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## G = D13⋊C8order 208 = 24·13

### The semidirect product of D13 and C8 acting via C8/C4=C2

Aliases: D13⋊C8, C52.3C4, D26.2C4, Dic13.4C22, C13⋊C83C2, C131(C2×C8), C4.3(C13⋊C4), C26.1(C2×C4), (C4×D13).5C2, C2.1(C2×C13⋊C4), SmallGroup(208,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — D13⋊C8
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — D13⋊C8
 Lower central C13 — D13⋊C8
 Upper central C1 — C4

Generators and relations for D13⋊C8
G = < a,b,c | a13=b2=c8=1, bab=a-1, cac-1=a5, cbc-1=a4b >

Character table of D13⋊C8

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 13A 13B 13C 26A 26B 26C 52A 52B 52C 52D 52E 52F size 1 1 13 13 1 1 13 13 13 13 13 13 13 13 13 13 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 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 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 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 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 -1 -1 i -i -i i i -i -i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -1 -1 -i i i -i -i i i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 -1 i i i -i -i -i -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 -i -i -i i i i i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 1 -1 1 -1 -i i -i i ζ85 ζ85 ζ8 ζ87 ζ83 ζ87 ζ83 ζ8 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 8 ρ10 1 -1 1 -1 i -i i -i ζ83 ζ83 ζ87 ζ8 ζ85 ζ8 ζ85 ζ87 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 8 ρ11 1 -1 -1 1 -i i i -i ζ83 ζ87 ζ83 ζ85 ζ8 ζ8 ζ85 ζ87 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 8 ρ12 1 -1 1 -1 -i i -i i ζ8 ζ8 ζ85 ζ83 ζ87 ζ83 ζ87 ζ85 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 8 ρ13 1 -1 -1 1 i -i -i i ζ85 ζ8 ζ85 ζ83 ζ87 ζ87 ζ83 ζ8 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 8 ρ14 1 -1 1 -1 i -i i -i ζ87 ζ87 ζ83 ζ85 ζ8 ζ85 ζ8 ζ83 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 8 ρ15 1 -1 -1 1 -i i i -i ζ87 ζ83 ζ87 ζ8 ζ85 ζ85 ζ8 ζ83 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 8 ρ16 1 -1 -1 1 i -i -i i ζ8 ζ85 ζ8 ζ87 ζ83 ζ83 ζ87 ζ85 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 8 ρ17 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ139-ζ137-ζ136-ζ134 orthogonal lifted from C2×C13⋊C4 ρ18 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1311-ζ1310-ζ133-ζ132 orthogonal lifted from C2×C13⋊C4 ρ19 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ20 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ21 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1312-ζ138-ζ135-ζ13 orthogonal lifted from C2×C13⋊C4 ρ22 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ23 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 complex faithful, Schur index 2 ρ24 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 complex faithful, Schur index 2 ρ25 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 complex faithful, Schur index 2 ρ26 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 complex faithful, Schur index 2 ρ27 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 complex faithful, Schur index 2 ρ28 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 complex faithful, Schur index 2

Smallest permutation representation of D13⋊C8
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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 63)(54 62)(55 61)(56 60)(57 59)(64 65)(66 68)(69 78)(70 77)(71 76)(72 75)(73 74)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)
(1 92 40 74 25 79 27 65)(2 100 52 66 26 87 39 57)(3 95 51 71 14 82 38 62)(4 103 50 76 15 90 37 54)(5 98 49 68 16 85 36 59)(6 93 48 73 17 80 35 64)(7 101 47 78 18 88 34 56)(8 96 46 70 19 83 33 61)(9 104 45 75 20 91 32 53)(10 99 44 67 21 86 31 58)(11 94 43 72 22 81 30 63)(12 102 42 77 23 89 29 55)(13 97 41 69 24 84 28 60)```

`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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,63)(54,62)(55,61)(56,60)(57,59)(64,65)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99), (1,92,40,74,25,79,27,65)(2,100,52,66,26,87,39,57)(3,95,51,71,14,82,38,62)(4,103,50,76,15,90,37,54)(5,98,49,68,16,85,36,59)(6,93,48,73,17,80,35,64)(7,101,47,78,18,88,34,56)(8,96,46,70,19,83,33,61)(9,104,45,75,20,91,32,53)(10,99,44,67,21,86,31,58)(11,94,43,72,22,81,30,63)(12,102,42,77,23,89,29,55)(13,97,41,69,24,84,28,60)>;`

`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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,63)(54,62)(55,61)(56,60)(57,59)(64,65)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99), (1,92,40,74,25,79,27,65)(2,100,52,66,26,87,39,57)(3,95,51,71,14,82,38,62)(4,103,50,76,15,90,37,54)(5,98,49,68,16,85,36,59)(6,93,48,73,17,80,35,64)(7,101,47,78,18,88,34,56)(8,96,46,70,19,83,33,61)(9,104,45,75,20,91,32,53)(10,99,44,67,21,86,31,58)(11,94,43,72,22,81,30,63)(12,102,42,77,23,89,29,55)(13,97,41,69,24,84,28,60) );`

`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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,63),(54,62),(55,61),(56,60),(57,59),(64,65),(66,68),(69,78),(70,77),(71,76),(72,75),(73,74),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99)], [(1,92,40,74,25,79,27,65),(2,100,52,66,26,87,39,57),(3,95,51,71,14,82,38,62),(4,103,50,76,15,90,37,54),(5,98,49,68,16,85,36,59),(6,93,48,73,17,80,35,64),(7,101,47,78,18,88,34,56),(8,96,46,70,19,83,33,61),(9,104,45,75,20,91,32,53),(10,99,44,67,21,86,31,58),(11,94,43,72,22,81,30,63),(12,102,42,77,23,89,29,55),(13,97,41,69,24,84,28,60)])`

D13⋊C8 is a maximal subgroup of   C8×C13⋊C4  C104⋊C4  D521C4  D13.Q16  D13⋊M4(2)  Dic26.C4  D52.C4
D13⋊C8 is a maximal quotient of   D13⋊C16  D26.C8  C4×C13⋊C8  D26⋊C8  Dic13⋊C8

Matrix representation of D13⋊C8 in GL4(𝔽5) generated by

 4 4 0 3 0 1 3 1 1 3 3 4 4 4 2 0
,
 1 3 1 3 1 3 3 4 0 1 3 1 4 4 4 3
,
 2 4 3 0 0 0 1 4 2 0 3 0 2 0 0 0
`G:=sub<GL(4,GF(5))| [4,0,1,4,4,1,3,4,0,3,3,2,3,1,4,0],[1,1,0,4,3,3,1,4,1,3,3,4,3,4,1,3],[2,0,2,2,4,0,0,0,3,1,3,0,0,4,0,0] >;`

D13⋊C8 in GAP, Magma, Sage, TeX

`D_{13}\rtimes C_8`
`% in TeX`

`G:=Group("D13:C8");`
`// GroupNames label`

`G:=SmallGroup(208,28);`
`// by ID`

`G=gap.SmallGroup(208,28);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-13,20,46,42,3204,1214]);`
`// Polycyclic`

`G:=Group<a,b,c|a^13=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;`
`// generators/relations`

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