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## G = Q8⋊D26order 416 = 25·13

### 2nd semidirect product of Q8 and D26 acting via D26/D13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Q8⋊D26
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D4×D13 — Q8⋊D26
 Lower central C13 — C26 — C52 — Q8⋊D26
 Upper central C1 — C2 — C4 — SD16

Generators and relations for Q8⋊D26
G = < a,b,c,d | a4=c26=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 632 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, SD16, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, C52, C52, D26, D26, C2×C26, C132C8, C104, C4×D13, C4×D13, D52, D52, C13⋊D4, D4×C13, Q8×C13, C22×D13, C8⋊D13, D104, D4⋊D13, Q8⋊D13, C13×SD16, D4×D13, D52⋊C2, Q8⋊D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C22×D13, D4×D13, Q8⋊D26

Smallest permutation representation of Q8⋊D26
On 104 points
Generators in S104
```(1 36 16 49)(2 50 17 37)(3 38 18 51)(4 52 19 39)(5 40 20 27)(6 28 21 41)(7 42 22 29)(8 30 23 43)(9 44 24 31)(10 32 25 45)(11 46 26 33)(12 34 14 47)(13 48 15 35)(53 66 96 83)(54 84 97 67)(55 68 98 85)(56 86 99 69)(57 70 100 87)(58 88 101 71)(59 72 102 89)(60 90 103 73)(61 74 104 91)(62 92 79 75)(63 76 80 93)(64 94 81 77)(65 78 82 95)
(1 65 16 82)(2 53 17 96)(3 67 18 84)(4 55 19 98)(5 69 20 86)(6 57 21 100)(7 71 22 88)(8 59 23 102)(9 73 24 90)(10 61 25 104)(11 75 26 92)(12 63 14 80)(13 77 15 94)(27 56 40 99)(28 87 41 70)(29 58 42 101)(30 89 43 72)(31 60 44 103)(32 91 45 74)(33 62 46 79)(34 93 47 76)(35 64 48 81)(36 95 49 78)(37 66 50 83)(38 97 51 54)(39 68 52 85)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(48 52)(49 51)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(101 104)(102 103)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 13A ··· 13F 26A ··· 26F 26G ··· 26L 52A ··· 52F 52G ··· 52L 104A ··· 104L order 1 2 2 2 2 2 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 4 26 52 52 2 4 26 4 52 2 ··· 2 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 D26 C8⋊C22 D4×D13 Q8⋊D26 kernel Q8⋊D26 C8⋊D13 D104 D4⋊D13 Q8⋊D13 C13×SD16 D4×D13 D52⋊C2 Dic13 D26 SD16 C8 D4 Q8 C13 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 6 6 6 6 1 6 12

Matrix representation of Q8⋊D26 in GL4(𝔽313) generated by

 1 0 108 79 0 1 306 161 291 108 312 0 244 98 0 312
,
 0 0 186 44 0 0 86 61 293 76 0 0 177 134 0 0
,
 208 187 184 215 179 182 217 249 0 0 304 126 0 0 216 245
,
 221 110 249 113 182 92 232 81 0 0 86 1 0 0 117 227
`G:=sub<GL(4,GF(313))| [1,0,291,244,0,1,108,98,108,306,312,0,79,161,0,312],[0,0,293,177,0,0,76,134,186,86,0,0,44,61,0,0],[208,179,0,0,187,182,0,0,184,217,304,216,215,249,126,245],[221,182,0,0,110,92,0,0,249,232,86,117,113,81,1,227] >;`

Q8⋊D26 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{26}`
`% in TeX`

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

`G:=SmallGroup(416,135);`
`// by ID`

`G=gap.SmallGroup(416,135);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,116,86,297,159,69,13829]);`
`// Polycyclic`

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

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