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G = C3×D33order 198 = 2·32·11

Direct product of C3 and D33

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

Aliases: C3×D33, C331C6, C332S3, C321D11, C11⋊(C3×S3), C3⋊(C3×D11), (C3×C33)⋊2C2, SmallGroup(198,7)

Series: Derived Chief Lower central Upper central

C1C33 — C3×D33
C1C11C33C3×C33 — C3×D33
C33 — C3×D33
C1C3

Generators and relations for C3×D33
 G = < a,b,c | a3=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >

33C2
2C3
11S3
33C6
3D11
2C33
11C3×S3
3C3×D11

Smallest permutation representation of C3×D33
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 36)(2 35)(3 34)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)

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

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

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

C3×D33 is a maximal subgroup of   C3×S3×D11  D33⋊S3

54 conjugacy classes

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

54 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3C3×S3D11C3×D11D33C3×D33
kernelC3×D33C3×C33D33C33C33C11C32C3C3C1
# reps1122125101020

Matrix representation of C3×D33 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] >;

C3×D33 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 C3×D33 in TeX

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