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## G = D33⋊S3order 396 = 22·32·11

### The semidirect product of D33 and S3 acting via S3/C3=C2

Aliases: D33⋊S3, C333D6, C322D22, C112S32, C3⋊S3⋊D11, C33(S3×D11), (C3×D33)⋊3C2, (C3×C33)⋊4C22, (C11×C3⋊S3)⋊2C2, SmallGroup(396,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C33 — D33⋊S3
 Chief series C1 — C11 — C33 — C3×C33 — C3×D33 — D33⋊S3
 Lower central C3×C33 — D33⋊S3
 Upper central C1

Generators and relations for D33⋊S3
G = < a,b,c,d | a33=b2=c3=d2=1, bab=a-1, ac=ca, dad=a23, bc=cb, dbd=a22b, dcd=c-1 >

9C2
33C2
33C2
2C3
99C22
3S3
3S3
6S3
11S3
11S3
33C6
33C6
3D11
3D11
9C22
2C33
33D6
33D6
11C3×S3
11C3×S3
9D22
11S32

Smallest permutation representation of D33⋊S3
On 66 points
Generators in S66
```(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)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)
(1 23 12)(2 24 13)(3 25 14)(4 26 15)(5 27 16)(6 28 17)(7 29 18)(8 30 19)(9 31 20)(10 32 21)(11 33 22)(34 45 56)(35 46 57)(36 47 58)(37 48 59)(38 49 60)(39 50 61)(40 51 62)(41 52 63)(42 53 64)(43 54 65)(44 55 66)
(2 24)(3 14)(5 27)(6 17)(8 30)(9 20)(11 33)(12 23)(15 26)(18 29)(21 32)(35 57)(36 47)(38 60)(39 50)(41 63)(42 53)(44 66)(45 56)(48 59)(51 62)(54 65)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 11A ··· 11E 22A ··· 22E 33A ··· 33T order 1 2 2 2 3 3 3 6 6 11 ··· 11 22 ··· 22 33 ··· 33 size 1 9 33 33 2 2 4 66 66 2 ··· 2 18 ··· 18 4 ··· 4

39 irreducible representations

 dim 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + image C1 C2 C2 S3 D6 D11 D22 S32 S3×D11 D33⋊S3 kernel D33⋊S3 C3×D33 C11×C3⋊S3 D33 C33 C3⋊S3 C32 C11 C3 C1 # reps 1 2 1 2 2 5 5 1 10 10

Matrix representation of D33⋊S3 in GL6(𝔽67)

 9 0 0 0 0 0 31 15 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 66 0 0 0 0 1 66
,
 6 66 0 0 0 0 35 61 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 66 0 0 0 0 0 66
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 66 1 0 0 0 0 66 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 66 0 0 0 0 0 0 66 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(67))| [9,31,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,66,66],[6,35,0,0,0,0,66,61,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,66,66],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,66,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[66,0,0,0,0,0,0,66,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D33⋊S3 in GAP, Magma, Sage, TeX

`D_{33}\rtimes S_3`
`% in TeX`

`G:=Group("D33:S3");`
`// GroupNames label`

`G:=SmallGroup(396,23);`
`// by ID`

`G=gap.SmallGroup(396,23);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-11,122,67,248,9004]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^33=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^23,b*c=c*b,d*b*d=a^22*b,d*c*d=c^-1>;`
`// generators/relations`

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