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## G = C2×GL3(𝔽2)  order 336 = 24·3·7

### Direct product of C2 and GL3(𝔽2)

Aliases: C2×GL3(𝔽2), SmallGroup(336,209)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C2×GL3(𝔽2)
 Derived series GL3(𝔽2) — C2×GL3(𝔽2)
 Lower central GL3(𝔽2) — C2×GL3(𝔽2)
 Upper central C1 — C2

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)

Permutation representations of C2×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 11 10 6)(2 5 13 12 8 3)(4 14)

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

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

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

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 12 6 13 7 11)(2 10 3 14 5 8)(4 9)

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

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

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

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 8 15 2 14 9)(3 13 5 16 7 11)(4 10)(6 12)

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

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

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

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

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

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

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

G:=TransitiveGroup(28,43);

Polynomial with Galois group C2×GL3(𝔽2) over ℚ
actionf(x)Disc(f)
14T17x14-18x12-964x10+32592x8-353912x6+1736792x4-3987152x2+3495368-253·66110·181755260737814
14T19x14-17x12+114x10-386x8+699x6-650x4+257x2-17214·179·4914

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

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