C₁₃⋊C₉ #1184

order	117
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	Aut(G) = C₃×AGL(1,𝔽₁₃) (order 468) · Inn(G) = C₁₃⋊C₃ · Out(G) = C₃×C₄
holomorph	G ⋊ Aut(G) = #72230380
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

C₁₃⋊C₉

C₁₃ ⋊ C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₁₃⋊C₃
commutator, hypocenter	C₁₃	C₉
Fitting, socle, prime radical, cosocle	C₃×C₁₃	C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃

6 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(117, 1);