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## G = C3×D33order 198 = 2·32·11

### Direct product of C3 and D33

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

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

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

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 3A 3B 3C 3D 3E 6A 6B 11A ··· 11E 33A ··· 33AN order 1 2 3 3 3 3 3 6 6 11 ··· 11 33 ··· 33 size 1 33 1 1 2 2 2 33 33 2 ··· 2 2 ··· 2

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 C3×S3 D11 C3×D11 D33 C3×D33 kernel C3×D33 C3×C33 D33 C33 C33 C11 C32 C3 C3 C1 # reps 1 1 2 2 1 2 5 10 10 20

Matrix representation of C3×D33 in GL2(𝔽67) generated by

 29 0 0 29
,
 49 0 0 26
,
 0 26 49 0
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

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