metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7⋊C8, C14.C4, C4.2D7, C2.Dic7, C28.2C2, SmallGroup(56,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C8 |
Generators and relations for C7⋊C8
G = < a,b | a7=b8=1, bab-1=a-1 >
Character table of C7⋊C8
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 28A | 28B | 28C | 28D | 28E | 28F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | -1 | -1 | i | -i | -i | -i | i | i | linear of order 8 |
| ρ6 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | -1 | -1 | -i | i | i | i | -i | -i | linear of order 8 |
| ρ7 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | -1 | -1 | i | -i | -i | -i | i | i | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | -1 | -1 | -i | i | i | i | -i | -i | linear of order 8 |
| ρ9 | 2 | 2 | 2 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 2 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | 2 | 2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ12 | 2 | 2 | -2 | -2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | symplectic lifted from Dic7, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | symplectic lifted from Dic7, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | symplectic lifted from Dic7, Schur index 2 |
| ρ15 | 2 | -2 | 2i | -2i | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ43ζ74+ζ43ζ73 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | complex faithful |
| ρ16 | 2 | -2 | 2i | -2i | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ43ζ76+ζ43ζ7 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | complex faithful |
| ρ17 | 2 | -2 | -2i | 2i | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ4ζ74+ζ4ζ73 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | complex faithful |
| ρ18 | 2 | -2 | -2i | 2i | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ4ζ75+ζ4ζ72 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | complex faithful |
| ρ19 | 2 | -2 | 2i | -2i | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ43ζ75+ζ43ζ72 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | complex faithful |
| ρ20 | 2 | -2 | -2i | 2i | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ4ζ76+ζ4ζ7 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | complex faithful |
►Regular action on 56 points
Generators in S56
(1 49 32 19 16 34 45)(2 46 35 9 20 25 50)(3 51 26 21 10 36 47)(4 48 37 11 22 27 52)(5 53 28 23 12 38 41)(6 42 39 13 24 29 54)(7 55 30 17 14 40 43)(8 44 33 15 18 31 56)
(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 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)
G:=sub<Sym(56)| (1,49,32,19,16,34,45)(2,46,35,9,20,25,50)(3,51,26,21,10,36,47)(4,48,37,11,22,27,52)(5,53,28,23,12,38,41)(6,42,39,13,24,29,54)(7,55,30,17,14,40,43)(8,44,33,15,18,31,56), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)>;
G:=Group( (1,49,32,19,16,34,45)(2,46,35,9,20,25,50)(3,51,26,21,10,36,47)(4,48,37,11,22,27,52)(5,53,28,23,12,38,41)(6,42,39,13,24,29,54)(7,55,30,17,14,40,43)(8,44,33,15,18,31,56), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56) );
G=PermutationGroup([[(1,49,32,19,16,34,45),(2,46,35,9,20,25,50),(3,51,26,21,10,36,47),(4,48,37,11,22,27,52),(5,53,28,23,12,38,41),(6,42,39,13,24,29,54),(7,55,30,17,14,40,43),(8,44,33,15,18,31,56)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56)]])
C7⋊C8 is a maximal subgroup of
C8×D7 C8⋊D7 C4.Dic7 D4⋊D7 D4.D7 Q8⋊D7 C7⋊Q16 C7⋊C24 C21⋊C8 C35⋊3C8 C35⋊C8 C49⋊C8 C72⋊4C8
C7⋊C8 is a maximal quotient of
C7⋊C16 C21⋊C8 C35⋊3C8 C35⋊C8 C49⋊C8 C72⋊4C8
Matrix representation of C7⋊C8 ►in GL2(𝔽13) generated by
| 5 | 4 |
| 6 | 5 |
| 0 | 8 |
| 1 | 0 |
G:=sub<GL(2,GF(13))| [5,6,4,5],[0,1,8,0] >;
C7⋊C8 in GAP, Magma, Sage, TeX
C_7\rtimes C_8
% in TeX
G:=Group("C7:C8"); // GroupNames label
G:=SmallGroup(56,1);
// by ID
G=gap.SmallGroup(56,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-7,8,21,771]);
// Polycyclic
G:=Group<a,b|a^7=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C7⋊C8 in TeX
Character table of C7⋊C8 in TeX