C₁₃⋊C₂₇ #66845

order	351
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	Aut(G) = C₉×AGL(1,𝔽₁₃) (order 1404) · Inn(G) = C₁₃⋊C₃ · Out(G) = C₄×C₉
homology	Abelianization (H₁) = C₂₇ · Schur multiplier (H₂) = C₁

Aliases

C₁₃⋊C₂₇

C₁₃ ⋊ C₂₇

	subgroup	G/H
center, Frattini, hypercenter	C₉	C₁₃⋊C₃
commutator, hypocenter	C₁₃	C₂₇
Fitting, cosocle	C₉×C₁₃	C₃
socle, prime radical	C₃×C₁₃	C₉
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₉

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 351

A minimal faithful permutation representation — 2 generators on 40 points:

(2,3,5)(4,7,10)(6,9,8)(11,13,12)(14,15,18,16,19,24,21,26,31,17,20,25,22,27,32,29,34,37,23,28,33,30,35,38,36,39,40);(1,2,4,3,6,10,7,11,13,5,8,12,9)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(351, 1);