D₁₁₇ #6616

order	234
isoclinism	isoclinic class of D₁₁₇
automorphism	Aut(G) = AGL(1,ℤ/9)×AGL(1,𝔽₁₃) (order 8424) · Inn(G) = G · Out(G) = C₃²×C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇
unambiguous semidirect product	C₉⋊D₁₃
unambiguous semidirect product	C₁₃⋊D₉

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₃	C₂
Frattini	C₃	D₃₉
socle	C₃×C₁₃	D₃
LCS	G ▷ C₉×C₁₃
UCS	C₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 2 generators on 22 points:

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

Polycyclic presentation:

2,3,13,3|b^3=d^2,b^a=b^2d,c^a=c^12,d^a=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 5);