direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C39, C3⋊C156, C6.C78, C78.8S3, C32⋊2C52, C39⋊11C12, C78.13C6, C2.(S3×C39), (C3×C39)⋊10C4, C26.4(C3×S3), C6.4(S3×C13), (C3×C6).1C26, (C3×C78).4C2, SmallGroup(468,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C39 |
Generators and relations for Dic3×C39
G = < a,b,c | a39=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C39
►On 156 points
Generators in S156
(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)(118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 44 14 57 27 70)(2 45 15 58 28 71)(3 46 16 59 29 72)(4 47 17 60 30 73)(5 48 18 61 31 74)(6 49 19 62 32 75)(7 50 20 63 33 76)(8 51 21 64 34 77)(9 52 22 65 35 78)(10 53 23 66 36 40)(11 54 24 67 37 41)(12 55 25 68 38 42)(13 56 26 69 39 43)(79 123 105 149 92 136)(80 124 106 150 93 137)(81 125 107 151 94 138)(82 126 108 152 95 139)(83 127 109 153 96 140)(84 128 110 154 97 141)(85 129 111 155 98 142)(86 130 112 156 99 143)(87 131 113 118 100 144)(88 132 114 119 101 145)(89 133 115 120 102 146)(90 134 116 121 103 147)(91 135 117 122 104 148)
(1 151 57 81)(2 152 58 82)(3 153 59 83)(4 154 60 84)(5 155 61 85)(6 156 62 86)(7 118 63 87)(8 119 64 88)(9 120 65 89)(10 121 66 90)(11 122 67 91)(12 123 68 92)(13 124 69 93)(14 125 70 94)(15 126 71 95)(16 127 72 96)(17 128 73 97)(18 129 74 98)(19 130 75 99)(20 131 76 100)(21 132 77 101)(22 133 78 102)(23 134 40 103)(24 135 41 104)(25 136 42 105)(26 137 43 106)(27 138 44 107)(28 139 45 108)(29 140 46 109)(30 141 47 110)(31 142 48 111)(32 143 49 112)(33 144 50 113)(34 145 51 114)(35 146 52 115)(36 147 53 116)(37 148 54 117)(38 149 55 79)(39 150 56 80)
G:=sub<Sym(156)| (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)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,44,14,57,27,70)(2,45,15,58,28,71)(3,46,16,59,29,72)(4,47,17,60,30,73)(5,48,18,61,31,74)(6,49,19,62,32,75)(7,50,20,63,33,76)(8,51,21,64,34,77)(9,52,22,65,35,78)(10,53,23,66,36,40)(11,54,24,67,37,41)(12,55,25,68,38,42)(13,56,26,69,39,43)(79,123,105,149,92,136)(80,124,106,150,93,137)(81,125,107,151,94,138)(82,126,108,152,95,139)(83,127,109,153,96,140)(84,128,110,154,97,141)(85,129,111,155,98,142)(86,130,112,156,99,143)(87,131,113,118,100,144)(88,132,114,119,101,145)(89,133,115,120,102,146)(90,134,116,121,103,147)(91,135,117,122,104,148), (1,151,57,81)(2,152,58,82)(3,153,59,83)(4,154,60,84)(5,155,61,85)(6,156,62,86)(7,118,63,87)(8,119,64,88)(9,120,65,89)(10,121,66,90)(11,122,67,91)(12,123,68,92)(13,124,69,93)(14,125,70,94)(15,126,71,95)(16,127,72,96)(17,128,73,97)(18,129,74,98)(19,130,75,99)(20,131,76,100)(21,132,77,101)(22,133,78,102)(23,134,40,103)(24,135,41,104)(25,136,42,105)(26,137,43,106)(27,138,44,107)(28,139,45,108)(29,140,46,109)(30,141,47,110)(31,142,48,111)(32,143,49,112)(33,144,50,113)(34,145,51,114)(35,146,52,115)(36,147,53,116)(37,148,54,117)(38,149,55,79)(39,150,56,80)>;
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)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,44,14,57,27,70)(2,45,15,58,28,71)(3,46,16,59,29,72)(4,47,17,60,30,73)(5,48,18,61,31,74)(6,49,19,62,32,75)(7,50,20,63,33,76)(8,51,21,64,34,77)(9,52,22,65,35,78)(10,53,23,66,36,40)(11,54,24,67,37,41)(12,55,25,68,38,42)(13,56,26,69,39,43)(79,123,105,149,92,136)(80,124,106,150,93,137)(81,125,107,151,94,138)(82,126,108,152,95,139)(83,127,109,153,96,140)(84,128,110,154,97,141)(85,129,111,155,98,142)(86,130,112,156,99,143)(87,131,113,118,100,144)(88,132,114,119,101,145)(89,133,115,120,102,146)(90,134,116,121,103,147)(91,135,117,122,104,148), (1,151,57,81)(2,152,58,82)(3,153,59,83)(4,154,60,84)(5,155,61,85)(6,156,62,86)(7,118,63,87)(8,119,64,88)(9,120,65,89)(10,121,66,90)(11,122,67,91)(12,123,68,92)(13,124,69,93)(14,125,70,94)(15,126,71,95)(16,127,72,96)(17,128,73,97)(18,129,74,98)(19,130,75,99)(20,131,76,100)(21,132,77,101)(22,133,78,102)(23,134,40,103)(24,135,41,104)(25,136,42,105)(26,137,43,106)(27,138,44,107)(28,139,45,108)(29,140,46,109)(30,141,47,110)(31,142,48,111)(32,143,49,112)(33,144,50,113)(34,145,51,114)(35,146,52,115)(36,147,53,116)(37,148,54,117)(38,149,55,79)(39,150,56,80) );
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),(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,44,14,57,27,70),(2,45,15,58,28,71),(3,46,16,59,29,72),(4,47,17,60,30,73),(5,48,18,61,31,74),(6,49,19,62,32,75),(7,50,20,63,33,76),(8,51,21,64,34,77),(9,52,22,65,35,78),(10,53,23,66,36,40),(11,54,24,67,37,41),(12,55,25,68,38,42),(13,56,26,69,39,43),(79,123,105,149,92,136),(80,124,106,150,93,137),(81,125,107,151,94,138),(82,126,108,152,95,139),(83,127,109,153,96,140),(84,128,110,154,97,141),(85,129,111,155,98,142),(86,130,112,156,99,143),(87,131,113,118,100,144),(88,132,114,119,101,145),(89,133,115,120,102,146),(90,134,116,121,103,147),(91,135,117,122,104,148)], [(1,151,57,81),(2,152,58,82),(3,153,59,83),(4,154,60,84),(5,155,61,85),(6,156,62,86),(7,118,63,87),(8,119,64,88),(9,120,65,89),(10,121,66,90),(11,122,67,91),(12,123,68,92),(13,124,69,93),(14,125,70,94),(15,126,71,95),(16,127,72,96),(17,128,73,97),(18,129,74,98),(19,130,75,99),(20,131,76,100),(21,132,77,101),(22,133,78,102),(23,134,40,103),(24,135,41,104),(25,136,42,105),(26,137,43,106),(27,138,44,107),(28,139,45,108),(29,140,46,109),(30,141,47,110),(31,142,48,111),(32,143,49,112),(33,144,50,113),(34,145,51,114),(35,146,52,115),(36,147,53,116),(37,148,54,117),(38,149,55,79),(39,150,56,80)]])
234 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X | 39Y | ··· | 39BH | 52A | ··· | 52X | 78A | ··· | 78X | 78Y | ··· | 78BH | 156A | ··· | 156AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
234 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 | C12 | C13 | C26 | C39 | C52 | C78 | C156 | S3 | Dic3 | C3×S3 | C3×Dic3 | S3×C13 | Dic3×C13 | S3×C39 | Dic3×C39 |
| kernel | Dic3×C39 | C3×C78 | Dic3×C13 | C3×C39 | C78 | C39 | C3×Dic3 | C3×C6 | Dic3 | C32 | C6 | C3 | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 24 | 48 | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 |
Matrix representation of Dic3×C39 ►in GL3(𝔽157) generated by
| 12 | 0 | 0 |
| 0 | 17 | 0 |
| 0 | 0 | 17 |
| 156 | 0 | 0 |
| 0 | 144 | 0 |
| 0 | 101 | 12 |
| 129 | 0 | 0 |
| 0 | 113 | 143 |
| 0 | 127 | 44 |
G:=sub<GL(3,GF(157))| [12,0,0,0,17,0,0,0,17],[156,0,0,0,144,101,0,0,12],[129,0,0,0,113,127,0,143,44] >;
Dic3×C39 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{39} % in TeX
G:=Group("Dic3xC39"); // GroupNames label
G:=SmallGroup(468,24);
// by ID
G=gap.SmallGroup(468,24);
# by ID
G:=PCGroup([5,-2,-3,-13,-2,-3,390,7804]);
// Polycyclic
G:=Group<a,b,c|a^39=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export