D₁₃⋊C₉ #6612

order	234
isoclinism	isoclinic class of D₁₃⋊C₃
automorphism	Aut(G) = C₃×AGL(1,𝔽₁₃) (order 468) · Inn(G) = D₁₃⋊C₃ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₁₃⋊C₉
unambiguous semidirect product	(C₁₃⋊C₉)⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₁₃⋊C₃
commutator, hypocenter	C₁₃	C₂×C₉
Fitting, socle, cosocle	C₃×C₁₃	C₂×C₃
prime radical	C₃×D₁₃	C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃

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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 1);