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## G = C2×F7order 84 = 22·3·7

### Direct product of C2 and F7

Aliases: C2×F7, C14⋊C6, D7⋊C6, D14⋊C3, C7⋊(C2×C6), C7⋊C3⋊C22, (C2×C7⋊C3)⋊C2, Aut(D14), Hol(C14), SmallGroup(84,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×F7
 Chief series C1 — C7 — C7⋊C3 — F7 — C2×F7
 Lower central C7 — C2×F7
 Upper central C1 — C2

Generators and relations for C2×F7
G = < a,b,c | a2=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C2×F7

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 7 14 size 1 1 7 7 7 7 7 7 7 7 7 7 6 6 ρ1 1 1 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 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ5 1 1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 linear of order 6 ρ6 1 -1 -1 1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 1 -1 linear of order 6 ρ7 1 -1 1 -1 ζ3 ζ32 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 1 -1 linear of order 6 ρ8 1 -1 1 -1 ζ32 ζ3 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 1 -1 linear of order 6 ρ9 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 linear of order 3 ρ10 1 -1 -1 1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 1 -1 linear of order 6 ρ11 1 1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 linear of order 6 ρ12 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 linear of order 3 ρ13 6 -6 0 0 0 0 0 0 0 0 0 0 -1 1 orthogonal faithful ρ14 6 6 0 0 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from F7

Permutation representations of C2×F7
On 14 points - transitive group 14T7
Generators in S14
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 8)(2 11 3 14 5 13)(4 10 7 12 6 9)

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

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

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

G:=TransitiveGroup(14,7);

On 28 points - transitive group 28T15
Generators in S28
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 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)(2 16 3 19 5 18)(4 15 7 17 6 21)(8 27)(9 23 10 26 12 25)(11 22 14 24 13 28)

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

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

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

G:=TransitiveGroup(28,15);

C2×F7 is a maximal subgroup of   C4⋊F7  Dic7⋊C6  Q8⋊F7
C2×F7 is a maximal quotient of   C4.F7  C4⋊F7  Dic7⋊C6

Polynomial with Galois group C2×F7 over ℚ
actionf(x)Disc(f)
14T7x14+3x12+3x10-x8-6x4+x2-3238·319·76

Matrix representation of C2×F7 in GL6(ℤ)

 -1 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 -1 0 0 0 0 0 0 -1
,
 -1 -1 -1 -1 -1 -1 1 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 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 1 0

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,-1,0,1,0,0,0,-1,0,0,1,0,0,-1,0,0,0,1,0,-1,0,0,0,0,1,-1,0,0,0,0,0],[1,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,1,0,1,0,0,-1,0] >;

C2×F7 in GAP, Magma, Sage, TeX

C_2\times F_7
% in TeX

G:=Group("C2xF7");
// GroupNames label

G:=SmallGroup(84,7);
// by ID

G=gap.SmallGroup(84,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,1155,203]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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