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## G = D26.D4order 416 = 25·13

### 1st non-split extension by D26 of D4 acting via D4/C2=C22

Aliases: D26.1D4, (C2×C52)⋊1C4, C131(C23⋊C4), (C2×D52).2C2, D13.D41C2, (C22×D13)⋊2C4, C26.1(C22⋊C4), C2.4(D13.D4), (C22×D13).14C22, (C2×C4)⋊(C13⋊C4), (C2×C26).2(C2×C4), C22.2(C2×C13⋊C4), SmallGroup(416,74)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D26.D4
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=a12bc-1 >

2C2
26C2
26C2
52C2
2C4
13C22
13C22
26C22
52C4
52C4
52C22
52C22
2D13
2D13
2C26
4D13
13C23
13C23
26C2×C4
26D4
26D4
26C2×C4
2C52
2D26
4D26
4D26
13C2×D4
13C22⋊C4
13C22⋊C4
2D52
2D52
13C23⋊C4

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

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 2 2 2 4 4 4 4 4 13 13 13 26 ··· 26 52 ··· 52 size 1 1 2 26 26 52 4 52 52 52 52 4 4 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.D4 kernel D26.D4 D13.D4 C2×D52 C2×C52 C22×D13 D26 C13 C2×C4 C22 C2 C1 # reps 1 2 1 2 2 2 1 3 3 6 12

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

 12 39 11 46 7 8 8 7 46 11 39 12 41 15 49 8
,
 12 39 11 46 8 49 15 41 46 5 38 45 1 7 41 7
,
 16 29 47 5 23 18 43 28 47 17 14 28 41 17 19 5
,
 1 0 0 0 7 48 15 8 12 38 4 45 0 0 1 0
`G:=sub<GL(4,GF(53))| [12,7,46,41,39,8,11,15,11,8,39,49,46,7,12,8],[12,8,46,1,39,49,5,7,11,15,38,41,46,41,45,7],[16,23,47,41,29,18,17,17,47,43,14,19,5,28,28,5],[1,7,12,0,0,48,38,0,0,15,4,1,0,8,45,0] >;`

D26.D4 in GAP, Magma, Sage, TeX

`D_{26}.D_4`
`% in TeX`

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

`G:=SmallGroup(416,74);`
`// by ID`

`G=gap.SmallGroup(416,74);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,188,579,9221,3473]);`
`// 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^12*b*c^-1>;`
`// generators/relations`

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