Aliases: AGL2(𝔽3), PSU3(𝔽2)⋊S3, ASL2(𝔽3)⋊C2, C32⋊GL2(𝔽3), C3⋊S3.S4, Aut(C3⋊S3), Hol(C32), SmallGroup(432,734)
Series: Derived ►Chief ►Lower central ►Upper central
| ASL2(𝔽3) — AGL2(𝔽3) |
Generators and relations for AGL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, dbd-1=ebe-1=ab=ba, cac-1=b-1, dad-1=a-1b, ae=ea, af=fa, cbc-1=a, fbf=a-1b-1, dcd-1=ede-1=c-1, ece-1=c-1d, fcf=cd, fdf=c2d, fef=e-1 >
9C2
36C2
4C3
12C3
24C3
27C4
54C22
12S3
12S3
12S3
24S3
36S3
36C6
36C6
4C32
8C32
9Q8
27C8
27D4
36D6
36D6
12C3×S3
12C3×S3
12C3×S3
24C3×S3
4He3
27SD16
6S32
12S32
3F9
Character table of AGL2(𝔽3)
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4 | 6A | 6B | 8A | 8B | |
| size | 1 | 9 | 36 | 8 | 24 | 48 | 54 | 72 | 72 | 54 | 54 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | -2 | 0 | 2 | -1 | -1 | 0 | 1 | 0 | -√-2 | √-2 | complex lifted from GL2(𝔽3) |
| ρ5 | 2 | -2 | 0 | 2 | -1 | -1 | 0 | 1 | 0 | √-2 | -√-2 | complex lifted from GL2(𝔽3) |
| ρ6 | 3 | 3 | 1 | 3 | 0 | 0 | -1 | 0 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | -1 | 3 | 0 | 0 | -1 | 0 | -1 | 1 | 1 | orthogonal lifted from S4 |
| ρ8 | 4 | -4 | 0 | 4 | 1 | 1 | 0 | -1 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ9 | 8 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | -1 | 0 | 0 | orthogonal faithful |
| ρ10 | 8 | 0 | -2 | -1 | 2 | -1 | 0 | 0 | 1 | 0 | 0 | orthogonal faithful |
| ρ11 | 16 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 9 points: primitive, doubly transitive - transitive group 9T26
Generators in S9
(1 9 7)(2 6 5)(3 8 4)
(1 6 8)(2 3 7)(4 9 5)
(2 3 4 5)(6 7 8 9)
(2 9 4 7)(3 8 5 6)
(2 5 6)(3 8 4)
(2 4)(3 6)(5 8)
G:=sub<Sym(9)| (1,9,7)(2,6,5)(3,8,4), (1,6,8)(2,3,7)(4,9,5), (2,3,4,5)(6,7,8,9), (2,9,4,7)(3,8,5,6), (2,5,6)(3,8,4), (2,4)(3,6)(5,8)>;
G:=Group( (1,9,7)(2,6,5)(3,8,4), (1,6,8)(2,3,7)(4,9,5), (2,3,4,5)(6,7,8,9), (2,9,4,7)(3,8,5,6), (2,5,6)(3,8,4), (2,4)(3,6)(5,8) );
G=PermutationGroup([[(1,9,7),(2,6,5),(3,8,4)], [(1,6,8),(2,3,7),(4,9,5)], [(2,3,4,5),(6,7,8,9)], [(2,9,4,7),(3,8,5,6)], [(2,5,6),(3,8,4)], [(2,4),(3,6),(5,8)]])
G:=TransitiveGroup(9,26);
►On 12 points - transitive group 12T157
Generators in S12
(1 6 8)(2 5 7)(4 11 9)
(1 6 8)(2 7 5)(3 12 10)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 11 7 9)(6 10 8 12)
(1 2 4)(5 11 6)(7 9 8)
(2 4)(5 11)(7 9)(10 12)
G:=sub<Sym(12)| (1,6,8)(2,5,7)(4,11,9), (1,6,8)(2,7,5)(3,12,10), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,11,7,9)(6,10,8,12), (1,2,4)(5,11,6)(7,9,8), (2,4)(5,11)(7,9)(10,12)>;
G:=Group( (1,6,8)(2,5,7)(4,11,9), (1,6,8)(2,7,5)(3,12,10), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,11,7,9)(6,10,8,12), (1,2,4)(5,11,6)(7,9,8), (2,4)(5,11)(7,9)(10,12) );
G=PermutationGroup([[(1,6,8),(2,5,7),(4,11,9)], [(1,6,8),(2,7,5),(3,12,10)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(5,11,7,9),(6,10,8,12)], [(1,2,4),(5,11,6),(7,9,8)], [(2,4),(5,11),(7,9),(10,12)]])
G:=TransitiveGroup(12,157);
►On 18 points - transitive group 18T157
Generators in S18
(1 16 18)(2 9 7)(3 15 4)(5 6 17)(8 13 12)(10 14 11)
(1 17 15)(2 10 8)(3 18 6)(4 16 5)(7 11 12)(9 14 13)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(3 15 5 17)(4 18 6 16)(7 11 9 13)(8 14 10 12)
(3 15 4)(5 17 6)(8 13 12)(10 11 14)
(1 2)(3 10)(4 11)(5 8)(6 13)(7 18)(9 16)(12 17)(14 15)
G:=sub<Sym(18)| (1,16,18)(2,9,7)(3,15,4)(5,6,17)(8,13,12)(10,14,11), (1,17,15)(2,10,8)(3,18,6)(4,16,5)(7,11,12)(9,14,13), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (3,15,5,17)(4,18,6,16)(7,11,9,13)(8,14,10,12), (3,15,4)(5,17,6)(8,13,12)(10,11,14), (1,2)(3,10)(4,11)(5,8)(6,13)(7,18)(9,16)(12,17)(14,15)>;
G:=Group( (1,16,18)(2,9,7)(3,15,4)(5,6,17)(8,13,12)(10,14,11), (1,17,15)(2,10,8)(3,18,6)(4,16,5)(7,11,12)(9,14,13), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (3,15,5,17)(4,18,6,16)(7,11,9,13)(8,14,10,12), (3,15,4)(5,17,6)(8,13,12)(10,11,14), (1,2)(3,10)(4,11)(5,8)(6,13)(7,18)(9,16)(12,17)(14,15) );
G=PermutationGroup([[(1,16,18),(2,9,7),(3,15,4),(5,6,17),(8,13,12),(10,14,11)], [(1,17,15),(2,10,8),(3,18,6),(4,16,5),(7,11,12),(9,14,13)], [(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(3,15,5,17),(4,18,6,16),(7,11,9,13),(8,14,10,12)], [(3,15,4),(5,17,6),(8,13,12),(10,11,14)], [(1,2),(3,10),(4,11),(5,8),(6,13),(7,18),(9,16),(12,17),(14,15)]])
G:=TransitiveGroup(18,157);
►On 24 points - transitive group 24T1325
Generators in S24
(1 14 16)(2 13 15)(3 24 22)(4 21 23)(5 11 9)(8 20 18)
(1 14 16)(2 15 13)(3 22 24)(4 21 23)(6 12 10)(7 17 19)
(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 6)(2 5)(3 8)(4 7)(9 13 11 15)(10 16 12 14)(17 21 19 23)(18 24 20 22)
(1 2 5)(3 8 4)(9 16 15)(11 14 13)(18 23 22)(20 21 24)
(1 4)(2 8)(3 5)(6 7)(9 22)(10 17)(11 24)(12 19)(13 20)(14 21)(15 18)(16 23)
G:=sub<Sym(24)| (1,14,16)(2,13,15)(3,24,22)(4,21,23)(5,11,9)(8,20,18), (1,14,16)(2,15,13)(3,22,24)(4,21,23)(6,12,10)(7,17,19), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,5)(3,8)(4,7)(9,13,11,15)(10,16,12,14)(17,21,19,23)(18,24,20,22), (1,2,5)(3,8,4)(9,16,15)(11,14,13)(18,23,22)(20,21,24), (1,4)(2,8)(3,5)(6,7)(9,22)(10,17)(11,24)(12,19)(13,20)(14,21)(15,18)(16,23)>;
G:=Group( (1,14,16)(2,13,15)(3,24,22)(4,21,23)(5,11,9)(8,20,18), (1,14,16)(2,15,13)(3,22,24)(4,21,23)(6,12,10)(7,17,19), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,5)(3,8)(4,7)(9,13,11,15)(10,16,12,14)(17,21,19,23)(18,24,20,22), (1,2,5)(3,8,4)(9,16,15)(11,14,13)(18,23,22)(20,21,24), (1,4)(2,8)(3,5)(6,7)(9,22)(10,17)(11,24)(12,19)(13,20)(14,21)(15,18)(16,23) );
G=PermutationGroup([[(1,14,16),(2,13,15),(3,24,22),(4,21,23),(5,11,9),(8,20,18)], [(1,14,16),(2,15,13),(3,22,24),(4,21,23),(6,12,10),(7,17,19)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6),(2,5),(3,8),(4,7),(9,13,11,15),(10,16,12,14),(17,21,19,23),(18,24,20,22)], [(1,2,5),(3,8,4),(9,16,15),(11,14,13),(18,23,22),(20,21,24)], [(1,4),(2,8),(3,5),(6,7),(9,22),(10,17),(11,24),(12,19),(13,20),(14,21),(15,18),(16,23)]])
G:=TransitiveGroup(24,1325);
►On 24 points - transitive group 24T1326
Generators in S24
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(8 9 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(2 22 21)(4 24 23)(5 20 6)(7 18 8)(9 12 13)(10 15 11)
(1 3)(2 21)(4 23)(5 20)(7 18)(10 15)(12 13)(14 16)(17 19)
G:=sub<Sym(24)| (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (2,22,21)(4,24,23)(5,20,6)(7,18,8)(9,12,13)(10,15,11), (1,3)(2,21)(4,23)(5,20)(7,18)(10,15)(12,13)(14,16)(17,19)>;
G:=Group( (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (2,22,21)(4,24,23)(5,20,6)(7,18,8)(9,12,13)(10,15,11), (1,3)(2,21)(4,23)(5,20)(7,18)(10,15)(12,13)(14,16)(17,19) );
G=PermutationGroup([[(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,19,14),(3,16,17),(5,10,21),(6,22,11),(7,23,12),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)], [(2,22,21),(4,24,23),(5,20,6),(7,18,8),(9,12,13),(10,15,11)], [(1,3),(2,21),(4,23),(5,20),(7,18),(10,15),(12,13),(14,16),(17,19)]])
G:=TransitiveGroup(24,1326);
►On 24 points - transitive group 24T1327
Generators in S24
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(8 9 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(2 22 21)(4 24 23)(5 20 6)(7 18 8)(9 12 13)(10 15 11)
(2 23)(4 21)(5 13)(6 9)(7 15)(8 11)(10 18)(12 20)(14 19)(16 17)(22 24)
G:=sub<Sym(24)| (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (2,22,21)(4,24,23)(5,20,6)(7,18,8)(9,12,13)(10,15,11), (2,23)(4,21)(5,13)(6,9)(7,15)(8,11)(10,18)(12,20)(14,19)(16,17)(22,24)>;
G:=Group( (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (2,22,21)(4,24,23)(5,20,6)(7,18,8)(9,12,13)(10,15,11), (2,23)(4,21)(5,13)(6,9)(7,15)(8,11)(10,18)(12,20)(14,19)(16,17)(22,24) );
G=PermutationGroup([[(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,19,14),(3,16,17),(5,10,21),(6,22,11),(7,23,12),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)], [(2,22,21),(4,24,23),(5,20,6),(7,18,8),(9,12,13),(10,15,11)], [(2,23),(4,21),(5,13),(6,9),(7,15),(8,11),(10,18),(12,20),(14,19),(16,17),(22,24)]])
G:=TransitiveGroup(24,1327);
►On 24 points - transitive group 24T1334
Generators in S24
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(8 9 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(1 14 19)(2 11 5)(3 16 17)(4 9 7)(6 21 15)(8 23 13)(10 20 22)(12 18 24)
(2 23)(4 21)(5 13)(6 9)(7 15)(8 11)(10 18)(12 20)(14 19)(16 17)(22 24)
G:=sub<Sym(24)| (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (1,14,19)(2,11,5)(3,16,17)(4,9,7)(6,21,15)(8,23,13)(10,20,22)(12,18,24), (2,23)(4,21)(5,13)(6,9)(7,15)(8,11)(10,18)(12,20)(14,19)(16,17)(22,24)>;
G:=Group( (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (1,14,19)(2,11,5)(3,16,17)(4,9,7)(6,21,15)(8,23,13)(10,20,22)(12,18,24), (2,23)(4,21)(5,13)(6,9)(7,15)(8,11)(10,18)(12,20)(14,19)(16,17)(22,24) );
G=PermutationGroup([[(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,19,14),(3,16,17),(5,10,21),(6,22,11),(7,23,12),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)], [(1,14,19),(2,11,5),(3,16,17),(4,9,7),(6,21,15),(8,23,13),(10,20,22),(12,18,24)], [(2,23),(4,21),(5,13),(6,9),(7,15),(8,11),(10,18),(12,20),(14,19),(16,17),(22,24)]])
G:=TransitiveGroup(24,1334);
►On 27 points - transitive group 27T139
Generators in S27
(1 5 7)(2 16 18)(3 11 9)(4 22 21)(6 23 20)(8 24 25)(10 27 26)(12 19 13)(14 15 17)
(1 6 4)(2 17 19)(3 8 10)(5 23 22)(7 20 21)(9 25 26)(11 24 27)(12 18 15)(13 16 14)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 23 6 21)(5 22 7 20)(8 26 10 24)(9 25 11 27)(12 19 14 17)(13 18 15 16)
(1 2 3)(4 13 26)(5 16 11)(6 15 24)(7 18 9)(8 20 14)(10 22 12)(17 25 23)(19 27 21)
(2 3)(4 23)(6 21)(8 12)(9 18)(10 14)(11 16)(13 25)(15 27)(17 26)(19 24)(20 22)
G:=sub<Sym(27)| (1,5,7)(2,16,18)(3,11,9)(4,22,21)(6,23,20)(8,24,25)(10,27,26)(12,19,13)(14,15,17), (1,6,4)(2,17,19)(3,8,10)(5,23,22)(7,20,21)(9,25,26)(11,24,27)(12,18,15)(13,16,14), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,23,6,21)(5,22,7,20)(8,26,10,24)(9,25,11,27)(12,19,14,17)(13,18,15,16), (1,2,3)(4,13,26)(5,16,11)(6,15,24)(7,18,9)(8,20,14)(10,22,12)(17,25,23)(19,27,21), (2,3)(4,23)(6,21)(8,12)(9,18)(10,14)(11,16)(13,25)(15,27)(17,26)(19,24)(20,22)>;
G:=Group( (1,5,7)(2,16,18)(3,11,9)(4,22,21)(6,23,20)(8,24,25)(10,27,26)(12,19,13)(14,15,17), (1,6,4)(2,17,19)(3,8,10)(5,23,22)(7,20,21)(9,25,26)(11,24,27)(12,18,15)(13,16,14), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,23,6,21)(5,22,7,20)(8,26,10,24)(9,25,11,27)(12,19,14,17)(13,18,15,16), (1,2,3)(4,13,26)(5,16,11)(6,15,24)(7,18,9)(8,20,14)(10,22,12)(17,25,23)(19,27,21), (2,3)(4,23)(6,21)(8,12)(9,18)(10,14)(11,16)(13,25)(15,27)(17,26)(19,24)(20,22) );
G=PermutationGroup([[(1,5,7),(2,16,18),(3,11,9),(4,22,21),(6,23,20),(8,24,25),(10,27,26),(12,19,13),(14,15,17)], [(1,6,4),(2,17,19),(3,8,10),(5,23,22),(7,20,21),(9,25,26),(11,24,27),(12,18,15),(13,16,14)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,23,6,21),(5,22,7,20),(8,26,10,24),(9,25,11,27),(12,19,14,17),(13,18,15,16)], [(1,2,3),(4,13,26),(5,16,11),(6,15,24),(7,18,9),(8,20,14),(10,22,12),(17,25,23),(19,27,21)], [(2,3),(4,23),(6,21),(8,12),(9,18),(10,14),(11,16),(13,25),(15,27),(17,26),(19,24),(20,22)]])
G:=TransitiveGroup(27,139);
Polynomial with Galois group AGL2(𝔽3) over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 9T26 | x9-60x7-30x6+441x5+90x4-1116x3+180x2+972x-480 | 220·315·54·114·134·7092 |
| 12T157 | x12-x11-53x10+53x9+847x8-482x7-5571x6+1485x5+16143x4-1208x3-17855x2-864x+2389 | 212·116·1392·27774·629212·108496732 |
Matrix representation of AGL2(𝔽3) ►in GL8(ℤ)
| -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 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 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())| [-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],[0,-1,0,0,0,1,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0],[1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1],[1,-1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,-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,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0],[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] >;
AGL2(𝔽3) in GAP, Magma, Sage, TeX
{\rm AGL}_2({\mathbb F}_3) % in TeX
G:=Group("AGL(2,3)"); // GroupNames label
G:=SmallGroup(432,734);
// by ID
G=gap.SmallGroup(432,734);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,-3,3,57,632,387,100,1179,262,185,80,14117,6060,1699,1034,201,14118,8245,1588,223,622]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=e^3=f^2=1,d^2=c^2,d*b*d^-1=e*b*e^-1=a*b=b*a,c*a*c^-1=b^-1,d*a*d^-1=a^-1*b,a*e=e*a,a*f=f*a,c*b*c^-1=a,f*b*f=a^-1*b^-1,d*c*d^-1=e*d*e^-1=c^-1,e*c*e^-1=c^-1*d,f*c*f=c*d,f*d*f=c^2*d,f*e*f=e^-1>;
// generators/relations
Export
Subgroup lattice of AGL2(𝔽3) in TeX
Character table of AGL2(𝔽3) in TeX