AGL(1,𝔽₉) #642

order	72
isoclinism	isoclinic class of AGL(1,𝔽₉)
automorphism	Aut(G) = AΓL(1,𝔽₉) (order 144) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72220433
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

AGL(1,𝔽₉)

C₃² ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²	C₈
prime radical	C₃⋊D₃	C₄
cosocle	#157	C₂
LCS	G ▷ C₃²
UCS	C₁

10 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:

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

Polycyclic presentation:

2^3,3^2|a^2=b,b^2=c,d^a=e,e^a=de^2,d^b=de^2,e^b=d^2e^2,d^c=d^2,e^c=e^2

Here ζ is a primitive root of unity of order 8 (ζ⁸ = 1); i = ζ².

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 39);