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## G = S3×D33order 396 = 22·32·11

### Direct product of S3 and D33

Aliases: S3×D33, C332D6, C31D66, C321D22, C111S32, (S3×C11)⋊S3, (C3×S3)⋊D11, C3⋊D332C2, C31(S3×D11), (S3×C33)⋊1C2, (C3×D33)⋊2C2, (C3×C33)⋊3C22, SmallGroup(396,22)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×D33
G = < a,b,c,d | a3=b2=c33=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
33C2
99C2
2C3
99C22
3C6
11S3
33S3
33C6
33S3
66S3
3C22
3D11
9D11
2C33
33D6
33D6
11C3⋊S3
11C3×S3
9D22
3D33
3D33
3C66
6D33
11S32
3D66

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 11A ··· 11E 22A ··· 22E 33A ··· 33J 33K ··· 33Y 66A ··· 66J order 1 2 2 2 3 3 3 6 6 11 ··· 11 22 ··· 22 33 ··· 33 33 ··· 33 66 ··· 66 size 1 3 33 99 2 2 4 6 66 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D6 D11 D22 D33 D66 S32 S3×D11 S3×D33 kernel S3×D33 S3×C33 C3×D33 C3⋊D33 S3×C11 D33 C33 C3×S3 C32 S3 C3 C11 C3 C1 # reps 1 1 1 1 1 1 2 5 5 10 10 1 5 10

Matrix representation of S3×D33 in GL6(𝔽67)

 1 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 0 66 1 0 0 0 0 66 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 66 0 0 0 0 0 0 66 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 28 41 0 0 0 0 26 7 0 0 0 0 0 0 0 66 0 0 0 0 1 66 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 53 0 0 0 0 58 64 0 0 0 0 0 0 66 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(67))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,66,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,0,0,0,0,0,0,66,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[28,26,0,0,0,0,41,7,0,0,0,0,0,0,0,1,0,0,0,0,66,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,58,0,0,0,0,53,64,0,0,0,0,0,0,66,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×D33 in GAP, Magma, Sage, TeX

S_3\times D_{33}
% in TeX

G:=Group("S3xD33");
// GroupNames label

G:=SmallGroup(396,22);
// by ID

G=gap.SmallGroup(396,22);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-11,67,483,9004]);
// Polycyclic

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

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