direct product, metacyclic, supersoluble, monomial, A-group
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
| C7 — C2×F7 |
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 |
►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
| action | f(x) | Disc(f) |
|---|---|---|
| 14T7 | x14+3x12+3x10-x8-6x4+x2-3 | 238·319·76 |
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
Export
Subgroup lattice of C2×F7 in TeX
Character table of C2×F7 in TeX