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G = C2×C180order 360 = 23·32·5

Abelian group of type [2,180]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C180, SmallGroup(360,30)

Series: Derived Chief Lower central Upper central

C1 — C2×C180
C1C3C6C30C90C180 — 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
(1 324)(2 325)(3 326)(4 327)(5 328)(6 329)(7 330)(8 331)(9 332)(10 333)(11 334)(12 335)(13 336)(14 337)(15 338)(16 339)(17 340)(18 341)(19 342)(20 343)(21 344)(22 345)(23 346)(24 347)(25 348)(26 349)(27 350)(28 351)(29 352)(30 353)(31 354)(32 355)(33 356)(34 357)(35 358)(36 359)(37 360)(38 181)(39 182)(40 183)(41 184)(42 185)(43 186)(44 187)(45 188)(46 189)(47 190)(48 191)(49 192)(50 193)(51 194)(52 195)(53 196)(54 197)(55 198)(56 199)(57 200)(58 201)(59 202)(60 203)(61 204)(62 205)(63 206)(64 207)(65 208)(66 209)(67 210)(68 211)(69 212)(70 213)(71 214)(72 215)(73 216)(74 217)(75 218)(76 219)(77 220)(78 221)(79 222)(80 223)(81 224)(82 225)(83 226)(84 227)(85 228)(86 229)(87 230)(88 231)(89 232)(90 233)(91 234)(92 235)(93 236)(94 237)(95 238)(96 239)(97 240)(98 241)(99 242)(100 243)(101 244)(102 245)(103 246)(104 247)(105 248)(106 249)(107 250)(108 251)(109 252)(110 253)(111 254)(112 255)(113 256)(114 257)(115 258)(116 259)(117 260)(118 261)(119 262)(120 263)(121 264)(122 265)(123 266)(124 267)(125 268)(126 269)(127 270)(128 271)(129 272)(130 273)(131 274)(132 275)(133 276)(134 277)(135 278)(136 279)(137 280)(138 281)(139 282)(140 283)(141 284)(142 285)(143 286)(144 287)(145 288)(146 289)(147 290)(148 291)(149 292)(150 293)(151 294)(152 295)(153 296)(154 297)(155 298)(156 299)(157 300)(158 301)(159 302)(160 303)(161 304)(162 305)(163 306)(164 307)(165 308)(166 309)(167 310)(168 311)(169 312)(170 313)(171 314)(172 315)(173 316)(174 317)(175 318)(176 319)(177 320)(178 321)(179 322)(180 323)
(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)| (1,324)(2,325)(3,326)(4,327)(5,328)(6,329)(7,330)(8,331)(9,332)(10,333)(11,334)(12,335)(13,336)(14,337)(15,338)(16,339)(17,340)(18,341)(19,342)(20,343)(21,344)(22,345)(23,346)(24,347)(25,348)(26,349)(27,350)(28,351)(29,352)(30,353)(31,354)(32,355)(33,356)(34,357)(35,358)(36,359)(37,360)(38,181)(39,182)(40,183)(41,184)(42,185)(43,186)(44,187)(45,188)(46,189)(47,190)(48,191)(49,192)(50,193)(51,194)(52,195)(53,196)(54,197)(55,198)(56,199)(57,200)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,211)(69,212)(70,213)(71,214)(72,215)(73,216)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(81,224)(82,225)(83,226)(84,227)(85,228)(86,229)(87,230)(88,231)(89,232)(90,233)(91,234)(92,235)(93,236)(94,237)(95,238)(96,239)(97,240)(98,241)(99,242)(100,243)(101,244)(102,245)(103,246)(104,247)(105,248)(106,249)(107,250)(108,251)(109,252)(110,253)(111,254)(112,255)(113,256)(114,257)(115,258)(116,259)(117,260)(118,261)(119,262)(120,263)(121,264)(122,265)(123,266)(124,267)(125,268)(126,269)(127,270)(128,271)(129,272)(130,273)(131,274)(132,275)(133,276)(134,277)(135,278)(136,279)(137,280)(138,281)(139,282)(140,283)(141,284)(142,285)(143,286)(144,287)(145,288)(146,289)(147,290)(148,291)(149,292)(150,293)(151,294)(152,295)(153,296)(154,297)(155,298)(156,299)(157,300)(158,301)(159,302)(160,303)(161,304)(162,305)(163,306)(164,307)(165,308)(166,309)(167,310)(168,311)(169,312)(170,313)(171,314)(172,315)(173,316)(174,317)(175,318)(176,319)(177,320)(178,321)(179,322)(180,323), (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( (1,324)(2,325)(3,326)(4,327)(5,328)(6,329)(7,330)(8,331)(9,332)(10,333)(11,334)(12,335)(13,336)(14,337)(15,338)(16,339)(17,340)(18,341)(19,342)(20,343)(21,344)(22,345)(23,346)(24,347)(25,348)(26,349)(27,350)(28,351)(29,352)(30,353)(31,354)(32,355)(33,356)(34,357)(35,358)(36,359)(37,360)(38,181)(39,182)(40,183)(41,184)(42,185)(43,186)(44,187)(45,188)(46,189)(47,190)(48,191)(49,192)(50,193)(51,194)(52,195)(53,196)(54,197)(55,198)(56,199)(57,200)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,211)(69,212)(70,213)(71,214)(72,215)(73,216)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(81,224)(82,225)(83,226)(84,227)(85,228)(86,229)(87,230)(88,231)(89,232)(90,233)(91,234)(92,235)(93,236)(94,237)(95,238)(96,239)(97,240)(98,241)(99,242)(100,243)(101,244)(102,245)(103,246)(104,247)(105,248)(106,249)(107,250)(108,251)(109,252)(110,253)(111,254)(112,255)(113,256)(114,257)(115,258)(116,259)(117,260)(118,261)(119,262)(120,263)(121,264)(122,265)(123,266)(124,267)(125,268)(126,269)(127,270)(128,271)(129,272)(130,273)(131,274)(132,275)(133,276)(134,277)(135,278)(136,279)(137,280)(138,281)(139,282)(140,283)(141,284)(142,285)(143,286)(144,287)(145,288)(146,289)(147,290)(148,291)(149,292)(150,293)(151,294)(152,295)(153,296)(154,297)(155,298)(156,299)(157,300)(158,301)(159,302)(160,303)(161,304)(162,305)(163,306)(164,307)(165,308)(166,309)(167,310)(168,311)(169,312)(170,313)(171,314)(172,315)(173,316)(174,317)(175,318)(176,319)(177,320)(178,321)(179,322)(180,323), (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([(1,324),(2,325),(3,326),(4,327),(5,328),(6,329),(7,330),(8,331),(9,332),(10,333),(11,334),(12,335),(13,336),(14,337),(15,338),(16,339),(17,340),(18,341),(19,342),(20,343),(21,344),(22,345),(23,346),(24,347),(25,348),(26,349),(27,350),(28,351),(29,352),(30,353),(31,354),(32,355),(33,356),(34,357),(35,358),(36,359),(37,360),(38,181),(39,182),(40,183),(41,184),(42,185),(43,186),(44,187),(45,188),(46,189),(47,190),(48,191),(49,192),(50,193),(51,194),(52,195),(53,196),(54,197),(55,198),(56,199),(57,200),(58,201),(59,202),(60,203),(61,204),(62,205),(63,206),(64,207),(65,208),(66,209),(67,210),(68,211),(69,212),(70,213),(71,214),(72,215),(73,216),(74,217),(75,218),(76,219),(77,220),(78,221),(79,222),(80,223),(81,224),(82,225),(83,226),(84,227),(85,228),(86,229),(87,230),(88,231),(89,232),(90,233),(91,234),(92,235),(93,236),(94,237),(95,238),(96,239),(97,240),(98,241),(99,242),(100,243),(101,244),(102,245),(103,246),(104,247),(105,248),(106,249),(107,250),(108,251),(109,252),(110,253),(111,254),(112,255),(113,256),(114,257),(115,258),(116,259),(117,260),(118,261),(119,262),(120,263),(121,264),(122,265),(123,266),(124,267),(125,268),(126,269),(127,270),(128,271),(129,272),(130,273),(131,274),(132,275),(133,276),(134,277),(135,278),(136,279),(137,280),(138,281),(139,282),(140,283),(141,284),(142,285),(143,286),(144,287),(145,288),(146,289),(147,290),(148,291),(149,292),(150,293),(151,294),(152,295),(153,296),(154,297),(155,298),(156,299),(157,300),(158,301),(159,302),(160,303),(161,304),(162,305),(163,306),(164,307),(165,308),(166,309),(167,310),(168,311),(169,312),(170,313),(171,314),(172,315),(173,316),(174,317),(175,318),(176,319),(177,320),(178,321),(179,322),(180,323)], [(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 2A2B2C3A3B4A4B4C4D5A5B5C5D6A···6F9A···9F10A···10L12A···12H15A···15H18A···18R20A···20P30A···30X36A···36X45A···45X60A···60AF90A···90BT180A···180CR
order122233444455556···69···910···1012···1215···1518···1820···2030···3036···3645···4560···6090···90180···180
size111111111111111···11···11···11···11···11···11···11···11···11···11···11···11···1

360 irreducible representations

dim111111111111111111111111
type+++
imageC1C2C2C3C4C5C6C6C9C10C10C12C15C18C18C20C30C30C36C45C60C90C90C180
kernelC2×C180C180C2×C90C2×C60C90C2×C36C60C2×C30C2×C20C36C2×C18C30C2×C12C20C2×C10C18C12C2×C6C10C2×C4C6C4C22C2
# reps121244426848812616168242432482496

Matrix representation of C2×C180 in GL2(𝔽181) generated by

10
0180
,
890
096
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

Subgroup lattice of C2×C180 in TeX

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