non-abelian, soluble, monomial
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
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 >
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 |
►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 5 9)(3 7 11)(4 8 12)
(1 9 5)(2 6 10)(4 8 12)
(1 2 3 4)(5 6 7 8 9 10 11 12)
(2 4)(6 8)(7 11)(10 12)
G:=sub<Sym(12)| (1,5,9)(3,7,11)(4,8,12), (1,9,5)(2,6,10)(4,8,12), (1,2,3,4)(5,6,7,8,9,10,11,12), (2,4)(6,8)(7,11)(10,12)>;
G:=Group( (1,5,9)(3,7,11)(4,8,12), (1,9,5)(2,6,10)(4,8,12), (1,2,3,4)(5,6,7,8,9,10,11,12), (2,4)(6,8)(7,11)(10,12) );
G=PermutationGroup([[(1,5,9),(3,7,11),(4,8,12)], [(1,9,5),(2,6,10),(4,8,12)], [(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 16 12)(2 8 4)(3 5 18)(6 17 15)(7 14 9)(10 11 13)
(1 9 5)(2 17 13)(3 12 14)(4 6 11)(7 18 16)(8 15 10)
(1 2)(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)
(1 2)(3 15)(4 18)(5 13)(6 16)(7 11)(8 14)(9 17)(10 12)
G:=sub<Sym(18)| (1,16,12)(2,8,4)(3,5,18)(6,17,15)(7,14,9)(10,11,13), (1,9,5)(2,17,13)(3,12,14)(4,6,11)(7,18,16)(8,15,10), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (1,2)(3,15)(4,18)(5,13)(6,16)(7,11)(8,14)(9,17)(10,12)>;
G:=Group( (1,16,12)(2,8,4)(3,5,18)(6,17,15)(7,14,9)(10,11,13), (1,9,5)(2,17,13)(3,12,14)(4,6,11)(7,18,16)(8,15,10), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (1,2)(3,15)(4,18)(5,13)(6,16)(7,11)(8,14)(9,17)(10,12) );
G=PermutationGroup([[(1,16,12),(2,8,4),(3,5,18),(6,17,15),(7,14,9),(10,11,13)], [(1,9,5),(2,17,13),(3,12,14),(4,6,11),(7,18,16),(8,15,10)], [(1,2),(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)], [(1,2),(3,15),(4,18),(5,13),(6,16),(7,11),(8,14),(9,17),(10,12)]])
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 11 15)(2 4 8)(3 17 5)(6 14 16)(7 9 13)(10 12 18)
(1 5 9)(2 12 16)(3 13 11)(4 18 6)(7 15 17)(8 10 14)
(1 2)(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)
(3 7)(4 10)(6 8)(11 17)(13 15)(14 18)
G:=sub<Sym(18)| (1,11,15)(2,4,8)(3,17,5)(6,14,16)(7,9,13)(10,12,18), (1,5,9)(2,12,16)(3,13,11)(4,18,6)(7,15,17)(8,10,14), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (3,7)(4,10)(6,8)(11,17)(13,15)(14,18)>;
G:=Group( (1,11,15)(2,4,8)(3,17,5)(6,14,16)(7,9,13)(10,12,18), (1,5,9)(2,12,16)(3,13,11)(4,18,6)(7,15,17)(8,10,14), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (3,7)(4,10)(6,8)(11,17)(13,15)(14,18) );
G=PermutationGroup([[(1,11,15),(2,4,8),(3,17,5),(6,14,16),(7,9,13),(10,12,18)], [(1,5,9),(2,12,16),(3,13,11),(4,18,6),(7,15,17),(8,10,14)], [(1,2),(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)], [(3,7),(4,10),(6,8),(11,17),(13,15),(14,18)]])
G:=TransitiveGroup(18,73);
►On 24 points - transitive group 24T278
Generators in S24
(1 12 16)(2 13 9)(3 10 14)(5 17 21)(7 19 23)(8 20 24)
(2 13 9)(3 14 10)(4 11 15)(5 21 17)(6 18 22)(8 20 24)
(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 22)(10 17)(11 20)(12 23)(13 18)(14 21)(15 24)(16 19)
G:=sub<Sym(24)| (1,12,16)(2,13,9)(3,10,14)(5,17,21)(7,19,23)(8,20,24), (2,13,9)(3,14,10)(4,11,15)(5,21,17)(6,18,22)(8,20,24), (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)(9,22)(10,17)(11,20)(12,23)(13,18)(14,21)(15,24)(16,19)>;
G:=Group( (1,12,16)(2,13,9)(3,10,14)(5,17,21)(7,19,23)(8,20,24), (2,13,9)(3,14,10)(4,11,15)(5,21,17)(6,18,22)(8,20,24), (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)(9,22)(10,17)(11,20)(12,23)(13,18)(14,21)(15,24)(16,19) );
G=PermutationGroup([[(1,12,16),(2,13,9),(3,10,14),(5,17,21),(7,19,23),(8,20,24)], [(2,13,9),(3,14,10),(4,11,15),(5,21,17),(6,18,22),(8,20,24)], [(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),(9,22),(10,17),(11,20),(12,23),(13,18),(14,21),(15,24),(16,19)]])
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 9)(2 22 10)(3 11 23)(5 13 17)(6 14 18)(7 19 15)
(2 22 10)(3 23 11)(4 12 24)(6 14 18)(7 15 19)(8 20 16)
(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 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)
G:=sub<Sym(24)| (1,21,9)(2,22,10)(3,11,23)(5,13,17)(6,14,18)(7,19,15), (2,22,10)(3,23,11)(4,12,24)(6,14,18)(7,15,19)(8,20,16), (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)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18)>;
G:=Group( (1,21,9)(2,22,10)(3,11,23)(5,13,17)(6,14,18)(7,19,15), (2,22,10)(3,23,11)(4,12,24)(6,14,18)(7,15,19)(8,20,16), (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)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18) );
G=PermutationGroup([[(1,21,9),(2,22,10),(3,11,23),(5,13,17),(6,14,18),(7,19,15)], [(2,22,10),(3,23,11),(4,12,24),(6,14,18),(7,15,19),(8,20,16)], [(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),(9,21),(10,24),(11,19),(12,22),(13,17),(14,20),(15,23),(16,18)]])
G:=TransitiveGroup(24,280);
AΓL1(𝔽9) is a maximal subgroup of
AGL2(𝔽3) C33⋊SD16 C33⋊3SD16 F9⋊S3
AΓL1(𝔽9) is a maximal quotient of C2.AΓL1(𝔽9) PSU3(𝔽2)⋊C4 F9⋊C4 He3⋊SD16 C33⋊SD16 C33⋊3SD16 F9⋊S3
| action | f(x) | Disc(f) |
|---|---|---|
| 9T19 | x9-3x8-32x7+80x6+298x5-558x4-616x3+616x2+255x-29 | 230·514·233·892·3112 |
| 12T84 | x12-4x11+2x10+12x9-20x8+16x6-6x4-8x3+4x2+8x+4 | 240·32·134·234 |
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
Export
Subgroup lattice of AΓL1(𝔽9) in TeX
Character table of AΓL1(𝔽9) in TeX