Copied to
clipboard

## G = D8⋊D13order 416 = 25·13

### 2nd semidirect product of D8 and D13 acting via D13/C13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D8⋊D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D4×D13 — D8⋊D13
 Lower central C13 — C26 — C52 — D8⋊D13
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8⋊D13
G = < a,b,c,d | a4=b2=c26=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c-1 >

Subgroups: 584 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, Dic13, C52, D26, D26, C2×C26, C132C8, C104, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, D4×C13, C22×D13, C8⋊D13, C104⋊C2, D4⋊D13, D4.D13, C13×D8, D4×D13, D42D13, D8⋊D13
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C22×D13, D4×D13, D8⋊D13

Smallest permutation representation of D8⋊D13
On 104 points
Generators in S104
```(1 31 19 44)(2 45 20 32)(3 33 21 46)(4 47 22 34)(5 35 23 48)(6 49 24 36)(7 37 25 50)(8 51 26 38)(9 39 14 52)(10 27 15 40)(11 41 16 28)(12 29 17 42)(13 43 18 30)(53 66 89 102)(54 103 90 67)(55 68 91 104)(56 79 92 69)(57 70 93 80)(58 81 94 71)(59 72 95 82)(60 83 96 73)(61 74 97 84)(62 85 98 75)(63 76 99 86)(64 87 100 77)(65 78 101 88)
(1 80)(2 94)(3 82)(4 96)(5 84)(6 98)(7 86)(8 100)(9 88)(10 102)(11 90)(12 104)(13 92)(14 78)(15 66)(16 54)(17 68)(18 56)(19 70)(20 58)(21 72)(22 60)(23 74)(24 62)(25 76)(26 64)(27 89)(28 67)(29 91)(30 69)(31 93)(32 71)(33 95)(34 73)(35 97)(36 75)(37 99)(38 77)(39 101)(40 53)(41 103)(42 55)(43 79)(44 57)(45 81)(46 59)(47 83)(48 61)(49 85)(50 63)(51 87)(52 65)
(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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 26)(11 25)(12 24)(13 23)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 84)(80 83)(81 82)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 C8⋊C22 D4×D13 D8⋊D13 kernel D8⋊D13 C8⋊D13 C104⋊C2 D4⋊D13 D4.D13 C13×D8 D4×D13 D4⋊2D13 Dic13 D26 D8 C8 D4 C13 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 6 6 12 1 6 12

Matrix representation of D8⋊D13 in GL6(𝔽313)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 289 0 0 0 0 287 312 0 0 0 0 29 278 312 259 0 0 98 0 58 1
,
 312 0 0 0 0 0 0 312 0 0 0 0 0 0 237 0 70 24 0 0 98 0 58 1 0 0 226 54 76 35 0 0 287 312 0 0
,
 14 13 0 0 0 0 272 275 0 0 0 0 0 0 1 289 0 0 0 0 0 312 0 0 0 0 29 278 312 0 0 0 98 0 58 1
,
 84 37 0 0 0 0 190 229 0 0 0 0 0 0 1 289 0 0 0 0 0 312 0 0 0 0 0 278 1 0 0 0 215 0 255 312

`G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,287,29,98,0,0,289,312,278,0,0,0,0,0,312,58,0,0,0,0,259,1],[312,0,0,0,0,0,0,312,0,0,0,0,0,0,237,98,226,287,0,0,0,0,54,312,0,0,70,58,76,0,0,0,24,1,35,0],[14,272,0,0,0,0,13,275,0,0,0,0,0,0,1,0,29,98,0,0,289,312,278,0,0,0,0,0,312,58,0,0,0,0,0,1],[84,190,0,0,0,0,37,229,0,0,0,0,0,0,1,0,0,215,0,0,289,312,278,0,0,0,0,0,1,255,0,0,0,0,0,312] >;`

D8⋊D13 in GAP, Magma, Sage, TeX

`D_8\rtimes D_{13}`
`% in TeX`

`G:=Group("D8:D13");`
`// GroupNames label`

`G:=SmallGroup(416,132);`
`// by ID`

`G=gap.SmallGroup(416,132);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,116,297,159,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^26=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽