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## G = D4⋊D26order 416 = 25·13

### 2nd semidirect product of D4 and D26 acting via D26/C26=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D4⋊D26
 Chief series C1 — C13 — C26 — C52 — D52 — C2×D52 — D4⋊D26
 Lower central C13 — C26 — C52 — D4⋊D26
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4⋊D26
G = < a,b,c,d | a4=b2=c26=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 536 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C13, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D13 [×2], C26, C26 [×2], C8⋊C22, C52 [×2], C52, D26 [×4], C2×C26, C2×C26, C132C8 [×2], D52 [×2], D52, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C22×D13, C52.4C4, D4⋊D13 [×2], Q8⋊D13 [×2], C2×D52, C13×C4○D4, D4⋊D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8⋊C22, D26 [×3], C13⋊D4 [×2], C22×D13, C2×C13⋊D4, D4⋊D26

Smallest permutation representation of D4⋊D26
On 104 points
Generators in S104
```(1 50 21 37)(2 51 22 38)(3 52 23 39)(4 40 24 27)(5 41 25 28)(6 42 26 29)(7 43 14 30)(8 44 15 31)(9 45 16 32)(10 46 17 33)(11 47 18 34)(12 48 19 35)(13 49 20 36)(53 82 66 95)(54 83 67 96)(55 84 68 97)(56 85 69 98)(57 86 70 99)(58 87 71 100)(59 88 72 101)(60 89 73 102)(61 90 74 103)(62 91 75 104)(63 92 76 79)(64 93 77 80)(65 94 78 81)
(1 101)(2 89)(3 103)(4 91)(5 79)(6 93)(7 81)(8 95)(9 83)(10 97)(11 85)(12 99)(13 87)(14 94)(15 82)(16 96)(17 84)(18 98)(19 86)(20 100)(21 88)(22 102)(23 90)(24 104)(25 92)(26 80)(27 75)(28 63)(29 77)(30 65)(31 53)(32 67)(33 55)(34 69)(35 57)(36 71)(37 59)(38 73)(39 61)(40 62)(41 76)(42 64)(43 78)(44 66)(45 54)(46 68)(47 56)(48 70)(49 58)(50 72)(51 60)(52 74)
(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)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 15)(16 26)(17 25)(18 24)(19 23)(20 22)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 52)(36 51)(37 50)(38 49)(39 48)(53 81)(54 80)(55 79)(56 104)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)```

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

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

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

71 conjugacy classes

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

71 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 D26 C13⋊D4 C13⋊D4 C8⋊C22 D4⋊D26 kernel D4⋊D26 C52.4C4 D4⋊D13 Q8⋊D13 C2×D52 C13×C4○D4 C52 C2×C26 C4○D4 C2×C4 D4 Q8 C4 C22 C13 C1 # reps 1 1 2 2 1 1 1 1 6 6 6 6 12 12 1 12

Matrix representation of D4⋊D26 in GL6(𝔽313)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 21 241 0 0 0 0 67 292 0 0 0 0 31 2 222 176 0 0 289 142 113 91
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 100 171 182 274 0 0 261 202 227 106 0 0 265 39 244 80 0 0 310 206 209 80
,
 81 81 0 0 0 0 232 147 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 210 2 312 0 0 0 6 299 0 312
,
 232 232 0 0 0 0 166 81 0 0 0 0 0 0 1 0 0 0 0 0 131 312 0 0 0 0 186 70 222 176 0 0 11 250 81 91

`G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,67,31,289,0,0,241,292,2,142,0,0,0,0,222,113,0,0,0,0,176,91],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,261,265,310,0,0,171,202,39,206,0,0,182,227,244,209,0,0,274,106,80,80],[81,232,0,0,0,0,81,147,0,0,0,0,0,0,1,0,210,6,0,0,0,1,2,299,0,0,0,0,312,0,0,0,0,0,0,312],[232,166,0,0,0,0,232,81,0,0,0,0,0,0,1,131,186,11,0,0,0,312,70,250,0,0,0,0,222,81,0,0,0,0,176,91] >;`

D4⋊D26 in GAP, Magma, Sage, TeX

`D_4\rtimes D_{26}`
`% in TeX`

`G:=Group("D4:D26");`
`// GroupNames label`

`G:=SmallGroup(416,170);`
`// by ID`

`G=gap.SmallGroup(416,170);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,579,159,69,13829]);`
`// Polycyclic`

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

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