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## G = C52.46D4order 416 = 25·13

### 3rd non-split extension by C52 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52.46D4
 Chief series C1 — C13 — C26 — C52 — C2×C52 — C2×D52 — C52.46D4
 Lower central C13 — C26 — C2×C26 — C52.46D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C52.46D4
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C52.46D4
On 104 points
Generators in S104
```(1 88 32 72 21 92 50 63)(2 89 33 73 22 93 51 64)(3 90 34 74 23 94 52 65)(4 91 35 75 24 95 40 53)(5 79 36 76 25 96 41 54)(6 80 37 77 26 97 42 55)(7 81 38 78 14 98 43 56)(8 82 39 66 15 99 44 57)(9 83 27 67 16 100 45 58)(10 84 28 68 17 101 46 59)(11 85 29 69 18 102 47 60)(12 86 30 70 19 103 48 61)(13 87 31 71 20 104 49 62)
(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(86 103)(87 104)(88 92)(89 93)(90 94)(91 95)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 41)(28 40)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(53 59)(54 58)(55 57)(60 65)(61 64)(62 63)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 104)(89 103)(90 102)(91 101)```

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

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

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

71 conjugacy classes

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

71 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D13 D26 D52 C13⋊D4 C4×D13 C4.D4 C52.46D4 kernel C52.46D4 C52.4C4 C13×M4(2) C2×D52 C22×D13 C52 M4(2) C2×C4 C4 C4 C22 C13 C1 # reps 1 1 1 1 4 2 6 6 12 12 12 1 12

Matrix representation of C52.46D4 in GL4(𝔽313) generated by

 155 296 311 0 90 221 0 311 59 137 158 17 222 101 223 92
,
 1 0 0 0 0 1 0 0 155 296 312 0 90 221 0 312
,
 6 1 0 0 277 255 0 0 261 198 6 1 127 52 277 255
,
 235 37 0 0 64 78 0 0 295 270 235 37 100 18 64 78
`G:=sub<GL(4,GF(313))| [155,90,59,222,296,221,137,101,311,0,158,223,0,311,17,92],[1,0,155,90,0,1,296,221,0,0,312,0,0,0,0,312],[6,277,261,127,1,255,198,52,0,0,6,277,0,0,1,255],[235,64,295,100,37,78,270,18,0,0,235,64,0,0,37,78] >;`

C52.46D4 in GAP, Magma, Sage, TeX

`C_{52}._{46}D_4`
`% in TeX`

`G:=Group("C52.46D4");`
`// GroupNames label`

`G:=SmallGroup(416,30);`
`// by ID`

`G=gap.SmallGroup(416,30);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,362,86,297,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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