D₉₉ #6005

order	198
isoclinism	isoclinic class of D₉₉
automorphism	Aut(G) = AGL(1,ℤ/9)×AGL(1,𝔽₁₁) (order 5940) · Inn(G) = G · Out(G) = C₂×C₃×C₅
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₉
unambiguous semidirect product	C₉⋊D₁₁
unambiguous semidirect product	C₁₁⋊D₉

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₁	C₂
Frattini	C₃	D₃₃
socle	C₃×C₁₁	D₃
LCS	G ▷ C₉×C₁₁
UCS	C₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 99

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(198, 3);