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## G = D4.D13order 208 = 24·13

### The non-split extension by D4 of D13 acting via D13/C13=C2

Aliases: D4.D13, C4.2D26, C26.8D4, C132SD16, Dic262C2, C52.2C22, C132C82C2, (D4×C13).1C2, C2.5(C13⋊D4), SmallGroup(208,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D4.D13
 Chief series C1 — C13 — C26 — C52 — Dic26 — D4.D13
 Lower central C13 — C26 — C52 — D4.D13
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D13
G = < a,b,c,d | a4=b2=c13=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4.D13
On 104 points
Generators in S104
```(1 51 16 30)(2 52 17 31)(3 40 18 32)(4 41 19 33)(5 42 20 34)(6 43 21 35)(7 44 22 36)(8 45 23 37)(9 46 24 38)(10 47 25 39)(11 48 26 27)(12 49 14 28)(13 50 15 29)(53 79 66 92)(54 80 67 93)(55 81 68 94)(56 82 69 95)(57 83 70 96)(58 84 71 97)(59 85 72 98)(60 86 73 99)(61 87 74 100)(62 88 75 101)(63 89 76 102)(64 90 77 103)(65 91 78 104)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 27)(12 28)(13 29)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)
(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 78 16 65)(2 77 17 64)(3 76 18 63)(4 75 19 62)(5 74 20 61)(6 73 21 60)(7 72 22 59)(8 71 23 58)(9 70 24 57)(10 69 25 56)(11 68 26 55)(12 67 14 54)(13 66 15 53)(27 94 48 81)(28 93 49 80)(29 92 50 79)(30 104 51 91)(31 103 52 90)(32 102 40 89)(33 101 41 88)(34 100 42 87)(35 99 43 86)(36 98 44 85)(37 97 45 84)(38 96 46 83)(39 95 47 82)```

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

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

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

D4.D13 is a maximal subgroup of   D8⋊D13  D83D13  SD16×D13  D4.D26  D526C22  C52.C23  D4.9D26
D4.D13 is a maximal quotient of   C52.Q8  C26.Q16  D4⋊Dic13

37 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 13A ··· 13F 26A ··· 26F 26G ··· 26R 52A ··· 52F order 1 2 2 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 4 2 52 26 26 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

37 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 D4 SD16 D13 D26 C13⋊D4 D4.D13 kernel D4.D13 C13⋊2C8 Dic26 D4×C13 C26 C13 D4 C4 C2 C1 # reps 1 1 1 1 1 2 6 6 12 6

Matrix representation of D4.D13 in GL4(𝔽313) generated by

 1 0 0 0 0 1 0 0 0 0 1 311 0 0 1 312
,
 1 0 0 0 0 1 0 0 0 0 1 311 0 0 0 312
,
 0 1 0 0 312 24 0 0 0 0 1 0 0 0 0 1
,
 81 166 0 0 232 232 0 0 0 0 0 183 0 0 248 0
`G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,311,312],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,311,312],[0,312,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[81,232,0,0,166,232,0,0,0,0,0,248,0,0,183,0] >;`

D4.D13 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(208,16);`
`// by ID`

`G=gap.SmallGroup(208,16);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,182,97,42,4804]);`
`// Polycyclic`

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

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