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## G = C52⋊5C16order 400 = 24·52

### 4th semidirect product of C52 and C16 acting via C16/C4=C4

Aliases: C525C16, C20.9F5, C52(C5⋊C16), C10.4(C5⋊C8), (C5×C10).5C8, (C5×C20).8C4, C2.(C525C8), C527C8.4C2, C4.2(C52⋊C4), SmallGroup(400,59)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊5C16
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C52⋊7C8 — C52⋊5C16
 Lower central C52 — C52⋊5C16
 Upper central C1 — C4

Generators and relations for C525C16
G = < a,b,c | a5=b5=c16=1, ab=ba, cac-1=a2, cbc-1=b3 >

Smallest permutation representation of C525C16
On 80 points
Generators in S80
```(1 20 62 69 44)(2 63 45 21 70)(3 46 71 64 22)(4 72 23 47 49)(5 24 50 73 48)(6 51 33 25 74)(7 34 75 52 26)(8 76 27 35 53)(9 28 54 77 36)(10 55 37 29 78)(11 38 79 56 30)(12 80 31 39 57)(13 32 58 65 40)(14 59 41 17 66)(15 42 67 60 18)(16 68 19 43 61)
(1 44 69 62 20)(2 63 45 21 70)(3 22 64 71 46)(4 72 23 47 49)(5 48 73 50 24)(6 51 33 25 74)(7 26 52 75 34)(8 76 27 35 53)(9 36 77 54 28)(10 55 37 29 78)(11 30 56 79 38)(12 80 31 39 57)(13 40 65 58 32)(14 59 41 17 66)(15 18 60 67 42)(16 68 19 43 61)
(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)```

`G:=sub<Sym(80)| (1,20,62,69,44)(2,63,45,21,70)(3,46,71,64,22)(4,72,23,47,49)(5,24,50,73,48)(6,51,33,25,74)(7,34,75,52,26)(8,76,27,35,53)(9,28,54,77,36)(10,55,37,29,78)(11,38,79,56,30)(12,80,31,39,57)(13,32,58,65,40)(14,59,41,17,66)(15,42,67,60,18)(16,68,19,43,61), (1,44,69,62,20)(2,63,45,21,70)(3,22,64,71,46)(4,72,23,47,49)(5,48,73,50,24)(6,51,33,25,74)(7,26,52,75,34)(8,76,27,35,53)(9,36,77,54,28)(10,55,37,29,78)(11,30,56,79,38)(12,80,31,39,57)(13,40,65,58,32)(14,59,41,17,66)(15,18,60,67,42)(16,68,19,43,61), (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)>;`

`G:=Group( (1,20,62,69,44)(2,63,45,21,70)(3,46,71,64,22)(4,72,23,47,49)(5,24,50,73,48)(6,51,33,25,74)(7,34,75,52,26)(8,76,27,35,53)(9,28,54,77,36)(10,55,37,29,78)(11,38,79,56,30)(12,80,31,39,57)(13,32,58,65,40)(14,59,41,17,66)(15,42,67,60,18)(16,68,19,43,61), (1,44,69,62,20)(2,63,45,21,70)(3,22,64,71,46)(4,72,23,47,49)(5,48,73,50,24)(6,51,33,25,74)(7,26,52,75,34)(8,76,27,35,53)(9,36,77,54,28)(10,55,37,29,78)(11,30,56,79,38)(12,80,31,39,57)(13,40,65,58,32)(14,59,41,17,66)(15,18,60,67,42)(16,68,19,43,61), (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) );`

`G=PermutationGroup([[(1,20,62,69,44),(2,63,45,21,70),(3,46,71,64,22),(4,72,23,47,49),(5,24,50,73,48),(6,51,33,25,74),(7,34,75,52,26),(8,76,27,35,53),(9,28,54,77,36),(10,55,37,29,78),(11,38,79,56,30),(12,80,31,39,57),(13,32,58,65,40),(14,59,41,17,66),(15,42,67,60,18),(16,68,19,43,61)], [(1,44,69,62,20),(2,63,45,21,70),(3,22,64,71,46),(4,72,23,47,49),(5,48,73,50,24),(6,51,33,25,74),(7,26,52,75,34),(8,76,27,35,53),(9,36,77,54,28),(10,55,37,29,78),(11,30,56,79,38),(12,80,31,39,57),(13,40,65,58,32),(14,59,41,17,66),(15,18,60,67,42),(16,68,19,43,61)], [(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)]])`

40 conjugacy classes

 class 1 2 4A 4B 5A ··· 5F 8A 8B 8C 8D 10A ··· 10F 16A ··· 16H 20A ··· 20L order 1 2 4 4 5 ··· 5 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 size 1 1 1 1 4 ··· 4 25 25 25 25 4 ··· 4 25 ··· 25 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 4 4 4 4 4 4 type + + + - + - image C1 C2 C4 C8 C16 F5 C5⋊C8 C5⋊C16 C52⋊C4 C52⋊5C8 C52⋊5C16 kernel C52⋊5C16 C52⋊7C8 C5×C20 C5×C10 C52 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 8 2 2 4 4 4 8

Matrix representation of C525C16 in GL5(𝔽241)

 1 0 0 0 0 0 51 240 0 0 0 1 0 0 0 0 0 0 240 51 0 0 0 190 190
,
 1 0 0 0 0 0 0 1 0 0 0 240 51 0 0 0 0 0 240 51 0 0 0 190 190
,
 197 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 207 240 0 0 0 193 34 0 0

`G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,51,1,0,0,0,240,0,0,0,0,0,0,240,190,0,0,0,51,190],[1,0,0,0,0,0,0,240,0,0,0,1,51,0,0,0,0,0,240,190,0,0,0,51,190],[197,0,0,0,0,0,0,0,207,193,0,0,0,240,34,0,1,0,0,0,0,0,1,0,0] >;`

C525C16 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_5C_{16}`
`% in TeX`

`G:=Group("C5^2:5C16");`
`// GroupNames label`

`G:=SmallGroup(400,59);`
`// by ID`

`G=gap.SmallGroup(400,59);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,1444,970,5765,5771]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;`
`// generators/relations`

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