#261⋊C₉ #90832

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

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₃	AGL(1,𝔽₁₃)
commutator, hypocenter	C₁₃	C₄×C₉
Fitting, socle	C₃×C₁₃	C₃×C₄
prime radical, cosocle	C₃×D₁₃	C₂×C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃

18 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:

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

Polycyclic presentation:

2,3,2,3,13|a^2=c,b^3=d,e^a=e^8,e^b=e^3,e^c=e^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 7);