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## G = C52.D4order 416 = 25·13

### 8th non-split extension by C52 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52.D4
 Chief series C1 — C13 — C26 — C52 — C2×C52 — C52.4C4 — C52.D4
 Lower central C13 — C26 — C2×C26 — C52.D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C52.D4
G = < a,b,c | a52=1, b4=a26, c2=a13, bab-1=a-1, cac-1=a25, cbc-1=a39b3 >

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

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

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

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

71 conjugacy classes

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

71 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C4 D4 D13 D26 Dic13 C13⋊D4 C4.D4 C52.D4 kernel C52.D4 C52.4C4 D4×C26 C22×C26 C52 C2×D4 C2×C4 C23 C4 C13 C1 # reps 1 2 1 4 2 6 6 12 24 1 12

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

 4 57 50 120 235 309 93 101 0 0 0 280 0 0 33 0
,
 110 59 306 291 0 0 0 312 0 1 0 0 243 181 252 203
,
 203 136 159 22 0 0 0 1 0 1 0 0 70 132 61 110
`G:=sub<GL(4,GF(313))| [4,235,0,0,57,309,0,0,50,93,0,33,120,101,280,0],[110,0,0,243,59,0,1,181,306,0,0,252,291,312,0,203],[203,0,0,70,136,0,1,132,159,0,0,61,22,1,0,110] >;`

C52.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(416,40);`
`// by ID`

`G=gap.SmallGroup(416,40);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,86,579,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^13,b*a*b^-1=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^39*b^3>;`
`// generators/relations`

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