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## G = D4⋊6D26order 416 = 25·13

### 2nd semidirect product of D4 and D26 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4⋊6D26
 Chief series C1 — C13 — C26 — D26 — C22×D13 — D4×D13 — D4⋊6D26
 Lower central C13 — C26 — D4⋊6D26
 Upper central C1 — C2 — C2×D4

Generators and relations for D46D26
G = < a,b,c,d | a4=b2=c26=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1120 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C13, C2×D4, C2×D4, C4○D4, D13, C26, C26, 2+ 1+4, Dic13, C52, D26, D26, C2×C26, C2×C26, C2×C26, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, D4×C13, C22×D13, C22×C26, D525C2, D4×D13, D42D13, C2×C13⋊D4, D4×C26, D46D26
Quotients: C1, C2, C22, C23, C24, D13, 2+ 1+4, D26, C22×D13, C23×D13, D46D26

Smallest permutation representation of D46D26
On 104 points
Generators in S104
```(1 85 26 98)(2 99 14 86)(3 87 15 100)(4 101 16 88)(5 89 17 102)(6 103 18 90)(7 91 19 104)(8 79 20 92)(9 93 21 80)(10 81 22 94)(11 95 23 82)(12 83 24 96)(13 97 25 84)(27 63 43 76)(28 77 44 64)(29 65 45 78)(30 53 46 66)(31 67 47 54)(32 55 48 68)(33 69 49 56)(34 57 50 70)(35 71 51 58)(36 59 52 72)(37 73 40 60)(38 61 41 74)(39 75 42 62)
(1 68)(2 56)(3 70)(4 58)(5 72)(6 60)(7 74)(8 62)(9 76)(10 64)(11 78)(12 66)(13 54)(14 69)(15 57)(16 71)(17 59)(18 73)(19 61)(20 75)(21 63)(22 77)(23 65)(24 53)(25 67)(26 55)(27 80)(28 94)(29 82)(30 96)(31 84)(32 98)(33 86)(34 100)(35 88)(36 102)(37 90)(38 104)(39 92)(40 103)(41 91)(42 79)(43 93)(44 81)(45 95)(46 83)(47 97)(48 85)(49 99)(50 87)(51 101)(52 89)
(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 2)(3 13)(4 12)(5 11)(6 10)(7 9)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(27 41)(28 40)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(53 58)(54 57)(55 56)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)```

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

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

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

77 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 13A ··· 13F 26A ··· 26R 26S ··· 26AP 52A ··· 52L order 1 2 2 ··· 2 2 2 2 2 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 2 ··· 2 26 26 26 26 2 2 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 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⋊6D26 kernel D4⋊6D26 D52⋊5C2 D4×D13 D4⋊2D13 C2×C13⋊D4 D4×C26 C2×D4 C2×C4 D4 C23 C13 C1 # reps 1 2 4 4 4 1 6 6 24 12 1 12

Matrix representation of D46D26 in GL4(𝔽53) generated by

 0 0 13 35 0 0 27 40 40 18 0 0 26 13 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 9 6 0 0 44 0 0 0 0 0 44 47 0 0 9 0
,
 41 14 0 0 39 12 0 0 0 0 12 39 0 0 14 41
`G:=sub<GL(4,GF(53))| [0,0,40,26,0,0,18,13,13,27,0,0,35,40,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[9,44,0,0,6,0,0,0,0,0,44,9,0,0,47,0],[41,39,0,0,14,12,0,0,0,0,12,14,0,0,39,41] >;`

D46D26 in GAP, Magma, Sage, TeX

`D_4\rtimes_6D_{26}`
`% in TeX`

`G:=Group("D4:6D26");`
`// GroupNames label`

`G:=SmallGroup(416,218);`
`// by ID`

`G=gap.SmallGroup(416,218);`
`# by ID`

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

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

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