direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C33, C3⋊C132, C6.C66, C33⋊3C12, C66.5C6, C66.8S3, C32⋊2C44, (C3×C33)⋊6C4, C2.(S3×C33), C6.4(S3×C11), C22.2(C3×S3), (C3×C6).1C22, (C3×C66).4C2, SmallGroup(396,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C33 |
Generators and relations for Dic3×C33
G = < a,b,c | a33=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C33
►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 52 23 41 12 63)(2 53 24 42 13 64)(3 54 25 43 14 65)(4 55 26 44 15 66)(5 56 27 45 16 34)(6 57 28 46 17 35)(7 58 29 47 18 36)(8 59 30 48 19 37)(9 60 31 49 20 38)(10 61 32 50 21 39)(11 62 33 51 22 40)(67 122 78 100 89 111)(68 123 79 101 90 112)(69 124 80 102 91 113)(70 125 81 103 92 114)(71 126 82 104 93 115)(72 127 83 105 94 116)(73 128 84 106 95 117)(74 129 85 107 96 118)(75 130 86 108 97 119)(76 131 87 109 98 120)(77 132 88 110 99 121)
(1 112 41 79)(2 113 42 80)(3 114 43 81)(4 115 44 82)(5 116 45 83)(6 117 46 84)(7 118 47 85)(8 119 48 86)(9 120 49 87)(10 121 50 88)(11 122 51 89)(12 123 52 90)(13 124 53 91)(14 125 54 92)(15 126 55 93)(16 127 56 94)(17 128 57 95)(18 129 58 96)(19 130 59 97)(20 131 60 98)(21 132 61 99)(22 100 62 67)(23 101 63 68)(24 102 64 69)(25 103 65 70)(26 104 66 71)(27 105 34 72)(28 106 35 73)(29 107 36 74)(30 108 37 75)(31 109 38 76)(32 110 39 77)(33 111 40 78)
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), (1,52,23,41,12,63)(2,53,24,42,13,64)(3,54,25,43,14,65)(4,55,26,44,15,66)(5,56,27,45,16,34)(6,57,28,46,17,35)(7,58,29,47,18,36)(8,59,30,48,19,37)(9,60,31,49,20,38)(10,61,32,50,21,39)(11,62,33,51,22,40)(67,122,78,100,89,111)(68,123,79,101,90,112)(69,124,80,102,91,113)(70,125,81,103,92,114)(71,126,82,104,93,115)(72,127,83,105,94,116)(73,128,84,106,95,117)(74,129,85,107,96,118)(75,130,86,108,97,119)(76,131,87,109,98,120)(77,132,88,110,99,121), (1,112,41,79)(2,113,42,80)(3,114,43,81)(4,115,44,82)(5,116,45,83)(6,117,46,84)(7,118,47,85)(8,119,48,86)(9,120,49,87)(10,121,50,88)(11,122,51,89)(12,123,52,90)(13,124,53,91)(14,125,54,92)(15,126,55,93)(16,127,56,94)(17,128,57,95)(18,129,58,96)(19,130,59,97)(20,131,60,98)(21,132,61,99)(22,100,62,67)(23,101,63,68)(24,102,64,69)(25,103,65,70)(26,104,66,71)(27,105,34,72)(28,106,35,73)(29,107,36,74)(30,108,37,75)(31,109,38,76)(32,110,39,77)(33,111,40,78)>;
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), (1,52,23,41,12,63)(2,53,24,42,13,64)(3,54,25,43,14,65)(4,55,26,44,15,66)(5,56,27,45,16,34)(6,57,28,46,17,35)(7,58,29,47,18,36)(8,59,30,48,19,37)(9,60,31,49,20,38)(10,61,32,50,21,39)(11,62,33,51,22,40)(67,122,78,100,89,111)(68,123,79,101,90,112)(69,124,80,102,91,113)(70,125,81,103,92,114)(71,126,82,104,93,115)(72,127,83,105,94,116)(73,128,84,106,95,117)(74,129,85,107,96,118)(75,130,86,108,97,119)(76,131,87,109,98,120)(77,132,88,110,99,121), (1,112,41,79)(2,113,42,80)(3,114,43,81)(4,115,44,82)(5,116,45,83)(6,117,46,84)(7,118,47,85)(8,119,48,86)(9,120,49,87)(10,121,50,88)(11,122,51,89)(12,123,52,90)(13,124,53,91)(14,125,54,92)(15,126,55,93)(16,127,56,94)(17,128,57,95)(18,129,58,96)(19,130,59,97)(20,131,60,98)(21,132,61,99)(22,100,62,67)(23,101,63,68)(24,102,64,69)(25,103,65,70)(26,104,66,71)(27,105,34,72)(28,106,35,73)(29,107,36,74)(30,108,37,75)(31,109,38,76)(32,110,39,77)(33,111,40,78) );
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)], [(1,52,23,41,12,63),(2,53,24,42,13,64),(3,54,25,43,14,65),(4,55,26,44,15,66),(5,56,27,45,16,34),(6,57,28,46,17,35),(7,58,29,47,18,36),(8,59,30,48,19,37),(9,60,31,49,20,38),(10,61,32,50,21,39),(11,62,33,51,22,40),(67,122,78,100,89,111),(68,123,79,101,90,112),(69,124,80,102,91,113),(70,125,81,103,92,114),(71,126,82,104,93,115),(72,127,83,105,94,116),(73,128,84,106,95,117),(74,129,85,107,96,118),(75,130,86,108,97,119),(76,131,87,109,98,120),(77,132,88,110,99,121)], [(1,112,41,79),(2,113,42,80),(3,114,43,81),(4,115,44,82),(5,116,45,83),(6,117,46,84),(7,118,47,85),(8,119,48,86),(9,120,49,87),(10,121,50,88),(11,122,51,89),(12,123,52,90),(13,124,53,91),(14,125,54,92),(15,126,55,93),(16,127,56,94),(17,128,57,95),(18,129,58,96),(19,130,59,97),(20,131,60,98),(21,132,61,99),(22,100,62,67),(23,101,63,68),(24,102,64,69),(25,103,65,70),(26,104,66,71),(27,105,34,72),(28,106,35,73),(29,107,36,74),(30,108,37,75),(31,109,38,76),(32,110,39,77),(33,111,40,78)]])
198 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 11A | ··· | 11J | 12A | 12B | 12C | 12D | 22A | ··· | 22J | 33A | ··· | 33T | 33U | ··· | 33AX | 44A | ··· | 44T | 66A | ··· | 66T | 66U | ··· | 66AX | 132A | ··· | 132AN |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
198 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C11 | C12 | C22 | C33 | C44 | C66 | C132 | S3 | Dic3 | C3×S3 | C3×Dic3 | S3×C11 | C11×Dic3 | S3×C33 | Dic3×C33 |
| kernel | Dic3×C33 | C3×C66 | C11×Dic3 | C3×C33 | C66 | C3×Dic3 | C33 | C3×C6 | Dic3 | C32 | C6 | C3 | C66 | C33 | C22 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 10 | 4 | 10 | 20 | 20 | 20 | 40 | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 |
Matrix representation of Dic3×C33 ►in GL3(𝔽397) generated by
| 362 | 0 | 0 |
| 0 | 147 | 0 |
| 0 | 0 | 147 |
| 396 | 0 | 0 |
| 0 | 362 | 0 |
| 0 | 0 | 34 |
| 334 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(397))| [362,0,0,0,147,0,0,0,147],[396,0,0,0,362,0,0,0,34],[334,0,0,0,0,1,0,1,0] >;
Dic3×C33 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{33} % in TeX
G:=Group("Dic3xC33"); // GroupNames label
G:=SmallGroup(396,12);
// by ID
G=gap.SmallGroup(396,12);
# by ID
G:=PCGroup([5,-2,-3,-11,-2,-3,330,6604]);
// Polycyclic
G:=Group<a,b,c|a^33=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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