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G = S3×C33order 198 = 2·32·11

Direct product of C33 and S3

direct product, metacyclic, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C3 — S3×C33
C1C3C33C3×C33 — S3×C33
C3 — S3×C33
C1C33

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

3C2
2C3
3C6
3C22
2C33
3C66

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

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

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

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

99 conjugacy classes

class 1  2 3A3B3C3D3E6A6B11A···11J22A···22J33A···33T33U···33AX66A···66T
order12333336611···1122···2233···3333···3366···66
size1311222331···13···31···12···23···3

99 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C11C22C33C66S3C3×S3S3×C11S3×C33
kernelS3×C33C3×C33S3×C11C33C3×S3C32S3C3C33C11C3C1
# reps112210102020121020

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

40
04
,
290
037
,
01
10
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

Export

Subgroup lattice of S3×C33 in TeX

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