Dic₁₆₅ #10620341

order	660
isoclinism	isoclinic class of D₁₆₅
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₆₅
unambiguous semidirect product	C₃⋊Dic₅₅
unambiguous semidirect product	C₁₁⋊Dic₁₅

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₆₅
commutator, hypocenter	C₃×C₅×C₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₁	C₂
LCS	G ▷ C₃×C₅×C₁₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 660

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23);(1,3,2,4)(6,7)(9,12)(10,11)(14,23)(15,22)(16,21)(17,20)(18,19)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(660, 11);