D₁₆₅ #66517

order	330
isoclinism	isoclinic class of D₁₆₅
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₅
unambiguous semidirect product	C₃⋊D₅₅
unambiguous semidirect product	C₅⋊D₃₃
unambiguous semidirect product	C₁₁⋊D₁₅

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₁	C₂
LCS	G ▷ C₃×C₅×C₁₁
UCS	C₁

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 165

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2,3,5,11|b^a=b^2,c^a=c^4,d^a=d^10

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(330, 11);