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G = C3xD33order 198 = 2·32·11

Direct product of C3 and D33

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

Aliases: C3xD33, C33:1C6, C33:2S3, C32:1D11, C11:(C3xS3), C3:(C3xD11), (C3xC33):2C2, SmallGroup(198,7)

Series: Derived Chief Lower central Upper central

C1C33 — C3xD33
C1C11C33C3xC33 — C3xD33
C33 — C3xD33
C1C3

Generators and relations for C3xD33
 G = < a,b,c | a3=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 108 in 18 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C3, S3, C6, C3xS3, D11, C3xD11, D33, C3xD33
33C2
2C3
11S3
33C6
3D11
2C33
11C3xS3
3C3xD11

Smallest permutation representation of C3xD33
On 66 points
Generators in S66
(1 12 23)(2 13 24)(3 14 25)(4 15 26)(5 16 27)(6 17 28)(7 18 29)(8 19 30)(9 20 31)(10 21 32)(11 22 33)(34 56 45)(35 57 46)(36 58 47)(37 59 48)(38 60 49)(39 61 50)(40 62 51)(41 63 52)(42 64 53)(43 65 54)(44 66 55)
(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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)

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

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

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

C3xD33 is a maximal subgroup of   C3xS3xD11  D33:S3

54 conjugacy classes

class 1  2 3A3B3C3D3E6A6B11A···11E33A···33AN
order12333336611···1133···33
size1331122233332···22···2

54 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3C3xS3D11C3xD11D33C3xD33
kernelC3xD33C3xC33D33C33C33C11C32C3C3C1
# reps1122125101020

Matrix representation of C3xD33 in GL2(F67) generated by

290
029
,
490
026
,
026
490
G:=sub<GL(2,GF(67))| [29,0,0,29],[49,0,0,26],[0,49,26,0] >;

C3xD33 in GAP, Magma, Sage, TeX

C_3\times D_{33}
% in TeX

G:=Group("C3xD33");
// GroupNames label

G:=SmallGroup(198,7);
// by ID

G=gap.SmallGroup(198,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-11,146,2883]);
// Polycyclic

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

Export

Subgroup lattice of C3xD33 in TeX

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