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## G = D13.D4order 208 = 24·13

### The non-split extension by D13 of D4 acting via D4/C22=C2

Aliases: D262C4, D13.2D4, D26.6C22, (C2×C26)⋊1C4, C22⋊(C13⋊C4), C13⋊(C22⋊C4), C26.7(C2×C4), (C22×D13).2C2, (C2×C13⋊C4)⋊C2, C2.7(C2×C13⋊C4), SmallGroup(208,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D13.D4
 Chief series C1 — C13 — D13 — D26 — C2×C13⋊C4 — D13.D4
 Lower central C13 — C26 — D13.D4
 Upper central C1 — C2 — C22

Generators and relations for D13.D4
G = < a,b,c,d | a13=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a5, cbc-1=dbd-1=a4b, dcd-1=a-1bc-1 >

2C2
13C2
13C2
26C2
13C22
13C22
26C4
26C4
26C22
26C22
2D13
2C26
13C23
13C2×C4
13C2×C4
2D26
2D26
13C22⋊C4

Character table of D13.D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 13A 13B 13C 26A 26B 26C 26D 26E 26F 26G 26H 26I size 1 1 2 13 13 26 26 26 26 26 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 trivial ρ2 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 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 linear of order 2 ρ5 1 1 1 -1 -1 -1 -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 -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 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 i -i -i i 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ9 2 -2 0 -2 2 0 0 0 0 0 2 2 2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 -2 0 2 -2 0 0 0 0 0 2 2 2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 orthogonal lifted from C2×C13⋊C4 ρ12 4 4 4 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ13 4 -4 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311-ζ1310-ζ133+ζ132 -ζ139+ζ137+ζ136-ζ134 ζ1312-ζ138-ζ135+ζ13 ζ139-ζ137-ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312+ζ138+ζ135-ζ13 -ζ1311+ζ1310+ζ133-ζ132 orthogonal faithful ρ14 4 4 4 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ15 4 4 -4 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 orthogonal lifted from C2×C13⋊C4 ρ16 4 -4 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1311+ζ1310+ζ133-ζ132 ζ139-ζ137-ζ136+ζ134 -ζ1312+ζ138+ζ135-ζ13 -ζ139+ζ137+ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ1312-ζ138-ζ135+ζ13 ζ1311-ζ1310-ζ133+ζ132 orthogonal faithful ρ17 4 4 -4 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 orthogonal lifted from C2×C13⋊C4 ρ18 4 -4 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ139+ζ137+ζ136-ζ134 -ζ1312+ζ138+ζ135-ζ13 -ζ1311+ζ1310+ζ133-ζ132 ζ1312-ζ138-ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ1311-ζ1310-ζ133+ζ132 ζ139-ζ137-ζ136+ζ134 orthogonal faithful ρ19 4 -4 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139-ζ137-ζ136+ζ134 ζ1312-ζ138-ζ135+ζ13 ζ1311-ζ1310-ζ133+ζ132 -ζ1312+ζ138+ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311+ζ1310+ζ133-ζ132 -ζ139+ζ137+ζ136-ζ134 orthogonal faithful ρ20 4 -4 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312-ζ138-ζ135+ζ13 -ζ1311+ζ1310+ζ133-ζ132 -ζ139+ζ137+ζ136-ζ134 ζ1311-ζ1310-ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ139-ζ137-ζ136+ζ134 -ζ1312+ζ138+ζ135-ζ13 orthogonal faithful ρ21 4 -4 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1312+ζ138+ζ135-ζ13 ζ1311-ζ1310-ζ133+ζ132 ζ139-ζ137-ζ136+ζ134 -ζ1311+ζ1310+ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139+ζ137+ζ136-ζ134 ζ1312-ζ138-ζ135+ζ13 orthogonal faithful ρ22 4 4 4 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4

Smallest permutation representation of D13.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 17)(2 16)(3 15)(4 14)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)
(1 31)(2 39 13 36)(3 34 12 28)(4 29 11 33)(5 37 10 38)(6 32 9 30)(7 27 8 35)(14 50 22 49)(15 45 21 41)(16 40 20 46)(17 48 19 51)(18 43)(23 44 26 42)(24 52 25 47)
(1 43 18 31)(2 51 17 36)(3 46 16 28)(4 41 15 33)(5 49 14 38)(6 44 26 30)(7 52 25 35)(8 47 24 27)(9 42 23 32)(10 50 22 37)(11 45 21 29)(12 40 20 34)(13 48 19 39)```

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

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

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

D13.D4 is a maximal subgroup of
D26.D4  D26.4D4  D26.C23  D4×C13⋊C4
D13.D4 is a maximal quotient of
D26.D4  D26⋊C8  Dic13.D4  D26.Q8  D521C4  Dic26⋊C4  D13.Q16  D52⋊C4  D26.4D4  C26.M4(2)  Dic13.4D4

Matrix representation of D13.D4 in GL4(𝔽53) generated by

 0 1 0 0 52 38 0 0 0 0 2 28 0 0 25 6
,
 0 52 0 0 52 0 0 0 0 0 51 25 0 0 2 2
,
 0 0 1 0 0 0 0 1 1 0 0 0 38 52 0 0
,
 0 0 1 0 0 0 0 1 52 0 0 0 15 1 0 0
`G:=sub<GL(4,GF(53))| [0,52,0,0,1,38,0,0,0,0,2,25,0,0,28,6],[0,52,0,0,52,0,0,0,0,0,51,2,0,0,25,2],[0,0,1,38,0,0,0,52,1,0,0,0,0,1,0,0],[0,0,52,15,0,0,0,1,1,0,0,0,0,1,0,0] >;`

D13.D4 in GAP, Magma, Sage, TeX

`D_{13}.D_4`
`% in TeX`

`G:=Group("D13.D4");`
`// GroupNames label`

`G:=SmallGroup(208,34);`
`// by ID`

`G=gap.SmallGroup(208,34);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,3204,1214]);`
`// Polycyclic`

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

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