direct product, abelian, monomial, 2-elementary
Aliases: C2×C130, SmallGroup(260,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C130 |
| C1 — C2×C130 |
| C1 — C2×C130 |
Generators and relations for C2×C130
G = < a,b | a2=b130=1, ab=ba >
Smallest permutation representation of C2×C130
►Regular action on 260 points
Generators in S260
(1 168)(2 169)(3 170)(4 171)(5 172)(6 173)(7 174)(8 175)(9 176)(10 177)(11 178)(12 179)(13 180)(14 181)(15 182)(16 183)(17 184)(18 185)(19 186)(20 187)(21 188)(22 189)(23 190)(24 191)(25 192)(26 193)(27 194)(28 195)(29 196)(30 197)(31 198)(32 199)(33 200)(34 201)(35 202)(36 203)(37 204)(38 205)(39 206)(40 207)(41 208)(42 209)(43 210)(44 211)(45 212)(46 213)(47 214)(48 215)(49 216)(50 217)(51 218)(52 219)(53 220)(54 221)(55 222)(56 223)(57 224)(58 225)(59 226)(60 227)(61 228)(62 229)(63 230)(64 231)(65 232)(66 233)(67 234)(68 235)(69 236)(70 237)(71 238)(72 239)(73 240)(74 241)(75 242)(76 243)(77 244)(78 245)(79 246)(80 247)(81 248)(82 249)(83 250)(84 251)(85 252)(86 253)(87 254)(88 255)(89 256)(90 257)(91 258)(92 259)(93 260)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 161)(125 162)(126 163)(127 164)(128 165)(129 166)(130 167)
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)
G:=sub<Sym(260)| (1,168)(2,169)(3,170)(4,171)(5,172)(6,173)(7,174)(8,175)(9,176)(10,177)(11,178)(12,179)(13,180)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,188)(22,189)(23,190)(24,191)(25,192)(26,193)(27,194)(28,195)(29,196)(30,197)(31,198)(32,199)(33,200)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,208)(42,209)(43,210)(44,211)(45,212)(46,213)(47,214)(48,215)(49,216)(50,217)(51,218)(52,219)(53,220)(54,221)(55,222)(56,223)(57,224)(58,225)(59,226)(60,227)(61,228)(62,229)(63,230)(64,231)(65,232)(66,233)(67,234)(68,235)(69,236)(70,237)(71,238)(72,239)(73,240)(74,241)(75,242)(76,243)(77,244)(78,245)(79,246)(80,247)(81,248)(82,249)(83,250)(84,251)(85,252)(86,253)(87,254)(88,255)(89,256)(90,257)(91,258)(92,259)(93,260)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,138)(102,139)(103,140)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,161)(125,162)(126,163)(127,164)(128,165)(129,166)(130,167), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)>;
G:=Group( (1,168)(2,169)(3,170)(4,171)(5,172)(6,173)(7,174)(8,175)(9,176)(10,177)(11,178)(12,179)(13,180)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,188)(22,189)(23,190)(24,191)(25,192)(26,193)(27,194)(28,195)(29,196)(30,197)(31,198)(32,199)(33,200)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,208)(42,209)(43,210)(44,211)(45,212)(46,213)(47,214)(48,215)(49,216)(50,217)(51,218)(52,219)(53,220)(54,221)(55,222)(56,223)(57,224)(58,225)(59,226)(60,227)(61,228)(62,229)(63,230)(64,231)(65,232)(66,233)(67,234)(68,235)(69,236)(70,237)(71,238)(72,239)(73,240)(74,241)(75,242)(76,243)(77,244)(78,245)(79,246)(80,247)(81,248)(82,249)(83,250)(84,251)(85,252)(86,253)(87,254)(88,255)(89,256)(90,257)(91,258)(92,259)(93,260)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,138)(102,139)(103,140)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,161)(125,162)(126,163)(127,164)(128,165)(129,166)(130,167), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260) );
G=PermutationGroup([[(1,168),(2,169),(3,170),(4,171),(5,172),(6,173),(7,174),(8,175),(9,176),(10,177),(11,178),(12,179),(13,180),(14,181),(15,182),(16,183),(17,184),(18,185),(19,186),(20,187),(21,188),(22,189),(23,190),(24,191),(25,192),(26,193),(27,194),(28,195),(29,196),(30,197),(31,198),(32,199),(33,200),(34,201),(35,202),(36,203),(37,204),(38,205),(39,206),(40,207),(41,208),(42,209),(43,210),(44,211),(45,212),(46,213),(47,214),(48,215),(49,216),(50,217),(51,218),(52,219),(53,220),(54,221),(55,222),(56,223),(57,224),(58,225),(59,226),(60,227),(61,228),(62,229),(63,230),(64,231),(65,232),(66,233),(67,234),(68,235),(69,236),(70,237),(71,238),(72,239),(73,240),(74,241),(75,242),(76,243),(77,244),(78,245),(79,246),(80,247),(81,248),(82,249),(83,250),(84,251),(85,252),(86,253),(87,254),(88,255),(89,256),(90,257),(91,258),(92,259),(93,260),(94,131),(95,132),(96,133),(97,134),(98,135),(99,136),(100,137),(101,138),(102,139),(103,140),(104,141),(105,142),(106,143),(107,144),(108,145),(109,146),(110,147),(111,148),(112,149),(113,150),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,157),(121,158),(122,159),(123,160),(124,161),(125,162),(126,163),(127,164),(128,165),(129,166),(130,167)], [(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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)]])
260 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 13A | ··· | 13L | 26A | ··· | 26AJ | 65A | ··· | 65AV | 130A | ··· | 130EN |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 13 | ··· | 13 | 26 | ··· | 26 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
260 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C10 | C13 | C26 | C65 | C130 |
| kernel | C2×C130 | C130 | C2×C26 | C26 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 3 | 4 | 12 | 12 | 36 | 48 | 144 |
Matrix representation of C2×C130 ►in GL2(𝔽131) generated by
| 130 | 0 |
| 0 | 1 |
| 26 | 0 |
| 0 | 51 |
G:=sub<GL(2,GF(131))| [130,0,0,1],[26,0,0,51] >;
C2×C130 in GAP, Magma, Sage, TeX
C_2\times C_{130} % in TeX
G:=Group("C2xC130"); // GroupNames label
G:=SmallGroup(260,15);
// by ID
G=gap.SmallGroup(260,15);
# by ID
G:=PCGroup([4,-2,-2,-5,-13]);
// Polycyclic
G:=Group<a,b|a^2=b^130=1,a*b=b*a>;
// generators/relations
Export