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## G = C13⋊D4order 104 = 23·13

### The semidirect product of C13 and D4 acting via D4/C22=C2

Aliases: C132D4, C22⋊D13, D262C2, Dic13⋊C2, C2.5D26, C26.5C22, (C2×C26)⋊2C2, SmallGroup(104,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C13⋊D4
 Chief series C1 — C13 — C26 — D26 — C13⋊D4
 Lower central C13 — C26 — C13⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C13⋊D4
G = < a,b,c | a13=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C13⋊D4

 class 1 2A 2B 2C 4 13A 13B 13C 13D 13E 13F 26A 26B 26C 26D 26E 26F 26G 26H 26I 26J 26K 26L 26M 26N 26O 26P 26Q 26R size 1 1 2 26 26 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 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 1 linear of order 2 ρ5 2 -2 0 0 0 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 0 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 ζ138+ζ135 ζ1312+ζ13 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 orthogonal lifted from D13 ρ7 2 2 2 0 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 ζ1311+ζ132 ζ1310+ζ133 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 orthogonal lifted from D13 ρ8 2 2 -2 0 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 -ζ1312-ζ13 -ζ1311-ζ132 -ζ1310-ζ133 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ137-ζ136 -ζ138-ζ135 -ζ139-ζ134 -ζ1310-ζ133 -ζ1311-ζ132 -ζ1312-ζ13 ζ1311+ζ132 ζ1310+ζ133 orthogonal lifted from D26 ρ9 2 2 -2 0 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 -ζ137-ζ136 -ζ1312-ζ13 -ζ138-ζ135 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ1310-ζ133 -ζ139-ζ134 -ζ1311-ζ132 -ζ138-ζ135 -ζ1312-ζ13 -ζ137-ζ136 ζ1312+ζ13 ζ138+ζ135 orthogonal lifted from D26 ρ10 2 2 2 0 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 ζ139+ζ134 ζ137+ζ136 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 orthogonal lifted from D13 ρ11 2 2 2 0 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 ζ1312+ζ13 ζ138+ζ135 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 orthogonal lifted from D13 ρ12 2 2 -2 0 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 -ζ139-ζ134 -ζ138-ζ135 -ζ1312-ζ13 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ1311-ζ132 -ζ137-ζ136 -ζ1310-ζ133 -ζ1312-ζ13 -ζ138-ζ135 -ζ139-ζ134 ζ138+ζ135 ζ1312+ζ13 orthogonal lifted from D26 ρ13 2 2 -2 0 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 -ζ1311-ζ132 -ζ139-ζ134 -ζ137-ζ136 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1312-ζ13 -ζ1310-ζ133 -ζ138-ζ135 -ζ137-ζ136 -ζ139-ζ134 -ζ1311-ζ132 ζ139+ζ134 ζ137+ζ136 orthogonal lifted from D26 ρ14 2 2 -2 0 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 -ζ138-ζ135 -ζ1310-ζ133 -ζ1311-ζ132 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ139-ζ134 -ζ1312-ζ13 -ζ137-ζ136 -ζ1311-ζ132 -ζ1310-ζ133 -ζ138-ζ135 ζ1310+ζ133 ζ1311+ζ132 orthogonal lifted from D26 ρ15 2 2 2 0 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 ζ137+ζ136 ζ139+ζ134 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 orthogonal lifted from D13 ρ16 2 2 -2 0 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 -ζ1310-ζ133 -ζ137-ζ136 -ζ139-ζ134 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ138-ζ135 -ζ1311-ζ132 -ζ1312-ζ13 -ζ139-ζ134 -ζ137-ζ136 -ζ1310-ζ133 ζ137+ζ136 ζ139+ζ134 orthogonal lifted from D26 ρ17 2 2 2 0 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 ζ1310+ζ133 ζ1311+ζ132 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 orthogonal lifted from D13 ρ18 2 -2 0 0 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ137-ζ136 -ζ137+ζ136 ζ1312-ζ13 -ζ138+ζ135 ζ1311-ζ132 -ζ139+ζ134 ζ1310-ζ133 -ζ1310+ζ133 ζ139-ζ134 -ζ1311+ζ132 ζ138-ζ135 -ζ1312+ζ13 ζ137-ζ136 -ζ1312-ζ13 -ζ138-ζ135 complex faithful ρ19 2 -2 0 0 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ138-ζ135 ζ138-ζ135 -ζ1310+ζ133 ζ1311-ζ132 -ζ137+ζ136 -ζ1312+ζ13 ζ139-ζ134 -ζ139+ζ134 ζ1312-ζ13 ζ137-ζ136 -ζ1311+ζ132 ζ1310-ζ133 -ζ138+ζ135 -ζ1310-ζ133 -ζ1311-ζ132 complex faithful ρ20 2 -2 0 0 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ137-ζ136 ζ137-ζ136 -ζ1312+ζ13 ζ138-ζ135 -ζ1311+ζ132 ζ139-ζ134 -ζ1310+ζ133 ζ1310-ζ133 -ζ139+ζ134 ζ1311-ζ132 -ζ138+ζ135 ζ1312-ζ13 -ζ137+ζ136 -ζ1312-ζ13 -ζ138-ζ135 complex faithful ρ21 2 -2 0 0 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ139-ζ134 -ζ139+ζ134 ζ138-ζ135 ζ1312-ζ13 -ζ1310+ζ133 ζ137-ζ136 ζ1311-ζ132 -ζ1311+ζ132 -ζ137+ζ136 ζ1310-ζ133 -ζ1312+ζ13 -ζ138+ζ135 ζ139-ζ134 -ζ138-ζ135 -ζ1312-ζ13 complex faithful ρ22 2 -2 0 0 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1311-ζ132 ζ1311-ζ132 ζ139-ζ134 ζ137-ζ136 -ζ138+ζ135 -ζ1310+ζ133 -ζ1312+ζ13 ζ1312-ζ13 ζ1310-ζ133 ζ138-ζ135 -ζ137+ζ136 -ζ139+ζ134 -ζ1311+ζ132 -ζ139-ζ134 -ζ137-ζ136 complex faithful ρ23 2 -2 0 0 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ1310-ζ133 ζ1310-ζ133 ζ137-ζ136 -ζ139+ζ134 -ζ1312+ζ13 ζ1311-ζ132 ζ138-ζ135 -ζ138+ζ135 -ζ1311+ζ132 ζ1312-ζ13 ζ139-ζ134 -ζ137+ζ136 -ζ1310+ζ133 -ζ137-ζ136 -ζ139-ζ134 complex faithful ρ24 2 -2 0 0 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ138-ζ135 -ζ138+ζ135 ζ1310-ζ133 -ζ1311+ζ132 ζ137-ζ136 ζ1312-ζ13 -ζ139+ζ134 ζ139-ζ134 -ζ1312+ζ13 -ζ137+ζ136 ζ1311-ζ132 -ζ1310+ζ133 ζ138-ζ135 -ζ1310-ζ133 -ζ1311-ζ132 complex faithful ρ25 2 -2 0 0 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ1312-ζ13 ζ1312-ζ13 ζ1311-ζ132 ζ1310-ζ133 ζ139-ζ134 ζ138-ζ135 ζ137-ζ136 -ζ137+ζ136 -ζ138+ζ135 -ζ139+ζ134 -ζ1310+ζ133 -ζ1311+ζ132 -ζ1312+ζ13 -ζ1311-ζ132 -ζ1310-ζ133 complex faithful ρ26 2 -2 0 0 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ139-ζ134 ζ139-ζ134 -ζ138+ζ135 -ζ1312+ζ13 ζ1310-ζ133 -ζ137+ζ136 -ζ1311+ζ132 ζ1311-ζ132 ζ137-ζ136 -ζ1310+ζ133 ζ1312-ζ13 ζ138-ζ135 -ζ139+ζ134 -ζ138-ζ135 -ζ1312-ζ13 complex faithful ρ27 2 -2 0 0 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1311-ζ132 -ζ1311+ζ132 -ζ139+ζ134 -ζ137+ζ136 ζ138-ζ135 ζ1310-ζ133 ζ1312-ζ13 -ζ1312+ζ13 -ζ1310+ζ133 -ζ138+ζ135 ζ137-ζ136 ζ139-ζ134 ζ1311-ζ132 -ζ139-ζ134 -ζ137-ζ136 complex faithful ρ28 2 -2 0 0 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ1310-ζ133 -ζ1310+ζ133 -ζ137+ζ136 ζ139-ζ134 ζ1312-ζ13 -ζ1311+ζ132 -ζ138+ζ135 ζ138-ζ135 ζ1311-ζ132 -ζ1312+ζ13 -ζ139+ζ134 ζ137-ζ136 ζ1310-ζ133 -ζ137-ζ136 -ζ139-ζ134 complex faithful ρ29 2 -2 0 0 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ1312-ζ13 -ζ1312+ζ13 -ζ1311+ζ132 -ζ1310+ζ133 -ζ139+ζ134 -ζ138+ζ135 -ζ137+ζ136 ζ137-ζ136 ζ138-ζ135 ζ139-ζ134 ζ1310-ζ133 ζ1311-ζ132 ζ1312-ζ13 -ζ1311-ζ132 -ζ1310-ζ133 complex faithful

Smallest permutation representation of C13⋊D4
On 52 points
Generators in S52
```(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)
(1 27 18 43)(2 39 19 42)(3 38 20 41)(4 37 21 40)(5 36 22 52)(6 35 23 51)(7 34 24 50)(8 33 25 49)(9 32 26 48)(10 31 14 47)(11 30 15 46)(12 29 16 45)(13 28 17 44)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 43)(28 42)(29 41)(30 40)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)```

`G:=sub<Sym(52)| (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), (1,27,18,43)(2,39,19,42)(3,38,20,41)(4,37,21,40)(5,36,22,52)(6,35,23,51)(7,34,24,50)(8,33,25,49)(9,32,26,48)(10,31,14,47)(11,30,15,46)(12,29,16,45)(13,28,17,44), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)>;`

`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), (1,27,18,43)(2,39,19,42)(3,38,20,41)(4,37,21,40)(5,36,22,52)(6,35,23,51)(7,34,24,50)(8,33,25,49)(9,32,26,48)(10,31,14,47)(11,30,15,46)(12,29,16,45)(13,28,17,44), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44) );`

`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)], [(1,27,18,43),(2,39,19,42),(3,38,20,41),(4,37,21,40),(5,36,22,52),(6,35,23,51),(7,34,24,50),(8,33,25,49),(9,32,26,48),(10,31,14,47),(11,30,15,46),(12,29,16,45),(13,28,17,44)], [(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,43),(28,42),(29,41),(30,40),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44)])`

C13⋊D4 is a maximal subgroup of
D525C2  D4×D13  D42D13  D26⋊C6  C39⋊D4  C13⋊D12  C397D4  C13⋊S4
C13⋊D4 is a maximal quotient of
C26.D4  D26⋊C4  D4⋊D13  D4.D13  Q8⋊D13  C13⋊Q16  C23.D13  C39⋊D4  C13⋊D12  C397D4

Matrix representation of C13⋊D4 in GL2(𝔽53) generated by

 0 1 52 26
,
 31 18 29 22
,
 1 0 26 52
`G:=sub<GL(2,GF(53))| [0,52,1,26],[31,29,18,22],[1,26,0,52] >;`

C13⋊D4 in GAP, Magma, Sage, TeX

`C_{13}\rtimes D_4`
`% in TeX`

`G:=Group("C13:D4");`
`// GroupNames label`

`G:=SmallGroup(104,8);`
`// by ID`

`G=gap.SmallGroup(104,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-13,49,1539]);`
`// Polycyclic`

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

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