direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D33, C33⋊1C6, C33⋊2S3, C32⋊1D11, C11⋊(C3×S3), C3⋊(C3×D11), (C3×C33)⋊2C2, SmallGroup(198,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C3×D33 |
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 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)]])
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
Export