direct product, non-abelian, not soluble
Aliases: C2×GL3(𝔽2), SmallGroup(336,209)
Series: Chief►Derived ►Lower central ►Upper central
| GL3(𝔽2) — C2×GL3(𝔽2) |
| GL3(𝔽2) — C2×GL3(𝔽2) |
21C2
21C2
28C3
8C7
7C22
7C22
21C22
21C22
21C4
21C4
21C22
28S3
28C6
28S3
8C14
7C23
7C23
21D4
21C2×C4
21D4
21D4
21D4
7A4
7A4
28D6
21C2×D4
7S4
7S4
7S4
7S4
Character table of C2×GL3(𝔽2)
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 6 | 7A | 7B | 14A | 14B | |
| size | 1 | 1 | 21 | 21 | 56 | 42 | 42 | 56 | 24 | 24 | 24 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 3 | -3 | -1 | 1 | 0 | -1 | 1 | 0 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1-√-7/2 | complex faithful |
| ρ4 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from GL3(𝔽2) |
| ρ5 | 3 | -3 | -1 | 1 | 0 | -1 | 1 | 0 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1+√-7/2 | complex faithful |
| ρ6 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from GL3(𝔽2) |
| ρ7 | 6 | 6 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from GL3(𝔽2) |
| ρ8 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal faithful |
| ρ9 | 7 | 7 | -1 | -1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from GL3(𝔽2) |
| ρ10 | 7 | -7 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ11 | 8 | -8 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | orthogonal faithful |
| ρ12 | 8 | 8 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from GL3(𝔽2) |
►On 14 points - transitive group 14T17
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 7 9 12 11 5)(2 4 14 13 8 3)(6 10)
G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,7,9,12,11,5)(2,4,14,13,8,3)(6,10)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,7,9,12,11,5)(2,4,14,13,8,3)(6,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,7,9,12,11,5),(2,4,14,13,8,3),(6,10)]])
G:=TransitiveGroup(14,17);
►On 14 points - transitive group 14T19
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 13 5 14 7 11)(2 10 3 8 4 9)(6 12)
G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,13,5,14,7,11)(2,10,3,8,4,9)(6,12)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,13,5,14,7,11)(2,10,3,8,4,9)(6,12) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,13,5,14,7,11),(2,10,3,8,4,9),(6,12)]])
G:=TransitiveGroup(14,19);
►On 16 points - transitive group 16T714
Generators in S16
(3 4 5 6 7 8 9)(10 11 12 13 14 15 16)
(1 14 7 2 6 15)(3 16 5 11 8 13)(4 12)(9 10)
G:=sub<Sym(16)| (3,4,5,6,7,8,9)(10,11,12,13,14,15,16), (1,14,7,2,6,15)(3,16,5,11,8,13)(4,12)(9,10)>;
G:=Group( (3,4,5,6,7,8,9)(10,11,12,13,14,15,16), (1,14,7,2,6,15)(3,16,5,11,8,13)(4,12)(9,10) );
G=PermutationGroup([[(3,4,5,6,7,8,9),(10,11,12,13,14,15,16)], [(1,14,7,2,6,15),(3,16,5,11,8,13),(4,12),(9,10)]])
G:=TransitiveGroup(16,714);
►On 28 points - transitive group 28T43
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 16 24 9 26 21)(2 20 22 10 23 19)(3 18 4 11 28 12)(5 17 7 13 27 8)(6 14)(15 25)
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), (1,16,24,9,26,21)(2,20,22,10,23,19)(3,18,4,11,28,12)(5,17,7,13,27,8)(6,14)(15,25)>;
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), (1,16,24,9,26,21)(2,20,22,10,23,19)(3,18,4,11,28,12)(5,17,7,13,27,8)(6,14)(15,25) );
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)], [(1,16,24,9,26,21),(2,20,22,10,23,19),(3,18,4,11,28,12),(5,17,7,13,27,8),(6,14),(15,25)]])
G:=TransitiveGroup(28,43);
Polynomial with Galois group C2×GL3(𝔽2) over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 14T17 | x14-18x12-964x10+32592x8-353912x6+1736792x4-3987152x2+3495368 | -253·66110·181755260737814 |
| 14T19 | x14-17x12+114x10-386x8+699x6-650x4+257x2-17 | 214·179·4914 |
Matrix representation of C2×GL3(𝔽2) ►in GL3(𝔽7) generated by
| 4 | 1 | 4 |
| 4 | 0 | 3 |
| 6 | 0 | 6 |
| 5 | 3 | 6 |
| 2 | 6 | 6 |
| 0 | 6 | 3 |
G:=sub<GL(3,GF(7))| [4,4,6,1,0,0,4,3,6],[5,2,0,3,6,6,6,6,3] >;
C2×GL3(𝔽2) in GAP, Magma, Sage, TeX
C_2\times {\rm GL}_3({\mathbb F}_2) % in TeX
G:=Group("C2xGL(3,2)"); // GroupNames label
G:=SmallGroup(336,209);
// by ID
G=gap.SmallGroup(336,209);
# by ID
Export
Subgroup lattice of C2×GL3(𝔽2) in TeX
Character table of C2×GL3(𝔽2) in TeX