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## G = D26.Q8order 416 = 25·13

### 3rd non-split extension by D26 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D26.Q8
 Chief series C1 — C13 — D13 — D26 — C22×D13 — C22×C13⋊C4 — D26.Q8
 Lower central C13 — C26 — D26.Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for D26.Q8
G = < a,b,c,d | a26=b2=c4=1, d2=a12bc2, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=a13c-1 >

Subgroups: 604 in 76 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×6], C2×C4, C2×C4 [×11], C23, C13, C22×C4 [×3], D13 [×4], C26 [×3], C2.C42, Dic13, C52, C13⋊C4 [×4], D26 [×2], D26 [×4], C2×C26, C4×D13 [×2], C2×Dic13, C2×C52, C2×C13⋊C4 [×4], C2×C13⋊C4 [×4], C22×D13, C2×C4×D13, C22×C13⋊C4 [×2], D26.Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C13⋊C4, C2×C13⋊C4, C4×C13⋊C4, C52⋊C4, D13.D4, D26.Q8

Smallest permutation representation of D26.Q8
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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 67)(28 66)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)
(1 36 104 72)(2 37 79 73)(3 38 80 74)(4 39 81 75)(5 40 82 76)(6 41 83 77)(7 42 84 78)(8 43 85 53)(9 44 86 54)(10 45 87 55)(11 46 88 56)(12 47 89 57)(13 48 90 58)(14 49 91 59)(15 50 92 60)(16 51 93 61)(17 52 94 62)(18 27 95 63)(19 28 96 64)(20 29 97 65)(21 30 98 66)(22 31 99 67)(23 32 100 68)(24 33 101 69)(25 34 102 70)(26 35 103 71)
(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 78 45 66)(28 73 44 71)(29 68 43 76)(30 63 42 55)(31 58 41 60)(32 53 40 65)(33 74 39 70)(34 69 38 75)(35 64 37 54)(36 59)(46 61 52 57)(47 56 51 62)(48 77 50 67)(49 72)(79 99 103 83)(80 94 102 88)(81 89 101 93)(82 84 100 98)(85 95 97 87)(86 90 96 92)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C ··· 4L 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 2 2 2 2 2 4 4 4 ··· 4 13 13 13 26 ··· 26 52 ··· 52 size 1 1 1 1 13 13 13 13 2 2 26 ··· 26 4 4 4 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + - + + + image C1 C2 C2 C4 C4 C4 D4 Q8 C13⋊C4 C2×C13⋊C4 C4×C13⋊C4 C52⋊C4 D13.D4 kernel D26.Q8 C2×C4×D13 C22×C13⋊C4 C2×Dic13 C2×C52 C2×C13⋊C4 D26 D26 C2×C4 C22 C2 C2 C2 # reps 1 1 2 2 2 8 3 1 3 3 6 6 6

Matrix representation of D26.Q8 in GL8(𝔽53)

 52 0 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 22 28 23 14 0 0 0 0 39 38 38 39 0 0 0 0 14 23 28 22 0 0 0 0 31 24 16 38
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 22 28 23 14 0 0 0 0 38 16 24 31 0 0 0 0 14 36 29 15 0 0 0 0 52 39 31 39
,
 32 43 0 0 0 0 0 0 44 21 0 0 0 0 0 0 0 0 10 40 0 0 0 0 0 0 20 43 0 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 0 52
,
 1 0 0 0 0 0 0 0 17 52 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 15 30 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 14 36 29 15 0 0 0 0 31 24 16 38 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(53))| [52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,39,14,31,0,0,0,0,28,38,23,24,0,0,0,0,23,38,28,16,0,0,0,0,14,39,22,38],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,38,14,52,0,0,0,0,28,16,36,39,0,0,0,0,23,24,29,31,0,0,0,0,14,31,15,39],[32,44,0,0,0,0,0,0,43,21,0,0,0,0,0,0,0,0,10,20,0,0,0,0,0,0,40,43,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52],[1,17,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,23,15,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,14,31,0,0,0,0,0,0,36,24,0,0,0,0,0,0,29,16,1,0,0,0,0,0,15,38,0] >;`

D26.Q8 in GAP, Magma, Sage, TeX

`D_{26}.Q_8`
`% in TeX`

`G:=Group("D26.Q8");`
`// GroupNames label`

`G:=SmallGroup(416,81);`
`// by ID`

`G=gap.SmallGroup(416,81);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,9221,3473]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^12*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^13*c^-1>;`
`// generators/relations`

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