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## G = C32⋊2D9.C3order 486 = 2·35

### 3rd non-split extension by C32⋊2D9 of C3 acting faithfully

Aliases: C32⋊C9.4S3, C32⋊C9.4C6, C33.9(C3×S3), C322D9.3C3, C33.7C321C2, C32.33(C32⋊C6), C3.5(He3.S3), C3.3(He3.C6), SmallGroup(486,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C32⋊2D9.C3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C33.7C32 — C32⋊2D9.C3
 Lower central C32⋊C9 — C32⋊2D9.C3
 Upper central C1 — C3

Generators and relations for C322D9.C3
G = < a,b,c,d,e | a3=b3=c9=d2=1, e3=b-1, ab=ba, cac-1=ab-1, dad=a-1b, eae-1=ac6, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=a-1b-1c4, ede-1=a-1b-1c3d >

Smallest permutation representation of C322D9.C3
On 54 points
Generators in S54
```(1 4 7)(3 9 6)(11 17 14)(12 15 18)(19 22 25)(21 27 24)(28 31 34)(29 35 32)(37 43 40)(39 42 45)(47 50 53)(48 54 51)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(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)
(1 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)
(1 26 17 7 23 14 4 20 11)(2 21 12 8 27 18 5 24 15)(3 22 10 9 19 16 6 25 13)(28 52 43 31 46 37 34 49 40)(29 50 44 32 53 38 35 47 41)(30 51 42 33 54 45 36 48 39)```

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

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

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

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A ··· 9F 9G ··· 9O 18A ··· 18F order 1 2 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 18 27 27 9 ··· 9 18 ··· 18 27 ··· 27

31 irreducible representations

 dim 1 1 1 1 2 2 3 6 6 6 type + + + + + image C1 C2 C3 C6 S3 C3×S3 He3.C6 C32⋊C6 He3.S3 C32⋊2D9.C3 kernel C32⋊2D9.C3 C33.7C32 C32⋊2D9 C32⋊C9 C32⋊C9 C33 C3 C32 C3 C1 # reps 1 1 2 2 1 2 12 1 3 6

Matrix representation of C322D9.C3 in GL6(𝔽19)

 11 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 0 1 0 0 0 0 0 0 1 0 0 0 11 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 11 0 0 0 0 0
,
 0 6 0 0 0 0 0 0 6 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 6 0 0

`G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,7,0,0],[0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,7,0,0,0,0,0],[0,0,6,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,6,0,0,0,0,0,0,6,0] >;`

C322D9.C3 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_2D_9.C_3`
`% in TeX`

`G:=Group("C3^2:2D9.C3");`
`// GroupNames label`

`G:=SmallGroup(486,12);`
`// by ID`

`G=gap.SmallGroup(486,12);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,2162,224,176,8643,873,1383,3244]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^9=d^2=1,e^3=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1*b,e*a*e^-1=a*c^6,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,e*c*e^-1=a^-1*b^-1*c^4,e*d*e^-1=a^-1*b^-1*c^3*d>;`
`// generators/relations`

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