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## G = C52⋊C9order 225 = 32·52

### The semidirect product of C52 and C9 acting via C9/C3=C3

Aliases: C52⋊C9, (C5×C15).C3, C3.(C52⋊C3), SmallGroup(225,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C9
 Chief series C1 — C52 — C5×C15 — C52⋊C9
 Lower central C52 — C52⋊C9
 Upper central C1 — C3

Generators and relations for C52⋊C9
G = < a,b,c | a5=b5=c9=1, cbc-1=ab=ba, cac-1=a3b2 >

Smallest permutation representation of C52⋊C9
On 45 points
Generators in S45
```(2 33 10 41 27)(3 19 42 11 34)(5 36 13 44 21)(6 22 45 14 28)(8 30 16 38 24)(9 25 39 17 31)
(1 40 32 26 18)(2 41 33 27 10)(3 19 42 11 34)(4 43 35 20 12)(5 44 36 21 13)(6 22 45 14 28)(7 37 29 23 15)(8 38 30 24 16)(9 25 39 17 31)
(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)```

`G:=sub<Sym(45)| (2,33,10,41,27)(3,19,42,11,34)(5,36,13,44,21)(6,22,45,14,28)(8,30,16,38,24)(9,25,39,17,31), (1,40,32,26,18)(2,41,33,27,10)(3,19,42,11,34)(4,43,35,20,12)(5,44,36,21,13)(6,22,45,14,28)(7,37,29,23,15)(8,38,30,24,16)(9,25,39,17,31), (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)>;`

`G:=Group( (2,33,10,41,27)(3,19,42,11,34)(5,36,13,44,21)(6,22,45,14,28)(8,30,16,38,24)(9,25,39,17,31), (1,40,32,26,18)(2,41,33,27,10)(3,19,42,11,34)(4,43,35,20,12)(5,44,36,21,13)(6,22,45,14,28)(7,37,29,23,15)(8,38,30,24,16)(9,25,39,17,31), (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) );`

`G=PermutationGroup([[(2,33,10,41,27),(3,19,42,11,34),(5,36,13,44,21),(6,22,45,14,28),(8,30,16,38,24),(9,25,39,17,31)], [(1,40,32,26,18),(2,41,33,27,10),(3,19,42,11,34),(4,43,35,20,12),(5,44,36,21,13),(6,22,45,14,28),(7,37,29,23,15),(8,38,30,24,16),(9,25,39,17,31)], [(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)]])`

C52⋊C9 is a maximal subgroup of   C52⋊D9  C52⋊C18

33 conjugacy classes

 class 1 3A 3B 5A ··· 5H 9A ··· 9F 15A ··· 15P order 1 3 3 5 ··· 5 9 ··· 9 15 ··· 15 size 1 1 1 3 ··· 3 25 ··· 25 3 ··· 3

33 irreducible representations

 dim 1 1 1 3 3 type + image C1 C3 C9 C52⋊C3 C52⋊C9 kernel C52⋊C9 C5×C15 C52 C3 C1 # reps 1 2 6 8 16

Matrix representation of C52⋊C9 in GL4(𝔽181) generated by

 1 0 0 0 0 1 0 0 0 51 135 0 0 68 0 59
,
 1 0 0 0 0 125 0 0 0 174 135 0 0 0 0 125
,
 39 0 0 0 0 58 124 0 0 58 123 1 0 59 123 0
`G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,51,68,0,0,135,0,0,0,0,59],[1,0,0,0,0,125,174,0,0,0,135,0,0,0,0,125],[39,0,0,0,0,58,58,59,0,124,123,123,0,0,1,0] >;`

C52⋊C9 in GAP, Magma, Sage, TeX

`C_5^2\rtimes C_9`
`% in TeX`

`G:=Group("C5^2:C9");`
`// GroupNames label`

`G:=SmallGroup(225,3);`
`// by ID`

`G=gap.SmallGroup(225,3);`
`# by ID`

`G:=PCGroup([4,-3,-3,-5,5,12,1730,2739]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^5=c^9=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^3*b^2>;`
`// generators/relations`

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