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## G = AΓL1(𝔽9)  order 144 = 24·32

### Affine semilinear group on 𝔽91

Aliases: AΓL1(𝔽9), F9⋊C2, C32⋊SD16, PSU3(𝔽2)⋊C2, C3⋊S3.D4, S3≀C2.C2, C32⋊C4.C22, Aut(C32⋊C4), SmallGroup(144,182)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C4 — AΓL1(𝔽9)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — F9 — AΓL1(𝔽9)
 Lower central C32 — C3⋊S3 — C32⋊C4 — AΓL1(𝔽9)
 Upper central C1

Generators and relations for AΓL1(𝔽9)
G = < a,b,c,d | a3=b3=c8=d2=1, cac-1=ab=ba, dad=a-1b, cbc-1=a, bd=db, dcd=c3 >

9C2
12C2
4C3
9C4
18C4
18C22
4S3
12C6
12S3
9C8
9D4
9Q8
12D6
9SD16
2S32

Character table of AΓL1(𝔽9)

 class 1 2A 2B 3 4A 4B 6 8A 8B size 1 9 12 8 18 36 24 18 18 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 -1 -1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ5 2 2 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 2 0 0 0 √-2 -√-2 complex lifted from SD16 ρ7 2 -2 0 2 0 0 0 -√-2 √-2 complex lifted from SD16 ρ8 8 0 -2 -1 0 0 1 0 0 orthogonal faithful ρ9 8 0 2 -1 0 0 -1 0 0 orthogonal faithful

Permutation representations of AΓL1(𝔽9)
On 9 points: primitive, doubly transitive - transitive group 9T19
Generators in S9
(1 7 3)(2 4 9)(5 8 6)
(1 8 4)(2 3 5)(6 9 7)
(2 3 4 5 6 7 8 9)
(2 6)(3 9)(5 7)

G:=sub<Sym(9)| (1,7,3)(2,4,9)(5,8,6), (1,8,4)(2,3,5)(6,9,7), (2,3,4,5,6,7,8,9), (2,6)(3,9)(5,7)>;

G:=Group( (1,7,3)(2,4,9)(5,8,6), (1,8,4)(2,3,5)(6,9,7), (2,3,4,5,6,7,8,9), (2,6)(3,9)(5,7) );

G=PermutationGroup([(1,7,3),(2,4,9),(5,8,6)], [(1,8,4),(2,3,5),(6,9,7)], [(2,3,4,5,6,7,8,9)], [(2,6),(3,9),(5,7)])

G:=TransitiveGroup(9,19);

On 12 points - transitive group 12T84
Generators in S12
(1 9 5)(3 11 7)(4 12 8)
(1 5 9)(2 10 6)(4 12 8)
(1 2 3 4)(5 6 7 8 9 10 11 12)
(2 4)(6 8)(7 11)(10 12)

G:=sub<Sym(12)| (1,9,5)(3,11,7)(4,12,8), (1,5,9)(2,10,6)(4,12,8), (1,2,3,4)(5,6,7,8,9,10,11,12), (2,4)(6,8)(7,11)(10,12)>;

G:=Group( (1,9,5)(3,11,7)(4,12,8), (1,5,9)(2,10,6)(4,12,8), (1,2,3,4)(5,6,7,8,9,10,11,12), (2,4)(6,8)(7,11)(10,12) );

G=PermutationGroup([(1,9,5),(3,11,7),(4,12,8)], [(1,5,9),(2,10,6),(4,12,8)], [(1,2,3,4),(5,6,7,8,9,10,11,12)], [(2,4),(6,8),(7,11),(10,12)])

G:=TransitiveGroup(12,84);

On 18 points - transitive group 18T68
Generators in S18
(1 5 9)(2 13 17)(3 14 12)(4 11 6)(7 16 18)(8 10 15)
(1 14 18)(2 6 10)(3 16 9)(4 15 13)(5 12 7)(8 17 11)
(1 2)(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)
(1 2)(3 13)(4 16)(5 11)(6 14)(7 17)(8 12)(9 15)(10 18)

G:=sub<Sym(18)| (1,5,9)(2,13,17)(3,14,12)(4,11,6)(7,16,18)(8,10,15), (1,14,18)(2,6,10)(3,16,9)(4,15,13)(5,12,7)(8,17,11), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (1,2)(3,13)(4,16)(5,11)(6,14)(7,17)(8,12)(9,15)(10,18)>;

G:=Group( (1,5,9)(2,13,17)(3,14,12)(4,11,6)(7,16,18)(8,10,15), (1,14,18)(2,6,10)(3,16,9)(4,15,13)(5,12,7)(8,17,11), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (1,2)(3,13)(4,16)(5,11)(6,14)(7,17)(8,12)(9,15)(10,18) );

G=PermutationGroup([(1,5,9),(2,13,17),(3,14,12),(4,11,6),(7,16,18),(8,10,15)], [(1,14,18),(2,6,10),(3,16,9),(4,15,13),(5,12,7),(8,17,11)], [(1,2),(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)], [(1,2),(3,13),(4,16),(5,11),(6,14),(7,17),(8,12),(9,15),(10,18)])

G:=TransitiveGroup(18,68);

On 18 points - transitive group 18T71
Generators in S18
(1 10 6)(2 13 17)(3 9 8)(4 5 7)(11 14 12)(15 16 18)
(1 3 7)(2 14 18)(4 10 9)(5 6 8)(11 16 17)(12 15 13)
(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)
(1 2)(3 14)(4 17)(5 12)(6 15)(7 18)(8 13)(9 16)(10 11)

G:=sub<Sym(18)| (1,10,6)(2,13,17)(3,9,8)(4,5,7)(11,14,12)(15,16,18), (1,3,7)(2,14,18)(4,10,9)(5,6,8)(11,16,17)(12,15,13), (3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (1,2)(3,14)(4,17)(5,12)(6,15)(7,18)(8,13)(9,16)(10,11)>;

G:=Group( (1,10,6)(2,13,17)(3,9,8)(4,5,7)(11,14,12)(15,16,18), (1,3,7)(2,14,18)(4,10,9)(5,6,8)(11,16,17)(12,15,13), (3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (1,2)(3,14)(4,17)(5,12)(6,15)(7,18)(8,13)(9,16)(10,11) );

G=PermutationGroup([(1,10,6),(2,13,17),(3,9,8),(4,5,7),(11,14,12),(15,16,18)], [(1,3,7),(2,14,18),(4,10,9),(5,6,8),(11,16,17),(12,15,13)], [(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)], [(1,2),(3,14),(4,17),(5,12),(6,15),(7,18),(8,13),(9,16),(10,11)])

G:=TransitiveGroup(18,71);

On 18 points - transitive group 18T73
Generators in S18
(1 4 8)(2 18 14)(3 16 5)(6 13 15)(7 9 12)(10 11 17)
(1 11 15)(2 5 9)(3 12 18)(4 17 6)(7 14 16)(8 10 13)
(1 2)(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)
(3 7)(4 10)(6 8)(12 14)(13 17)(16 18)

G:=sub<Sym(18)| (1,4,8)(2,18,14)(3,16,5)(6,13,15)(7,9,12)(10,11,17), (1,11,15)(2,5,9)(3,12,18)(4,17,6)(7,14,16)(8,10,13), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (3,7)(4,10)(6,8)(12,14)(13,17)(16,18)>;

G:=Group( (1,4,8)(2,18,14)(3,16,5)(6,13,15)(7,9,12)(10,11,17), (1,11,15)(2,5,9)(3,12,18)(4,17,6)(7,14,16)(8,10,13), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (3,7)(4,10)(6,8)(12,14)(13,17)(16,18) );

G=PermutationGroup([(1,4,8),(2,18,14),(3,16,5),(6,13,15),(7,9,12),(10,11,17)], [(1,11,15),(2,5,9),(3,12,18),(4,17,6),(7,14,16),(8,10,13)], [(1,2),(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)], [(3,7),(4,10),(6,8),(12,14),(13,17),(16,18)])

G:=TransitiveGroup(18,73);

On 24 points - transitive group 24T278
Generators in S24
(1 16 12)(2 9 13)(3 14 10)(5 23 19)(7 17 21)(8 18 22)
(2 9 13)(3 10 14)(4 15 11)(5 19 23)(6 24 20)(8 18 22)
(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 7)(2 6)(3 5)(4 8)(9 24)(10 19)(11 22)(12 17)(13 20)(14 23)(15 18)(16 21)

G:=sub<Sym(24)| (1,16,12)(2,9,13)(3,14,10)(5,23,19)(7,17,21)(8,18,22), (2,9,13)(3,10,14)(4,15,11)(5,19,23)(6,24,20)(8,18,22), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,7)(2,6)(3,5)(4,8)(9,24)(10,19)(11,22)(12,17)(13,20)(14,23)(15,18)(16,21)>;

G:=Group( (1,16,12)(2,9,13)(3,14,10)(5,23,19)(7,17,21)(8,18,22), (2,9,13)(3,10,14)(4,15,11)(5,19,23)(6,24,20)(8,18,22), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,7)(2,6)(3,5)(4,8)(9,24)(10,19)(11,22)(12,17)(13,20)(14,23)(15,18)(16,21) );

G=PermutationGroup([(1,16,12),(2,9,13),(3,14,10),(5,23,19),(7,17,21),(8,18,22)], [(2,9,13),(3,10,14),(4,15,11),(5,19,23),(6,24,20),(8,18,22)], [(1,2,3,4),(5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,7),(2,6),(3,5),(4,8),(9,24),(10,19),(11,22),(12,17),(13,20),(14,23),(15,18),(16,21)])

G:=TransitiveGroup(24,278);

On 24 points - transitive group 24T279
Generators in S24
(1 21 9)(3 23 11)(4 24 12)(5 13 17)(7 15 19)(8 16 20)
(1 9 21)(2 22 10)(4 24 12)(5 17 13)(6 14 18)(8 16 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)

G:=sub<Sym(24)| (1,21,9)(3,23,11)(4,24,12)(5,13,17)(7,15,19)(8,16,20), (1,9,21)(2,22,10)(4,24,12)(5,17,13)(6,14,18)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)>;

G:=Group( (1,21,9)(3,23,11)(4,24,12)(5,13,17)(7,15,19)(8,16,20), (1,9,21)(2,22,10)(4,24,12)(5,17,13)(6,14,18)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24) );

G=PermutationGroup([(1,21,9),(3,23,11),(4,24,12),(5,13,17),(7,15,19),(8,16,20)], [(1,9,21),(2,22,10),(4,24,12),(5,17,13),(6,14,18),(8,16,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24)])

G:=TransitiveGroup(24,279);

On 24 points - transitive group 24T280
Generators in S24
(1 21 13)(2 22 14)(3 15 23)(5 9 17)(6 10 18)(7 19 11)
(2 22 14)(3 23 15)(4 16 24)(6 10 18)(7 11 19)(8 20 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(9 17)(10 20)(11 23)(12 18)(13 21)(14 24)(15 19)(16 22)

G:=sub<Sym(24)| (1,21,13)(2,22,14)(3,15,23)(5,9,17)(6,10,18)(7,19,11), (2,22,14)(3,23,15)(4,16,24)(6,10,18)(7,11,19)(8,20,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,17)(10,20)(11,23)(12,18)(13,21)(14,24)(15,19)(16,22)>;

G:=Group( (1,21,13)(2,22,14)(3,15,23)(5,9,17)(6,10,18)(7,19,11), (2,22,14)(3,23,15)(4,16,24)(6,10,18)(7,11,19)(8,20,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,17)(10,20)(11,23)(12,18)(13,21)(14,24)(15,19)(16,22) );

G=PermutationGroup([(1,21,13),(2,22,14),(3,15,23),(5,9,17),(6,10,18),(7,19,11)], [(2,22,14),(3,23,15),(4,16,24),(6,10,18),(7,11,19),(8,20,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,17),(10,20),(11,23),(12,18),(13,21),(14,24),(15,19),(16,22)])

G:=TransitiveGroup(24,280);

AΓL1(𝔽9) is a maximal subgroup of   AGL2(𝔽3)  C33⋊SD16  C333SD16  F9⋊S3
AΓL1(𝔽9) is a maximal quotient of   C2.AΓL1(𝔽9)  PSU3(𝔽2)⋊C4  F9⋊C4  He3⋊SD16  C33⋊SD16  C333SD16  F9⋊S3

Polynomial with Galois group AΓL1(𝔽9) over ℚ
actionf(x)Disc(f)
9T19x9-3x8-32x7+80x6+298x5-558x4-616x3+616x2+255x-29230·514·233·892·3112
12T84x12-4x11+2x10+12x9-20x8+16x6-6x4-8x3+4x2+8x+4240·32·134·234

Matrix representation of AΓL1(𝔽9) in GL8(ℤ)

 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
,
 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1
,
 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0

G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0],[-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0] >;

AΓL1(𝔽9) in GAP, Magma, Sage, TeX

{\rm AGammaL}_1({\mathbb F}_9)
% in TeX

G:=Group("AGammaL(1,9)");
// GroupNames label

G:=SmallGroup(144,182);
// by ID

G=gap.SmallGroup(144,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,169,116,50,1444,1690,856,142,4037,1739,1169,455]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a^-1*b,c*b*c^-1=a,b*d=d*b,d*c*d=c^3>;
// generators/relations

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