AΓL(1,𝔽₉) #3839

order	144
isoclinism	isoclinic class of AΓL(1,𝔽₉)
automorphism	Aut(G) = AΓL(1,𝔽₉) (order 144) · Inn(G) = G · Out(G) = C₁
holomorph	G ⋊ Aut(G) = AΓL(1,𝔽₉)²
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

affine general semilinear	AΓL(1,𝔽₉)
unambiguous semidirect product	AGL(1,𝔽₉)⋊C₂
unambiguous semidirect product	PSU(3,𝔽₂)⋊C₂

C₃² ⋊ SD₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	#157	C₂²
Fitting, socle, hypocenter	C₃²	SD₁₆
prime radical	GO⁺(4,𝔽₂)	C₂
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁

25 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:

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

Polycyclic presentation:

2^4,3^2|a^2=c,c^2=d,b^a=bcd,e^a=f,f^a=ef^2,c^b=cd,e^b=ef^2,f^b=f^2,e^c=ef^2,f^c=e^2f^2,e^d=e^2,f^d=f^2

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 182);