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## G = C52⋊D9order 450 = 2·32·52

### The semidirect product of C52 and D9 acting via D9/C3=S3

Aliases: C52⋊D9, (C5×C15).S3, C52⋊C91C2, C3.(C52⋊S3), SmallGroup(450,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C9 — C52⋊D9
 Chief series C1 — C52 — C5×C15 — C52⋊C9 — C52⋊D9
 Lower central C52⋊C9 — C52⋊D9
 Upper central C1

Generators and relations for C52⋊D9
G = < a,b,c,d | a5=b5=c9=d2=1, cbc-1=ab=ba, cac-1=dad=a3b2, dbd=ab2, dcd=c-1 >

45C2
3C5
3C5
15S3
25C9
9D5
45C10
3C15
3C15
25D9
3D15
15C5×S3

Character table of C52⋊D9

 class 1 2 3 5A 5B 5C 5D 5E 5F 9A 9B 9C 10A 10B 10C 10D 15A 15B 15C 15D 15E 15F 15G 15H size 1 45 2 3 3 3 3 6 6 50 50 50 45 45 45 45 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 2 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 2 2 2 2 2 orthogonal lifted from S3 ρ4 2 0 -1 2 2 2 2 2 2 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from D9 ρ5 2 0 -1 2 2 2 2 2 2 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from D9 ρ6 2 0 -1 2 2 2 2 2 2 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from D9 ρ7 3 1 3 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 0 0 0 ζ52 ζ5 ζ54 ζ53 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ8 3 -1 3 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ9 3 1 3 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 0 0 0 ζ5 ζ53 ζ52 ζ54 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ10 3 -1 3 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ11 3 -1 3 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ12 3 1 3 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 0 0 0 ζ53 ζ54 ζ5 ζ52 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ13 3 -1 3 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ14 3 1 3 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 0 0 0 ζ54 ζ52 ζ53 ζ5 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ15 6 0 6 1+√5 1-√5 1+√5 1-√5 -3+√5/2 -3-√5/2 0 0 0 0 0 0 0 1-√5 1+√5 1+√5 1-√5 -3+√5/2 -3-√5/2 -3+√5/2 -3-√5/2 orthogonal lifted from C52⋊S3 ρ16 6 0 -3 1+√5 1-√5 1+√5 1-√5 -3+√5/2 -3-√5/2 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 orthogonal faithful ρ17 6 0 -3 1+√5 1-√5 1+√5 1-√5 -3+√5/2 -3-√5/2 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 orthogonal faithful ρ18 6 0 -3 1-√5 1+√5 1-√5 1+√5 -3-√5/2 -3+√5/2 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 orthogonal faithful ρ19 6 0 6 1-√5 1+√5 1-√5 1+√5 -3-√5/2 -3+√5/2 0 0 0 0 0 0 0 1+√5 1-√5 1-√5 1+√5 -3-√5/2 -3+√5/2 -3-√5/2 -3+√5/2 orthogonal lifted from C52⋊S3 ρ20 6 0 -3 1-√5 1+√5 1-√5 1+√5 -3-√5/2 -3+√5/2 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 orthogonal faithful ρ21 6 0 -3 4ζ52+2ζ5 4ζ54+2ζ52 2ζ54+4ζ53 2ζ53+4ζ5 1+√5 1-√5 0 0 0 0 0 0 0 -ζ53-2ζ5 -2ζ52-ζ5 -ζ54-2ζ53 -2ζ54-ζ52 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 complex faithful ρ22 6 0 -3 2ζ54+4ζ53 2ζ53+4ζ5 4ζ52+2ζ5 4ζ54+2ζ52 1+√5 1-√5 0 0 0 0 0 0 0 -2ζ54-ζ52 -ζ54-2ζ53 -2ζ52-ζ5 -ζ53-2ζ5 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 complex faithful ρ23 6 0 -3 4ζ54+2ζ52 2ζ54+4ζ53 2ζ53+4ζ5 4ζ52+2ζ5 1-√5 1+√5 0 0 0 0 0 0 0 -2ζ52-ζ5 -2ζ54-ζ52 -ζ53-2ζ5 -ζ54-2ζ53 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 complex faithful ρ24 6 0 -3 2ζ53+4ζ5 4ζ52+2ζ5 4ζ54+2ζ52 2ζ54+4ζ53 1-√5 1+√5 0 0 0 0 0 0 0 -ζ54-2ζ53 -ζ53-2ζ5 -2ζ54-ζ52 -2ζ52-ζ5 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 complex faithful

Smallest permutation representation of C52⋊D9
On 45 points
Generators in S45
```(1 42 32 11 26)(2 33 27 43 12)(3 34 19 44 13)(4 45 35 14 20)(5 36 21 37 15)(6 28 22 38 16)(7 39 29 17 23)(8 30 24 40 18)(9 31 25 41 10)
(1 32 26 42 11)(2 12 43 27 33)(4 35 20 45 14)(5 15 37 21 36)(7 29 23 39 17)(8 18 40 24 30)
(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)
(1 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 23)(20 22)(24 27)(25 26)(28 35)(29 34)(30 33)(31 32)(38 45)(39 44)(40 43)(41 42)```

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

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

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

Matrix representation of C52⋊D9 in GL5(𝔽181)

 1 0 0 0 0 0 1 0 0 0 0 0 135 0 102 0 0 0 125 0 0 0 0 0 125
,
 1 0 0 0 0 0 1 0 0 0 0 0 125 0 14 0 0 0 1 33 0 0 0 0 42
,
 127 131 0 0 0 50 177 0 0 0 0 0 161 1 38 0 0 33 0 100 0 0 41 0 20
,
 50 54 0 0 0 4 131 0 0 0 0 0 0 1 87 0 0 1 0 94 0 0 0 0 1

`G:=sub<GL(5,GF(181))| [1,0,0,0,0,0,1,0,0,0,0,0,135,0,0,0,0,0,125,0,0,0,102,0,125],[1,0,0,0,0,0,1,0,0,0,0,0,125,0,0,0,0,0,1,0,0,0,14,33,42],[127,50,0,0,0,131,177,0,0,0,0,0,161,33,41,0,0,1,0,0,0,0,38,100,20],[50,4,0,0,0,54,131,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,87,94,1] >;`

C52⋊D9 in GAP, Magma, Sage, TeX

`C_5^2\rtimes D_9`
`% in TeX`

`G:=Group("C5^2:D9");`
`// GroupNames label`

`G:=SmallGroup(450,11);`
`// by ID`

`G=gap.SmallGroup(450,11);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-5,5,101,66,182,2888,10804,4284]);`
`// Polycyclic`

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

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