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## G = S3×C33order 198 = 2·32·11

### Direct product of C33 and S3

Aliases: S3×C33, C3⋊C66, C333C6, C321C22, (C3×C33)⋊4C2, SmallGroup(198,6)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C33
G = < a,b,c | a33=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C33
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 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 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)

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

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

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

99 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 11A ··· 11J 22A ··· 22J 33A ··· 33T 33U ··· 33AX 66A ··· 66T order 1 2 3 3 3 3 3 6 6 11 ··· 11 22 ··· 22 33 ··· 33 33 ··· 33 66 ··· 66 size 1 3 1 1 2 2 2 3 3 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C11 C22 C33 C66 S3 C3×S3 S3×C11 S3×C33 kernel S3×C33 C3×C33 S3×C11 C33 C3×S3 C32 S3 C3 C33 C11 C3 C1 # reps 1 1 2 2 10 10 20 20 1 2 10 20

Matrix representation of S3×C33 in GL2(𝔽67) generated by

 4 0 0 4
,
 29 0 0 37
,
 0 1 1 0
G:=sub<GL(2,GF(67))| [4,0,0,4],[29,0,0,37],[0,1,1,0] >;

S3×C33 in GAP, Magma, Sage, TeX

S_3\times C_{33}
% in TeX

G:=Group("S3xC33");
// GroupNames label

G:=SmallGroup(198,6);
// by ID

G=gap.SmallGroup(198,6);
# by ID

G:=PCGroup([4,-2,-3,-11,-3,2115]);
// Polycyclic

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

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