D₃₃ #590

order	66
isoclinism	isoclinic class of D₃₃
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₁) (order 660) · Inn(G) = G · Out(G) = C₂×C₅
holomorph	G ⋊ Aut(G) = #72225758
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₃₃
unambiguous semidirect product	C₃⋊D₁₁
unambiguous semidirect product	C₁₁⋊D₃

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

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 33

A minimal faithful permutation representation — 2 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(66, 3);