Dic₃₃ #3606

order	132
isoclinism	isoclinic class of D₃₃
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₁₁) (order 1320) · Inn(G) = D₃₃ · Out(G) = C₂²×C₅
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃
unambiguous semidirect product	C₃⋊Dic₁₁
unambiguous semidirect product	C₁₁⋊Dic₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₃
commutator, hypocenter	C₃×C₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₁	C₂
LCS	G ▷ C₃×C₁₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 132

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(132, 3);