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## G = C8⋊D26order 416 = 25·13

### 1st semidirect product of C8 and D26 acting via D26/C13=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C8⋊D26
 Chief series C1 — C13 — C26 — C52 — D52 — C2×D52 — C8⋊D26
 Lower central C13 — C26 — C52 — C8⋊D26
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C8⋊D26
G = < a,b,c | a8=b26=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 680 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C13, M4(2), D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, C52, D26, C2×C26, C104, Dic26, C4×D13, D52, D52, D52, C13⋊D4, C2×C52, C22×D13, C104⋊C2, D104, C13×M4(2), C2×D52, D525C2, C8⋊D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, D52, C22×D13, C2×D52, C8⋊D26

Smallest permutation representation of C8⋊D26
On 104 points
Generators in S104
```(1 94 52 65 23 81 27 78)(2 82 40 53 24 95 28 66)(3 96 41 67 25 83 29 54)(4 84 42 55 26 97 30 68)(5 98 43 69 14 85 31 56)(6 86 44 57 15 99 32 70)(7 100 45 71 16 87 33 58)(8 88 46 59 17 101 34 72)(9 102 47 73 18 89 35 60)(10 90 48 61 19 103 36 74)(11 104 49 75 20 91 37 62)(12 92 50 63 21 79 38 76)(13 80 51 77 22 93 39 64)
(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 4)(2 3)(5 13)(6 12)(7 11)(8 10)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 42)(28 41)(29 40)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(53 83)(54 82)(55 81)(56 80)(57 79)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)```

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

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

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

71 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 13A ··· 13F 26A ··· 26F 26G ··· 26L 52A ··· 52L 52M ··· 52R 104A ··· 104X order 1 2 2 2 2 2 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 52 52 52 2 2 52 4 4 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

71 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 D52 D52 C8⋊C22 C8⋊D26 kernel C8⋊D26 C104⋊C2 D104 C13×M4(2) C2×D52 D52⋊5C2 C52 C2×C26 M4(2) C8 C2×C4 C4 C22 C13 C1 # reps 1 2 2 1 1 1 1 1 6 12 6 12 12 1 12

Matrix representation of C8⋊D26 in GL4(𝔽313) generated by

 238 303 218 5 117 143 166 304 278 276 110 70 163 15 58 135
,
 214 300 0 0 70 123 0 0 88 77 29 13 273 248 291 260
,
 127 1 0 0 148 186 0 0 129 7 289 227 61 62 54 24
`G:=sub<GL(4,GF(313))| [238,117,278,163,303,143,276,15,218,166,110,58,5,304,70,135],[214,70,88,273,300,123,77,248,0,0,29,291,0,0,13,260],[127,148,129,61,1,186,7,62,0,0,289,54,0,0,227,24] >;`

C8⋊D26 in GAP, Magma, Sage, TeX

`C_8\rtimes D_{26}`
`% in TeX`

`G:=Group("C8:D26");`
`// GroupNames label`

`G:=SmallGroup(416,129);`
`// by ID`

`G=gap.SmallGroup(416,129);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,50,579,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^26=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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