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## G = C13⋊C8order 104 = 23·13

### The semidirect product of C13 and C8 acting via C8/C2=C4

Aliases: C13⋊C8, C26.C4, Dic13.2C2, C2.(C13⋊C4), SmallGroup(104,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C13⋊C8
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8
 Lower central C13 — C13⋊C8
 Upper central C1 — C2

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

Smallest permutation representation of C13⋊C8
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`

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