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## G = D4⋊8D26order 416 = 25·13

### 4th semidirect product of D4 and D26 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4⋊8D26
 Chief series C1 — C13 — C26 — D26 — C22×D13 — D4×D13 — D4⋊8D26
 Lower central C13 — C26 — D4⋊8D26
 Upper central C1 — C2 — C4○D4

Generators and relations for D48D26
G = < a,b,c,d | a4=b2=c26=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 1264 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C13, C2×D4 [×9], C4○D4, C4○D4 [×5], D13 [×6], C26, C26 [×3], 2+ 1+4, Dic13 [×2], C52, C52 [×3], D26 [×6], D26 [×6], C2×C26 [×3], Dic26, C4×D13 [×6], D52 [×9], C13⋊D4 [×6], C2×C52 [×3], D4×C13 [×3], Q8×C13, C22×D13 [×6], C2×D52 [×3], D525C2 [×3], D4×D13 [×6], D52⋊C2 [×2], C13×C4○D4, D48D26
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, D13, 2+ 1+4, D26 [×7], C22×D13 [×7], C23×D13, D48D26

Smallest permutation representation of D48D26
On 104 points
Generators in S104
```(1 43 18 30)(2 44 19 31)(3 45 20 32)(4 46 21 33)(5 47 22 34)(6 48 23 35)(7 49 24 36)(8 50 25 37)(9 51 26 38)(10 52 14 39)(11 40 15 27)(12 41 16 28)(13 42 17 29)(53 85 66 98)(54 86 67 99)(55 87 68 100)(56 88 69 101)(57 89 70 102)(58 90 71 103)(59 91 72 104)(60 92 73 79)(61 93 74 80)(62 94 75 81)(63 95 76 82)(64 96 77 83)(65 97 78 84)
(1 88)(2 102)(3 90)(4 104)(5 92)(6 80)(7 94)(8 82)(9 96)(10 84)(11 98)(12 86)(13 100)(14 97)(15 85)(16 99)(17 87)(18 101)(19 89)(20 103)(21 91)(22 79)(23 93)(24 81)(25 95)(26 83)(27 53)(28 67)(29 55)(30 69)(31 57)(32 71)(33 59)(34 73)(35 61)(36 75)(37 63)(38 77)(39 65)(40 66)(41 54)(42 68)(43 56)(44 70)(45 58)(46 72)(47 60)(48 74)(49 62)(50 76)(51 64)(52 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 3)(4 13)(5 12)(6 11)(7 10)(8 9)(14 24)(15 23)(16 22)(17 21)(18 20)(25 26)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 52)(37 51)(38 50)(39 49)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(75 78)(76 77)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(100 104)(101 103)```

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

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

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

77 conjugacy classes

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

77 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D13 D26 D26 D26 2+ 1+4 D4⋊8D26 kernel D4⋊8D26 C2×D52 D52⋊5C2 D4×D13 D52⋊C2 C13×C4○D4 C4○D4 C2×C4 D4 Q8 C13 C1 # reps 1 3 3 6 2 1 6 18 18 6 1 12

Matrix representation of D48D26 in GL4(𝔽53) generated by

 52 10 0 0 21 1 0 0 25 50 29 43 16 16 10 24
,
 52 10 19 40 21 1 34 33 25 50 29 43 16 16 10 24
,
 41 6 0 0 2 21 0 0 40 5 50 47 43 2 6 47
,
 1 0 0 0 32 52 0 0 28 35 39 34 37 17 27 14
`G:=sub<GL(4,GF(53))| [52,21,25,16,10,1,50,16,0,0,29,10,0,0,43,24],[52,21,25,16,10,1,50,16,19,34,29,10,40,33,43,24],[41,2,40,43,6,21,5,2,0,0,50,6,0,0,47,47],[1,32,28,37,0,52,35,17,0,0,39,27,0,0,34,14] >;`

D48D26 in GAP, Magma, Sage, TeX

`D_4\rtimes_8D_{26}`
`% in TeX`

`G:=Group("D4:8D26");`
`// GroupNames label`

`G:=SmallGroup(416,223);`
`// by ID`

`G=gap.SmallGroup(416,223);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,579,69,13829]);`
`// 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,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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