metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic33, C33⋊1C4, C22.S3, C2.D33, C6.D11, C3⋊Dic11, C11⋊Dic3, C66.1C2, SmallGroup(132,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — Dic33 |
Generators and relations for Dic33
G = < a,b | a66=1, b2=a33, bab-1=a-1 >
Smallest permutation representation of Dic33
►Regular action on 132 points
Generators in S132
(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)(67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132)
(1 80 34 113)(2 79 35 112)(3 78 36 111)(4 77 37 110)(5 76 38 109)(6 75 39 108)(7 74 40 107)(8 73 41 106)(9 72 42 105)(10 71 43 104)(11 70 44 103)(12 69 45 102)(13 68 46 101)(14 67 47 100)(15 132 48 99)(16 131 49 98)(17 130 50 97)(18 129 51 96)(19 128 52 95)(20 127 53 94)(21 126 54 93)(22 125 55 92)(23 124 56 91)(24 123 57 90)(25 122 58 89)(26 121 59 88)(27 120 60 87)(28 119 61 86)(29 118 62 85)(30 117 63 84)(31 116 64 83)(32 115 65 82)(33 114 66 81)
G:=sub<Sym(132)| (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)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132), (1,80,34,113)(2,79,35,112)(3,78,36,111)(4,77,37,110)(5,76,38,109)(6,75,39,108)(7,74,40,107)(8,73,41,106)(9,72,42,105)(10,71,43,104)(11,70,44,103)(12,69,45,102)(13,68,46,101)(14,67,47,100)(15,132,48,99)(16,131,49,98)(17,130,50,97)(18,129,51,96)(19,128,52,95)(20,127,53,94)(21,126,54,93)(22,125,55,92)(23,124,56,91)(24,123,57,90)(25,122,58,89)(26,121,59,88)(27,120,60,87)(28,119,61,86)(29,118,62,85)(30,117,63,84)(31,116,64,83)(32,115,65,82)(33,114,66,81)>;
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)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132), (1,80,34,113)(2,79,35,112)(3,78,36,111)(4,77,37,110)(5,76,38,109)(6,75,39,108)(7,74,40,107)(8,73,41,106)(9,72,42,105)(10,71,43,104)(11,70,44,103)(12,69,45,102)(13,68,46,101)(14,67,47,100)(15,132,48,99)(16,131,49,98)(17,130,50,97)(18,129,51,96)(19,128,52,95)(20,127,53,94)(21,126,54,93)(22,125,55,92)(23,124,56,91)(24,123,57,90)(25,122,58,89)(26,121,59,88)(27,120,60,87)(28,119,61,86)(29,118,62,85)(30,117,63,84)(31,116,64,83)(32,115,65,82)(33,114,66,81) );
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),(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)], [(1,80,34,113),(2,79,35,112),(3,78,36,111),(4,77,37,110),(5,76,38,109),(6,75,39,108),(7,74,40,107),(8,73,41,106),(9,72,42,105),(10,71,43,104),(11,70,44,103),(12,69,45,102),(13,68,46,101),(14,67,47,100),(15,132,48,99),(16,131,49,98),(17,130,50,97),(18,129,51,96),(19,128,52,95),(20,127,53,94),(21,126,54,93),(22,125,55,92),(23,124,56,91),(24,123,57,90),(25,122,58,89),(26,121,59,88),(27,120,60,87),(28,119,61,86),(29,118,62,85),(30,117,63,84),(31,116,64,83),(32,115,65,82),(33,114,66,81)]])
Dic33 is a maximal subgroup of
Dic3×D11 S3×Dic11 C33⋊D4 C33⋊Q8 Dic66 C4×D33 C33⋊7D4 Dic99 C3⋊Dic33
Dic33 is a maximal quotient of
C33⋊C8 Dic99 C3⋊Dic33
36 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 11A | ··· | 11E | 22A | ··· | 22E | 33A | ··· | 33J | 66A | ··· | 66J |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 2 | 33 | 33 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D11 | Dic11 | D33 | Dic33 |
| kernel | Dic33 | C66 | C33 | C22 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 5 | 5 | 10 | 10 |
Matrix representation of Dic33 ►in GL2(𝔽397) generated by
| 21 | 287 |
| 110 | 388 |
| 45 | 348 |
| 252 | 352 |
G:=sub<GL(2,GF(397))| [21,110,287,388],[45,252,348,352] >;
Dic33 in GAP, Magma, Sage, TeX
{\rm Dic}_{33} % in TeX
G:=Group("Dic33"); // GroupNames label
G:=SmallGroup(132,3);
// by ID
G=gap.SmallGroup(132,3);
# by ID
G:=PCGroup([4,-2,-2,-3,-11,8,98,1923]);
// Polycyclic
G:=Group<a,b|a^66=1,b^2=a^33,b*a*b^-1=a^-1>;
// generators/relations
Export