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## G = D26.4D4order 416 = 25·13

### 4th non-split extension by D26 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — D26.4D4
 Chief series C1 — C13 — C26 — D26 — C22×D13 — D13.D4 — D26.4D4
 Lower central C13 — C26 — C2×C26 — D26.4D4
 Upper central C1 — C2 — C22 — C23

Generators and relations for D26.4D4
G = < a,b,c,d | a26=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a5, cbc-1=a17b, dbd-1=a4b, dcd-1=a-1bc-1 >

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

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4E 13A 13B 13C 26A ··· 26U order 1 2 2 2 2 2 4 ··· 4 13 13 13 26 ··· 26 size 1 1 2 4 26 26 52 ··· 52 4 4 4 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C4 C4 D4 C23⋊C4 C13⋊C4 C2×C13⋊C4 D13.D4 D26.4D4 kernel D26.4D4 D13.D4 C2×C13⋊D4 C2×Dic13 C22×C26 D26 C13 C23 C22 C2 C1 # reps 1 2 1 2 2 2 1 3 3 6 12

Matrix representation of D26.4D4 in GL4(𝔽53) generated by

 5 40 4 20 33 34 34 33 20 4 40 5 48 14 29 34
,
 17 34 8 40 3 40 9 36 47 24 43 50 10 13 36 6
,
 1 0 0 0 33 28 14 34 5 39 24 19 0 0 1 0
,
 17 49 8 29 18 28 18 24 32 11 32 24 48 18 39 29
`G:=sub<GL(4,GF(53))| [5,33,20,48,40,34,4,14,4,34,40,29,20,33,5,34],[17,3,47,10,34,40,24,13,8,9,43,36,40,36,50,6],[1,33,5,0,0,28,39,0,0,14,24,1,0,34,19,0],[17,18,32,48,49,28,11,18,8,18,32,39,29,24,24,29] >;`

D26.4D4 in GAP, Magma, Sage, TeX

`D_{26}._4D_4`
`% in TeX`

`G:=Group("D26.4D4");`
`// GroupNames label`

`G:=SmallGroup(416,86);`
`// by ID`

`G=gap.SmallGroup(416,86);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,579,9221,3473]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^17*b,d*b*d^-1=a^4*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

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