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## G = C32⋊C54order 486 = 2·35

### The semidirect product of C32 and C54 acting via C54/C9=C6

Aliases: C32⋊C54, C33.1C18, C3⋊S3⋊C27, (C3×C27)⋊1S3, C3.2(S3×C27), C32⋊C271C2, (C32×C9).1C6, C9.6(C32⋊C6), C32.13(S3×C9), C3.5(C32⋊C18), (C3×C3⋊S3).C9, (C9×C3⋊S3).C3, (C3×C9).48(C3×S3), SmallGroup(486,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C54
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32⋊C27 — C32⋊C54
 Lower central C32 — C32⋊C54
 Upper central C1 — C9

Generators and relations for C32⋊C54
G = < a,b,c | a3=b3=c54=1, ab=ba, cac-1=a-1b-1, cbc-1=b-1 >

Smallest permutation representation of C32⋊C54
On 54 points
Generators in S54
```(2 20 38)(3 21 39)(5 41 23)(6 42 24)(8 26 44)(9 27 45)(11 47 29)(12 48 30)(14 32 50)(15 33 51)(17 53 35)(18 54 36)
(1 37 19)(2 20 38)(3 39 21)(4 22 40)(5 41 23)(6 24 42)(7 43 25)(8 26 44)(9 45 27)(10 28 46)(11 47 29)(12 30 48)(13 49 31)(14 32 50)(15 51 33)(16 34 52)(17 53 35)(18 36 54)
(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)```

`G:=sub<Sym(54)| (2,20,38)(3,21,39)(5,41,23)(6,42,24)(8,26,44)(9,27,45)(11,47,29)(12,48,30)(14,32,50)(15,33,51)(17,53,35)(18,54,36), (1,37,19)(2,20,38)(3,39,21)(4,22,40)(5,41,23)(6,24,42)(7,43,25)(8,26,44)(9,45,27)(10,28,46)(11,47,29)(12,30,48)(13,49,31)(14,32,50)(15,51,33)(16,34,52)(17,53,35)(18,36,54), (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)>;`

`G:=Group( (2,20,38)(3,21,39)(5,41,23)(6,42,24)(8,26,44)(9,27,45)(11,47,29)(12,48,30)(14,32,50)(15,33,51)(17,53,35)(18,54,36), (1,37,19)(2,20,38)(3,39,21)(4,22,40)(5,41,23)(6,24,42)(7,43,25)(8,26,44)(9,45,27)(10,28,46)(11,47,29)(12,30,48)(13,49,31)(14,32,50)(15,51,33)(16,34,52)(17,53,35)(18,36,54), (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) );`

`G=PermutationGroup([[(2,20,38),(3,21,39),(5,41,23),(6,42,24),(8,26,44),(9,27,45),(11,47,29),(12,48,30),(14,32,50),(15,33,51),(17,53,35),(18,54,36)], [(1,37,19),(2,20,38),(3,39,21),(4,22,40),(5,41,23),(6,24,42),(7,43,25),(8,26,44),(9,45,27),(10,28,46),(11,47,29),(12,30,48),(13,49,31),(14,32,50),(15,51,33),(16,34,52),(17,53,35),(18,36,54)], [(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)]])`

90 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A ··· 9F 9G ··· 9L 9M ··· 9R 18A ··· 18F 27A ··· 27R 27S ··· 27AJ 54A ··· 54R order 1 2 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 9 1 1 2 2 2 6 6 6 9 9 1 ··· 1 2 ··· 2 6 ··· 6 9 ··· 9 3 ··· 3 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 type + + + + image C1 C2 C3 C6 C9 C18 C27 C54 S3 C3×S3 S3×C9 S3×C27 C32⋊C6 C32⋊C18 C32⋊C54 kernel C32⋊C54 C32⋊C27 C9×C3⋊S3 C32×C9 C3×C3⋊S3 C33 C3⋊S3 C32 C3×C27 C3×C9 C32 C3 C9 C3 C1 # reps 1 1 2 2 6 6 18 18 1 2 6 18 1 2 6

Matrix representation of C32⋊C54 in GL6(𝔽109)

 1 0 0 0 0 0 90 45 0 0 0 0 5 0 63 0 0 0 0 0 0 1 0 0 107 0 0 0 63 0 102 0 0 0 0 45
,
 45 0 0 0 0 0 0 45 0 0 0 0 0 0 45 0 0 0 45 0 0 63 0 0 17 0 0 0 63 0 12 0 0 0 0 63
,
 83 0 0 0 0 70 59 0 0 38 0 55 29 0 0 0 38 26 0 0 66 0 0 43 59 0 0 0 0 55 29 38 0 0 0 26

`G:=sub<GL(6,GF(109))| [1,90,5,0,107,102,0,45,0,0,0,0,0,0,63,0,0,0,0,0,0,1,0,0,0,0,0,0,63,0,0,0,0,0,0,45],[45,0,0,45,17,12,0,45,0,0,0,0,0,0,45,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63],[83,59,29,0,59,29,0,0,0,0,0,38,0,0,0,66,0,0,0,38,0,0,0,0,0,0,38,0,0,0,70,55,26,43,55,26] >;`

C32⋊C54 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_{54}`
`% in TeX`

`G:=Group("C3^2:C54");`
`// GroupNames label`

`G:=SmallGroup(486,16);`
`// by ID`

`G=gap.SmallGroup(486,16);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,3244,3250,11669]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^3=c^54=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=b^-1>;`
`// generators/relations`

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