C₃⋊D₃₃ #6011

order	198
isoclinism	isoclinic class of C₃⋊D₃₃
automorphism	Aut(G) = AGL(1,𝔽₁₁)×AGL(2,𝔽₃) (order 47520) · Inn(G) = G · Out(G) = C₅×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₃₃
unambiguous semidirect product	C₁₁⋊(C₃⋊D₃)

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₁	C₂
LCS	G ▷ C₃²×C₁₁
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 99

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(198, 9);