metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13⋊C8, C26.C4, Dic13.2C2, C2.(C13⋊C4), SmallGroup(104,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C8 |
Generators and relations for C13⋊C8
G = < a,b | a13=b8=1, bab-1=a5 >
Character table of C13⋊C8
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | 26B | 26C | |
| size | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | ζ85 | ζ87 | ζ8 | ζ83 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | ζ8 | ζ83 | ζ85 | ζ87 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | ζ83 | ζ8 | ζ87 | ζ85 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | ζ87 | ζ85 | ζ83 | ζ8 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ10 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | symplectic faithful, Schur index 2 |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | symplectic faithful, Schur index 2 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | symplectic faithful, Schur index 2 |
►Regular action on 104 points
Generators in S104
(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 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 103 40 73 14 80 27 64)(2 98 52 78 15 88 39 56)(3 93 51 70 16 83 38 61)(4 101 50 75 17 91 37 53)(5 96 49 67 18 86 36 58)(6 104 48 72 19 81 35 63)(7 99 47 77 20 89 34 55)(8 94 46 69 21 84 33 60)(9 102 45 74 22 79 32 65)(10 97 44 66 23 87 31 57)(11 92 43 71 24 82 30 62)(12 100 42 76 25 90 29 54)(13 95 41 68 26 85 28 59)
G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,103,40,73,14,80,27,64)(2,98,52,78,15,88,39,56)(3,93,51,70,16,83,38,61)(4,101,50,75,17,91,37,53)(5,96,49,67,18,86,36,58)(6,104,48,72,19,81,35,63)(7,99,47,77,20,89,34,55)(8,94,46,69,21,84,33,60)(9,102,45,74,22,79,32,65)(10,97,44,66,23,87,31,57)(11,92,43,71,24,82,30,62)(12,100,42,76,25,90,29,54)(13,95,41,68,26,85,28,59)>;
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)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,103,40,73,14,80,27,64)(2,98,52,78,15,88,39,56)(3,93,51,70,16,83,38,61)(4,101,50,75,17,91,37,53)(5,96,49,67,18,86,36,58)(6,104,48,72,19,81,35,63)(7,99,47,77,20,89,34,55)(8,94,46,69,21,84,33,60)(9,102,45,74,22,79,32,65)(10,97,44,66,23,87,31,57)(11,92,43,71,24,82,30,62)(12,100,42,76,25,90,29,54)(13,95,41,68,26,85,28,59) );
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),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,103,40,73,14,80,27,64),(2,98,52,78,15,88,39,56),(3,93,51,70,16,83,38,61),(4,101,50,75,17,91,37,53),(5,96,49,67,18,86,36,58),(6,104,48,72,19,81,35,63),(7,99,47,77,20,89,34,55),(8,94,46,69,21,84,33,60),(9,102,45,74,22,79,32,65),(10,97,44,66,23,87,31,57),(11,92,43,71,24,82,30,62),(12,100,42,76,25,90,29,54),(13,95,41,68,26,85,28,59)]])
C13⋊C8 is a maximal subgroup of
D13⋊C8 C52.C4 C13⋊M4(2) C13⋊C24 C39⋊C8
C13⋊C8 is a maximal quotient of C13⋊C16 C39⋊C8
Matrix representation of C13⋊C8 ►in GL4(𝔽5) generated by
| 2 | 1 | 4 | 0 |
| 3 | 3 | 2 | 3 |
| 4 | 4 | 0 | 2 |
| 1 | 3 | 2 | 3 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 4 | 4 |
| 1 | 0 | 2 | 3 |
| 0 | 1 | 2 | 3 |
G:=sub<GL(4,GF(5))| [2,3,4,1,1,3,4,3,4,2,0,2,0,3,2,3],[0,0,1,0,0,0,0,1,1,4,2,2,2,4,3,3] >;
C13⋊C8 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_8 % in TeX
G:=Group("C13:C8"); // GroupNames label
G:=SmallGroup(104,3);
// by ID
G=gap.SmallGroup(104,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-13,8,21,1027,775]);
// Polycyclic
G:=Group<a,b|a^13=b^8=1,b*a*b^-1=a^5>;
// generators/relations
Export
Subgroup lattice of C13⋊C8 in TeX
Character table of C13⋊C8 in TeX