₃×C₅×C₇	D₃
LCS	G ▷ C₅×C₇×C₉
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 315

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,21)(17,20)(18,19)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(630, 11);