non-abelian, almost simple, not soluble
Aliases: PGL2(𝔽7), SO3(𝔽7), PSO3(𝔽7), PO3(𝔽7), PU2(𝔽7), GL3(𝔽2)⋊C2, PGL(2,7), Aut(GL3(𝔽2)), SmallGroup(336,208)
Series: Chief►Derived ►Lower central ►Upper central
| GL3(𝔽2) — PGL2(𝔽7) |
| GL3(𝔽2) — PGL2(𝔽7) |
Character table of PGL2(𝔽7)
| class | 1 | 2A | 2B | 3 | 4 | 6 | 7 | 8A | 8B | |
| size | 1 | 21 | 28 | 56 | 42 | 56 | 48 | 42 | 42 | |
| 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 | 6 | -2 | 0 | 0 | 2 | 0 | -1 | 0 | 0 | orthogonal faithful |
| ρ4 | 6 | 2 | 0 | 0 | 0 | 0 | -1 | -√2 | √2 | orthogonal faithful |
| ρ5 | 6 | 2 | 0 | 0 | 0 | 0 | -1 | √2 | -√2 | orthogonal faithful |
| ρ6 | 7 | -1 | 1 | 1 | -1 | 1 | 0 | -1 | -1 | orthogonal faithful |
| ρ7 | 7 | -1 | -1 | 1 | -1 | -1 | 0 | 1 | 1 | orthogonal faithful |
| ρ8 | 8 | 0 | -2 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal faithful |
| ρ9 | 8 | 0 | 2 | -1 | 0 | -1 | 1 | 0 | 0 | orthogonal faithful |
►On 8 points: primitive, sharply triply transitive - transitive group 8T43
Generators in S8
(1 2 3 4 5 6 7 8)
(1 8 2 7 4 6 5)
G:=sub<Sym(8)| (1,2,3,4,5,6,7,8), (1,8,2,7,4,6,5)>;
G:=Group( (1,2,3,4,5,6,7,8), (1,8,2,7,4,6,5) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8)], [(1,8,2,7,4,6,5)]])
G:=TransitiveGroup(8,43);
►On 14 points - transitive group 14T16
Generators in S14
(1 2)(3 4 5 6)(7 8 9 10 11 12 13 14)
(1 14 4 6 12 10 8)(2 13 11 9 5 3 7)
G:=sub<Sym(14)| (1,2)(3,4,5,6)(7,8,9,10,11,12,13,14), (1,14,4,6,12,10,8)(2,13,11,9,5,3,7)>;
G:=Group( (1,2)(3,4,5,6)(7,8,9,10,11,12,13,14), (1,14,4,6,12,10,8)(2,13,11,9,5,3,7) );
G=PermutationGroup([[(1,2),(3,4,5,6),(7,8,9,10,11,12,13,14)], [(1,14,4,6,12,10,8),(2,13,11,9,5,3,7)]])
G:=TransitiveGroup(14,16);
►On 16 points - transitive group 16T713
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13 9 3 5 15 7)(2 12 14 8 4 16 10)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,9,3,5,15,7)(2,12,14,8,4,16,10)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,9,3,5,15,7)(2,12,14,8,4,16,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13,9,3,5,15,7),(2,12,14,8,4,16,10)]])
G:=TransitiveGroup(16,713);
►On 21 points: primitive - transitive group 21T20
Generators in S21
(2 3 4 5)(6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21)
(1 18 11 9 12 10 20)(2 17 15 21 5 6 7)(3 19 4 13 16 14 8)
G:=sub<Sym(21)| (2,3,4,5)(6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21), (1,18,11,9,12,10,20)(2,17,15,21,5,6,7)(3,19,4,13,16,14,8)>;
G:=Group( (2,3,4,5)(6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21), (1,18,11,9,12,10,20)(2,17,15,21,5,6,7)(3,19,4,13,16,14,8) );
G=PermutationGroup([[(2,3,4,5),(6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21)], [(1,18,11,9,12,10,20),(2,17,15,21,5,6,7),(3,19,4,13,16,14,8)]])
G:=TransitiveGroup(21,20);
►On 24 points - transitive group 24T707
Generators in S24
(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 19 2 15 4 17 5)(6 13 9 8 10 18 12)(7 23 14 24 20 16 21)
G:=sub<Sym(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), (1,19,2,15,4,17,5)(6,13,9,8,10,18,12)(7,23,14,24,20,16,21)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,19,2,15,4,17,5)(6,13,9,8,10,18,12)(7,23,14,24,20,16,21) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,19,2,15,4,17,5),(6,13,9,8,10,18,12),(7,23,14,24,20,16,21)]])
G:=TransitiveGroup(24,707);
►On 28 points - transitive group 28T42
Generators in S28
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28)
(1 24 7 5 17 19 11)(2 10 16 18 8 6 27)(3 9 22 28 13 15 26)(4 25 20 14 23 21 12)
G:=sub<Sym(28)| (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28), (1,24,7,5,17,19,11)(2,10,16,18,8,6,27)(3,9,22,28,13,15,26)(4,25,20,14,23,21,12)>;
G:=Group( (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28), (1,24,7,5,17,19,11)(2,10,16,18,8,6,27)(3,9,22,28,13,15,26)(4,25,20,14,23,21,12) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28)], [(1,24,7,5,17,19,11),(2,10,16,18,8,6,27),(3,9,22,28,13,15,26),(4,25,20,14,23,21,12)]])
G:=TransitiveGroup(28,42);
►On 28 points: primitive - transitive group 28T46
Generators in S28
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28)
(1 20 23 10 8 28 19)(2 4 12 21 5 22 6)(3 17 7 27 24 11 14)(9 15 13 26 25 18 16)
G:=sub<Sym(28)| (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28), (1,20,23,10,8,28,19)(2,4,12,21,5,22,6)(3,17,7,27,24,11,14)(9,15,13,26,25,18,16)>;
G:=Group( (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28), (1,20,23,10,8,28,19)(2,4,12,21,5,22,6)(3,17,7,27,24,11,14)(9,15,13,26,25,18,16) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28)], [(1,20,23,10,8,28,19),(2,4,12,21,5,22,6),(3,17,7,27,24,11,14),(9,15,13,26,25,18,16)]])
G:=TransitiveGroup(28,46);
Polynomial with Galois group PGL2(𝔽7) over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 8T43 | x8-4x7+21x4-18x+9 | -28·312·77 |
| 14T16 | x14-504x12-364x11+106400x10+102858x9-12009193x8-9963017x7+768587344x6+299534410x5-26524109654x4+7649980758x3+334176753379x2-351070982458x+93084845681 | 54·747·132·192·372·432·1077·1932·570892·35329732·136218232 |
Matrix representation of PGL2(𝔽7) ►in GL3(𝔽7) generated by
| 4 | 4 | 6 |
| 5 | 4 | 4 |
| 4 | 5 | 4 |
| 3 | 1 | 3 |
| 1 | 6 | 3 |
| 6 | 4 | 1 |
G:=sub<GL(3,GF(7))| [4,5,4,4,4,5,6,4,4],[3,1,6,1,6,4,3,3,1] >;
PGL2(𝔽7) in GAP, Magma, Sage, TeX
{\rm PGL}_2({\mathbb F}_7) % in TeX
G:=Group("PGL(2,7)"); // GroupNames label
G:=SmallGroup(336,208);
// by ID
G=gap.SmallGroup(336,208);
# by ID
Export
Subgroup lattice of PGL2(𝔽7) in TeX
Character table of PGL2(𝔽7) in TeX