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G = PGL2(𝔽7)  order 336 = 24·3·7

Projective linear group on 𝔽72

Aliases: PGL2(𝔽7), SO3(𝔽7), PSO3(𝔽7), PO3(𝔽7), PU2(𝔽7), GL3(𝔽2)⋊C2, PGL(2,7), Aut(GL3(𝔽2)), SmallGroup(336,208)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — GL3(𝔽2) — PGL2(𝔽7)
 Derived series GL3(𝔽2) — PGL2(𝔽7)
 Lower central GL3(𝔽2) — PGL2(𝔽7)
 Upper central C1

21C2
28C2
28C3
8C7
14C22
21C4
42C22
28S3
28S3
28C6
8D7
21C8
21D4
21D4
14A4
28D6
8F7
21D8
14S4

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

Permutation representations of PGL2(𝔽7)
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 10 8 14 6 4 12)(2 11 5 3 9 7 13)

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

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

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

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 3 11 15 5 7 13)(2 6 14 16 4 10 12)

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

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

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

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 9 19 17 20 18 11)(2 14 15 3 8 6 12)(4 10 5 21 7 13 16)

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

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

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

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 9 19 17 21 23 15)(2 14 28 22 20 18 12)(3 13 7 5 25 27 11)(4 10 24 26 8 6 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), (1,9,19,17,21,23,15)(2,14,28,22,20,18,12)(3,13,7,5,25,27,11)(4,10,24,26,8,6,16)>;

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), (1,9,19,17,21,23,15)(2,14,28,22,20,18,12)(3,13,7,5,25,27,11)(4,10,24,26,8,6,16) );

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)], [(1,9,19,17,21,23,15),(2,14,28,22,20,18,12),(3,13,7,5,25,27,11),(4,10,24,26,8,6,16)])

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 14 7 15 8 16 3)(2 27 9 20 18 6 26)(4 24 17 5 10 13 21)(11 25 23 19 22 28 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), (1,14,7,15,8,16,3)(2,27,9,20,18,6,26)(4,24,17,5,10,13,21)(11,25,23,19,22,28,12)>;

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), (1,14,7,15,8,16,3)(2,27,9,20,18,6,26)(4,24,17,5,10,13,21)(11,25,23,19,22,28,12) );

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)], [(1,14,7,15,8,16,3),(2,27,9,20,18,6,26),(4,24,17,5,10,13,21),(11,25,23,19,22,28,12)])

G:=TransitiveGroup(28,46);

Polynomial with Galois group PGL2(𝔽7) over ℚ
actionf(x)Disc(f)
8T43x8-4x7+21x4-18x+9-28·312·77
14T16x14-504x12-364x11+106400x10+102858x9-12009193x8-9963017x7+768587344x6+299534410x5-26524109654x4+7649980758x3+334176753379x2-351070982458x+9308484568154·747·132·192·372·432·1077·1932·570892·35329732·136218232

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

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