direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15×D13, C65⋊6C6, C39⋊2C10, C195⋊5C2, C13⋊3C30, SmallGroup(390,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C15×D13 |
Generators and relations for C15×D13
G = < a,b,c | a15=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C15×D13
►On 195 points
Generators in S195
(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)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)(166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)
(1 150 91 112 38 46 151 184 123 175 24 90 71)(2 136 92 113 39 47 152 185 124 176 25 76 72)(3 137 93 114 40 48 153 186 125 177 26 77 73)(4 138 94 115 41 49 154 187 126 178 27 78 74)(5 139 95 116 42 50 155 188 127 179 28 79 75)(6 140 96 117 43 51 156 189 128 180 29 80 61)(7 141 97 118 44 52 157 190 129 166 30 81 62)(8 142 98 119 45 53 158 191 130 167 16 82 63)(9 143 99 120 31 54 159 192 131 168 17 83 64)(10 144 100 106 32 55 160 193 132 169 18 84 65)(11 145 101 107 33 56 161 194 133 170 19 85 66)(12 146 102 108 34 57 162 195 134 171 20 86 67)(13 147 103 109 35 58 163 181 135 172 21 87 68)(14 148 104 110 36 59 164 182 121 173 22 88 69)(15 149 105 111 37 60 165 183 122 174 23 89 70)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 98)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 131)(32 132)(33 133)(34 134)(35 135)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 184)(47 185)(48 186)(49 187)(50 188)(51 189)(52 190)(53 191)(54 192)(55 193)(56 194)(57 195)(58 181)(59 182)(60 183)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(106 169)(107 170)(108 171)(109 172)(110 173)(111 174)(112 175)(113 176)(114 177)(115 178)(116 179)(117 180)(118 166)(119 167)(120 168)
G:=sub<Sym(195)| (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,150,91,112,38,46,151,184,123,175,24,90,71)(2,136,92,113,39,47,152,185,124,176,25,76,72)(3,137,93,114,40,48,153,186,125,177,26,77,73)(4,138,94,115,41,49,154,187,126,178,27,78,74)(5,139,95,116,42,50,155,188,127,179,28,79,75)(6,140,96,117,43,51,156,189,128,180,29,80,61)(7,141,97,118,44,52,157,190,129,166,30,81,62)(8,142,98,119,45,53,158,191,130,167,16,82,63)(9,143,99,120,31,54,159,192,131,168,17,83,64)(10,144,100,106,32,55,160,193,132,169,18,84,65)(11,145,101,107,33,56,161,194,133,170,19,85,66)(12,146,102,108,34,57,162,195,134,171,20,86,67)(13,147,103,109,35,58,163,181,135,172,21,87,68)(14,148,104,110,36,59,164,182,121,173,22,88,69)(15,149,105,111,37,60,165,183,122,174,23,89,70), (1,71)(2,72)(3,73)(4,74)(5,75)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,131)(32,132)(33,133)(34,134)(35,135)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,184)(47,185)(48,186)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,193)(56,194)(57,195)(58,181)(59,182)(60,183)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,166)(119,167)(120,168)>;
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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,150,91,112,38,46,151,184,123,175,24,90,71)(2,136,92,113,39,47,152,185,124,176,25,76,72)(3,137,93,114,40,48,153,186,125,177,26,77,73)(4,138,94,115,41,49,154,187,126,178,27,78,74)(5,139,95,116,42,50,155,188,127,179,28,79,75)(6,140,96,117,43,51,156,189,128,180,29,80,61)(7,141,97,118,44,52,157,190,129,166,30,81,62)(8,142,98,119,45,53,158,191,130,167,16,82,63)(9,143,99,120,31,54,159,192,131,168,17,83,64)(10,144,100,106,32,55,160,193,132,169,18,84,65)(11,145,101,107,33,56,161,194,133,170,19,85,66)(12,146,102,108,34,57,162,195,134,171,20,86,67)(13,147,103,109,35,58,163,181,135,172,21,87,68)(14,148,104,110,36,59,164,182,121,173,22,88,69)(15,149,105,111,37,60,165,183,122,174,23,89,70), (1,71)(2,72)(3,73)(4,74)(5,75)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,131)(32,132)(33,133)(34,134)(35,135)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,184)(47,185)(48,186)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,193)(56,194)(57,195)(58,181)(59,182)(60,183)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,166)(119,167)(120,168) );
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),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165),(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)], [(1,150,91,112,38,46,151,184,123,175,24,90,71),(2,136,92,113,39,47,152,185,124,176,25,76,72),(3,137,93,114,40,48,153,186,125,177,26,77,73),(4,138,94,115,41,49,154,187,126,178,27,78,74),(5,139,95,116,42,50,155,188,127,179,28,79,75),(6,140,96,117,43,51,156,189,128,180,29,80,61),(7,141,97,118,44,52,157,190,129,166,30,81,62),(8,142,98,119,45,53,158,191,130,167,16,82,63),(9,143,99,120,31,54,159,192,131,168,17,83,64),(10,144,100,106,32,55,160,193,132,169,18,84,65),(11,145,101,107,33,56,161,194,133,170,19,85,66),(12,146,102,108,34,57,162,195,134,171,20,86,67),(13,147,103,109,35,58,163,181,135,172,21,87,68),(14,148,104,110,36,59,164,182,121,173,22,88,69),(15,149,105,111,37,60,165,183,122,174,23,89,70)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,98),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,131),(32,132),(33,133),(34,134),(35,135),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,184),(47,185),(48,186),(49,187),(50,188),(51,189),(52,190),(53,191),(54,192),(55,193),(56,194),(57,195),(58,181),(59,182),(60,183),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141),(82,142),(83,143),(84,144),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(106,169),(107,170),(108,171),(109,172),(110,173),(111,174),(112,175),(113,176),(114,177),(115,178),(116,179),(117,180),(118,166),(119,167),(120,168)]])
120 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 13A | ··· | 13F | 15A | ··· | 15H | 30A | ··· | 30H | 39A | ··· | 39L | 65A | ··· | 65X | 195A | ··· | 195AV |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 13 | ··· | 13 | 15 | ··· | 15 | 30 | ··· | 30 | 39 | ··· | 39 | 65 | ··· | 65 | 195 | ··· | 195 |
| size | 1 | 13 | 1 | 1 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | D13 | C3×D13 | C5×D13 | C15×D13 |
| kernel | C15×D13 | C195 | C5×D13 | C3×D13 | C65 | C39 | D13 | C13 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 6 | 12 | 24 | 48 |
Matrix representation of C15×D13 ►in GL3(𝔽1171) generated by
| 750 | 0 | 0 |
| 0 | 1068 | 0 |
| 0 | 0 | 1068 |
| 1 | 0 | 0 |
| 0 | 82 | 1 |
| 0 | 1170 | 0 |
| 1170 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(1171))| [750,0,0,0,1068,0,0,0,1068],[1,0,0,0,82,1170,0,1,0],[1170,0,0,0,0,1,0,1,0] >;
C15×D13 in GAP, Magma, Sage, TeX
C_{15}\times D_{13} % in TeX
G:=Group("C15xD13"); // GroupNames label
G:=SmallGroup(390,5);
// by ID
G=gap.SmallGroup(390,5);
# by ID
G:=PCGroup([4,-2,-3,-5,-13,5763]);
// Polycyclic
G:=Group<a,b,c|a^15=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export