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## G = D52⋊6C22order 416 = 25·13

### 4th semidirect product of D52 and C22 acting via C22/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D52⋊6C22
 Chief series C1 — C13 — C26 — C52 — D52 — D52⋊5C2 — D52⋊6C22
 Lower central C13 — C26 — C52 — D52⋊6C22
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for D526C22
G = < a,b,c,d | a52=b2=c2=d2=1, bab=a-1, ac=ca, dad=a27, cbc=a26b, dbd=a39b, cd=dc >

Subgroups: 392 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, C52, D26, C2×C26, C2×C26, C132C8, Dic26, C4×D13, D52, C13⋊D4, C2×C52, D4×C13, D4×C13, C22×C26, C52.4C4, D4⋊D13, D4.D13, D525C2, D4×C26, D526C22
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C13⋊D4, C22×D13, C2×C13⋊D4, D526C22

Smallest permutation representation of D526C22
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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 104)(13 103)(14 102)(15 101)(16 100)(17 99)(18 98)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 80)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)
(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
(1 40)(2 15)(3 42)(4 17)(5 44)(6 19)(7 46)(8 21)(9 48)(10 23)(11 50)(12 25)(13 52)(14 27)(16 29)(18 31)(20 33)(22 35)(24 37)(26 39)(28 41)(30 43)(32 45)(34 47)(36 49)(38 51)(53 79)(55 81)(57 83)(59 85)(61 87)(63 89)(65 91)(67 93)(69 95)(71 97)(73 99)(75 101)(77 103)```

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

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

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

71 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 13A ··· 13F 26A ··· 26R 26S ··· 26AP 52A ··· 52L order 1 2 2 2 2 2 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 2 4 4 52 2 2 52 52 52 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

71 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 C13⋊D4 C13⋊D4 C8⋊C22 D52⋊6C22 kernel D52⋊6C22 C52.4C4 D4⋊D13 D4.D13 D52⋊5C2 D4×C26 C52 C2×C26 C2×D4 C2×C4 D4 C4 C22 C13 C1 # reps 1 1 2 2 1 1 1 1 6 6 12 12 12 1 12

Matrix representation of D526C22 in GL4(𝔽313) generated by

 58 140 0 0 256 255 0 0 221 99 286 175 290 229 65 27
,
 221 99 286 175 56 35 237 84 58 140 0 0 165 61 154 57
,
 1 0 0 0 0 1 0 0 0 0 312 0 103 260 0 312
,
 1 24 0 0 0 312 0 0 0 0 1 0 103 297 299 312
`G:=sub<GL(4,GF(313))| [58,256,221,290,140,255,99,229,0,0,286,65,0,0,175,27],[221,56,58,165,99,35,140,61,286,237,0,154,175,84,0,57],[1,0,0,103,0,1,0,260,0,0,312,0,0,0,0,312],[1,0,0,103,24,312,0,297,0,0,1,299,0,0,0,312] >;`

D526C22 in GAP, Magma, Sage, TeX

`D_{52}\rtimes_6C_2^2`
`% in TeX`

`G:=Group("D52:6C2^2");`
`// GroupNames label`

`G:=SmallGroup(416,153);`
`// by ID`

`G=gap.SmallGroup(416,153);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,579,159,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^52=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^27,c*b*c=a^26*b,d*b*d=a^39*b,c*d=d*c>;`
`// generators/relations`

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