direct product, abelian, monomial, 2-elementary
Aliases: C2×C180, SmallGroup(360,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C180 |
| C1 — C2×C180 |
| C1 — C2×C180 |
Generators and relations for C2×C180
G = < a,b | a2=b180=1, ab=ba >
Smallest permutation representation of C2×C180
►Regular action on 360 points
Generators in S360
(1 209)(2 210)(3 211)(4 212)(5 213)(6 214)(7 215)(8 216)(9 217)(10 218)(11 219)(12 220)(13 221)(14 222)(15 223)(16 224)(17 225)(18 226)(19 227)(20 228)(21 229)(22 230)(23 231)(24 232)(25 233)(26 234)(27 235)(28 236)(29 237)(30 238)(31 239)(32 240)(33 241)(34 242)(35 243)(36 244)(37 245)(38 246)(39 247)(40 248)(41 249)(42 250)(43 251)(44 252)(45 253)(46 254)(47 255)(48 256)(49 257)(50 258)(51 259)(52 260)(53 261)(54 262)(55 263)(56 264)(57 265)(58 266)(59 267)(60 268)(61 269)(62 270)(63 271)(64 272)(65 273)(66 274)(67 275)(68 276)(69 277)(70 278)(71 279)(72 280)(73 281)(74 282)(75 283)(76 284)(77 285)(78 286)(79 287)(80 288)(81 289)(82 290)(83 291)(84 292)(85 293)(86 294)(87 295)(88 296)(89 297)(90 298)(91 299)(92 300)(93 301)(94 302)(95 303)(96 304)(97 305)(98 306)(99 307)(100 308)(101 309)(102 310)(103 311)(104 312)(105 313)(106 314)(107 315)(108 316)(109 317)(110 318)(111 319)(112 320)(113 321)(114 322)(115 323)(116 324)(117 325)(118 326)(119 327)(120 328)(121 329)(122 330)(123 331)(124 332)(125 333)(126 334)(127 335)(128 336)(129 337)(130 338)(131 339)(132 340)(133 341)(134 342)(135 343)(136 344)(137 345)(138 346)(139 347)(140 348)(141 349)(142 350)(143 351)(144 352)(145 353)(146 354)(147 355)(148 356)(149 357)(150 358)(151 359)(152 360)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)
G:=sub<Sym(360)| (1,209)(2,210)(3,211)(4,212)(5,213)(6,214)(7,215)(8,216)(9,217)(10,218)(11,219)(12,220)(13,221)(14,222)(15,223)(16,224)(17,225)(18,226)(19,227)(20,228)(21,229)(22,230)(23,231)(24,232)(25,233)(26,234)(27,235)(28,236)(29,237)(30,238)(31,239)(32,240)(33,241)(34,242)(35,243)(36,244)(37,245)(38,246)(39,247)(40,248)(41,249)(42,250)(43,251)(44,252)(45,253)(46,254)(47,255)(48,256)(49,257)(50,258)(51,259)(52,260)(53,261)(54,262)(55,263)(56,264)(57,265)(58,266)(59,267)(60,268)(61,269)(62,270)(63,271)(64,272)(65,273)(66,274)(67,275)(68,276)(69,277)(70,278)(71,279)(72,280)(73,281)(74,282)(75,283)(76,284)(77,285)(78,286)(79,287)(80,288)(81,289)(82,290)(83,291)(84,292)(85,293)(86,294)(87,295)(88,296)(89,297)(90,298)(91,299)(92,300)(93,301)(94,302)(95,303)(96,304)(97,305)(98,306)(99,307)(100,308)(101,309)(102,310)(103,311)(104,312)(105,313)(106,314)(107,315)(108,316)(109,317)(110,318)(111,319)(112,320)(113,321)(114,322)(115,323)(116,324)(117,325)(118,326)(119,327)(120,328)(121,329)(122,330)(123,331)(124,332)(125,333)(126,334)(127,335)(128,336)(129,337)(130,338)(131,339)(132,340)(133,341)(134,342)(135,343)(136,344)(137,345)(138,346)(139,347)(140,348)(141,349)(142,350)(143,351)(144,352)(145,353)(146,354)(147,355)(148,356)(149,357)(150,358)(151,359)(152,360)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)>;
G:=Group( (1,209)(2,210)(3,211)(4,212)(5,213)(6,214)(7,215)(8,216)(9,217)(10,218)(11,219)(12,220)(13,221)(14,222)(15,223)(16,224)(17,225)(18,226)(19,227)(20,228)(21,229)(22,230)(23,231)(24,232)(25,233)(26,234)(27,235)(28,236)(29,237)(30,238)(31,239)(32,240)(33,241)(34,242)(35,243)(36,244)(37,245)(38,246)(39,247)(40,248)(41,249)(42,250)(43,251)(44,252)(45,253)(46,254)(47,255)(48,256)(49,257)(50,258)(51,259)(52,260)(53,261)(54,262)(55,263)(56,264)(57,265)(58,266)(59,267)(60,268)(61,269)(62,270)(63,271)(64,272)(65,273)(66,274)(67,275)(68,276)(69,277)(70,278)(71,279)(72,280)(73,281)(74,282)(75,283)(76,284)(77,285)(78,286)(79,287)(80,288)(81,289)(82,290)(83,291)(84,292)(85,293)(86,294)(87,295)(88,296)(89,297)(90,298)(91,299)(92,300)(93,301)(94,302)(95,303)(96,304)(97,305)(98,306)(99,307)(100,308)(101,309)(102,310)(103,311)(104,312)(105,313)(106,314)(107,315)(108,316)(109,317)(110,318)(111,319)(112,320)(113,321)(114,322)(115,323)(116,324)(117,325)(118,326)(119,327)(120,328)(121,329)(122,330)(123,331)(124,332)(125,333)(126,334)(127,335)(128,336)(129,337)(130,338)(131,339)(132,340)(133,341)(134,342)(135,343)(136,344)(137,345)(138,346)(139,347)(140,348)(141,349)(142,350)(143,351)(144,352)(145,353)(146,354)(147,355)(148,356)(149,357)(150,358)(151,359)(152,360)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360) );
G=PermutationGroup([[(1,209),(2,210),(3,211),(4,212),(5,213),(6,214),(7,215),(8,216),(9,217),(10,218),(11,219),(12,220),(13,221),(14,222),(15,223),(16,224),(17,225),(18,226),(19,227),(20,228),(21,229),(22,230),(23,231),(24,232),(25,233),(26,234),(27,235),(28,236),(29,237),(30,238),(31,239),(32,240),(33,241),(34,242),(35,243),(36,244),(37,245),(38,246),(39,247),(40,248),(41,249),(42,250),(43,251),(44,252),(45,253),(46,254),(47,255),(48,256),(49,257),(50,258),(51,259),(52,260),(53,261),(54,262),(55,263),(56,264),(57,265),(58,266),(59,267),(60,268),(61,269),(62,270),(63,271),(64,272),(65,273),(66,274),(67,275),(68,276),(69,277),(70,278),(71,279),(72,280),(73,281),(74,282),(75,283),(76,284),(77,285),(78,286),(79,287),(80,288),(81,289),(82,290),(83,291),(84,292),(85,293),(86,294),(87,295),(88,296),(89,297),(90,298),(91,299),(92,300),(93,301),(94,302),(95,303),(96,304),(97,305),(98,306),(99,307),(100,308),(101,309),(102,310),(103,311),(104,312),(105,313),(106,314),(107,315),(108,316),(109,317),(110,318),(111,319),(112,320),(113,321),(114,322),(115,323),(116,324),(117,325),(118,326),(119,327),(120,328),(121,329),(122,330),(123,331),(124,332),(125,333),(126,334),(127,335),(128,336),(129,337),(130,338),(131,339),(132,340),(133,341),(134,342),(135,343),(136,344),(137,345),(138,346),(139,347),(140,348),(141,349),(142,350),(143,351),(144,352),(145,353),(146,354),(147,355),(148,356),(149,357),(150,358),(151,359),(152,360),(153,181),(154,182),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188),(161,189),(162,190),(163,191),(164,192),(165,193),(166,194),(167,195),(168,196),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)]])
360 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 9A | ··· | 9F | 10A | ··· | 10L | 12A | ··· | 12H | 15A | ··· | 15H | 18A | ··· | 18R | 20A | ··· | 20P | 30A | ··· | 30X | 36A | ··· | 36X | 45A | ··· | 45X | 60A | ··· | 60AF | 90A | ··· | 90BT | 180A | ··· | 180CR |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 9 | ··· | 9 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 18 | ··· | 18 | 20 | ··· | 20 | 30 | ··· | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
360 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C9 | C10 | C10 | C12 | C15 | C18 | C18 | C20 | C30 | C30 | C36 | C45 | C60 | C90 | C90 | C180 |
| kernel | C2×C180 | C180 | C2×C90 | C2×C60 | C90 | C2×C36 | C60 | C2×C30 | C2×C20 | C36 | C2×C18 | C30 | C2×C12 | C20 | C2×C10 | C18 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 6 | 8 | 4 | 8 | 8 | 12 | 6 | 16 | 16 | 8 | 24 | 24 | 32 | 48 | 24 | 96 |
Matrix representation of C2×C180 ►in GL2(𝔽181) generated by
| 1 | 0 |
| 0 | 180 |
| 89 | 0 |
| 0 | 96 |
G:=sub<GL(2,GF(181))| [1,0,0,180],[89,0,0,96] >;
C2×C180 in GAP, Magma, Sage, TeX
C_2\times C_{180} % in TeX
G:=Group("C2xC180"); // GroupNames label
G:=SmallGroup(360,30);
// by ID
G=gap.SmallGroup(360,30);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,554]);
// Polycyclic
G:=Group<a,b|a^2=b^180=1,a*b=b*a>;
// generators/relations
Export