AGL(2,𝔽₃) #89087

order	432
isoclinism	isoclinic class of AGL(2,𝔽₃)
automorphism	Aut(G) = AGL(2,𝔽₃) (order 432) · Inn(G) = G · Out(G) = C₁
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

affine general linear	AGL(2,𝔽₃)
unambiguous semidirect product	(PSU(3,𝔽₂)⋊C₃)⋊C₂

Fitting, socle	C₃² ⋊ GL(2,𝔽₃)
derived	PSU(3,𝔽₂)⋊C₃ ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	PSU(3,𝔽₂)⋊C₃	C₂
Fitting, socle	C₃²	GL(2,𝔽₃)
LCS	G ▷ PSU(3,𝔽₂)⋊C₃
UCS	C₁

46 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,8,7,4,5,3,9);(2,5,6)(3,4,7)

Polycyclic presentation:

2,3,2^3,3^2|c^2=e,d^2=e,b^a=b^2,c^a=ce,d^a=cde,f^a=fg^2,g^a=g^2,c^b=de,d^b=cd,f^b=g^2,g^b=fg^2,d^c=de,f^c=f^2g,g^c=fg,f^d=g^2,g^d=f,f^e=f^2,g^e=g^2

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 734);