metabelian, supersoluble, monomial, A-group
Aliases: C60.6S3, C12.6D15, C30.7Dic3, C6.3Dic15, C15⋊3(C3⋊C8), C3⋊(C15⋊3C8), (C3×C15)⋊10C8, (C3×C30).7C4, (C3×C60).3C2, (C3×C12).4D5, C20.2(C3⋊S3), C4.2(C3⋊D15), C32⋊4(C5⋊2C8), C5⋊2(C32⋊4C8), C2.(C3⋊Dic15), (C3×C6).3Dic5, C10.2(C3⋊Dic3), SmallGroup(360,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C60.S3 |
Generators and relations for C60.S3
G = < a,b,c | a60=b3=1, c2=a45, ab=ba, cac-1=a29, cbc-1=b-1 >
Smallest permutation representation of C60.S3
►Regular action on 360 points
Generators in S360
(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)
(1 228 332)(2 229 333)(3 230 334)(4 231 335)(5 232 336)(6 233 337)(7 234 338)(8 235 339)(9 236 340)(10 237 341)(11 238 342)(12 239 343)(13 240 344)(14 181 345)(15 182 346)(16 183 347)(17 184 348)(18 185 349)(19 186 350)(20 187 351)(21 188 352)(22 189 353)(23 190 354)(24 191 355)(25 192 356)(26 193 357)(27 194 358)(28 195 359)(29 196 360)(30 197 301)(31 198 302)(32 199 303)(33 200 304)(34 201 305)(35 202 306)(36 203 307)(37 204 308)(38 205 309)(39 206 310)(40 207 311)(41 208 312)(42 209 313)(43 210 314)(44 211 315)(45 212 316)(46 213 317)(47 214 318)(48 215 319)(49 216 320)(50 217 321)(51 218 322)(52 219 323)(53 220 324)(54 221 325)(55 222 326)(56 223 327)(57 224 328)(58 225 329)(59 226 330)(60 227 331)(61 167 291)(62 168 292)(63 169 293)(64 170 294)(65 171 295)(66 172 296)(67 173 297)(68 174 298)(69 175 299)(70 176 300)(71 177 241)(72 178 242)(73 179 243)(74 180 244)(75 121 245)(76 122 246)(77 123 247)(78 124 248)(79 125 249)(80 126 250)(81 127 251)(82 128 252)(83 129 253)(84 130 254)(85 131 255)(86 132 256)(87 133 257)(88 134 258)(89 135 259)(90 136 260)(91 137 261)(92 138 262)(93 139 263)(94 140 264)(95 141 265)(96 142 266)(97 143 267)(98 144 268)(99 145 269)(100 146 270)(101 147 271)(102 148 272)(103 149 273)(104 150 274)(105 151 275)(106 152 276)(107 153 277)(108 154 278)(109 155 279)(110 156 280)(111 157 281)(112 158 282)(113 159 283)(114 160 284)(115 161 285)(116 162 286)(117 163 287)(118 164 288)(119 165 289)(120 166 290)
(1 290 46 275 31 260 16 245)(2 259 47 244 32 289 17 274)(3 288 48 273 33 258 18 243)(4 257 49 242 34 287 19 272)(5 286 50 271 35 256 20 241)(6 255 51 300 36 285 21 270)(7 284 52 269 37 254 22 299)(8 253 53 298 38 283 23 268)(9 282 54 267 39 252 24 297)(10 251 55 296 40 281 25 266)(11 280 56 265 41 250 26 295)(12 249 57 294 42 279 27 264)(13 278 58 263 43 248 28 293)(14 247 59 292 44 277 29 262)(15 276 60 261 45 246 30 291)(61 346 106 331 91 316 76 301)(62 315 107 360 92 345 77 330)(63 344 108 329 93 314 78 359)(64 313 109 358 94 343 79 328)(65 342 110 327 95 312 80 357)(66 311 111 356 96 341 81 326)(67 340 112 325 97 310 82 355)(68 309 113 354 98 339 83 324)(69 338 114 323 99 308 84 353)(70 307 115 352 100 337 85 322)(71 336 116 321 101 306 86 351)(72 305 117 350 102 335 87 320)(73 334 118 319 103 304 88 349)(74 303 119 348 104 333 89 318)(75 332 120 317 105 302 90 347)(121 228 166 213 151 198 136 183)(122 197 167 182 152 227 137 212)(123 226 168 211 153 196 138 181)(124 195 169 240 154 225 139 210)(125 224 170 209 155 194 140 239)(126 193 171 238 156 223 141 208)(127 222 172 207 157 192 142 237)(128 191 173 236 158 221 143 206)(129 220 174 205 159 190 144 235)(130 189 175 234 160 219 145 204)(131 218 176 203 161 188 146 233)(132 187 177 232 162 217 147 202)(133 216 178 201 163 186 148 231)(134 185 179 230 164 215 149 200)(135 214 180 199 165 184 150 229)
G:=sub<Sym(360)| (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), (1,228,332)(2,229,333)(3,230,334)(4,231,335)(5,232,336)(6,233,337)(7,234,338)(8,235,339)(9,236,340)(10,237,341)(11,238,342)(12,239,343)(13,240,344)(14,181,345)(15,182,346)(16,183,347)(17,184,348)(18,185,349)(19,186,350)(20,187,351)(21,188,352)(22,189,353)(23,190,354)(24,191,355)(25,192,356)(26,193,357)(27,194,358)(28,195,359)(29,196,360)(30,197,301)(31,198,302)(32,199,303)(33,200,304)(34,201,305)(35,202,306)(36,203,307)(37,204,308)(38,205,309)(39,206,310)(40,207,311)(41,208,312)(42,209,313)(43,210,314)(44,211,315)(45,212,316)(46,213,317)(47,214,318)(48,215,319)(49,216,320)(50,217,321)(51,218,322)(52,219,323)(53,220,324)(54,221,325)(55,222,326)(56,223,327)(57,224,328)(58,225,329)(59,226,330)(60,227,331)(61,167,291)(62,168,292)(63,169,293)(64,170,294)(65,171,295)(66,172,296)(67,173,297)(68,174,298)(69,175,299)(70,176,300)(71,177,241)(72,178,242)(73,179,243)(74,180,244)(75,121,245)(76,122,246)(77,123,247)(78,124,248)(79,125,249)(80,126,250)(81,127,251)(82,128,252)(83,129,253)(84,130,254)(85,131,255)(86,132,256)(87,133,257)(88,134,258)(89,135,259)(90,136,260)(91,137,261)(92,138,262)(93,139,263)(94,140,264)(95,141,265)(96,142,266)(97,143,267)(98,144,268)(99,145,269)(100,146,270)(101,147,271)(102,148,272)(103,149,273)(104,150,274)(105,151,275)(106,152,276)(107,153,277)(108,154,278)(109,155,279)(110,156,280)(111,157,281)(112,158,282)(113,159,283)(114,160,284)(115,161,285)(116,162,286)(117,163,287)(118,164,288)(119,165,289)(120,166,290), (1,290,46,275,31,260,16,245)(2,259,47,244,32,289,17,274)(3,288,48,273,33,258,18,243)(4,257,49,242,34,287,19,272)(5,286,50,271,35,256,20,241)(6,255,51,300,36,285,21,270)(7,284,52,269,37,254,22,299)(8,253,53,298,38,283,23,268)(9,282,54,267,39,252,24,297)(10,251,55,296,40,281,25,266)(11,280,56,265,41,250,26,295)(12,249,57,294,42,279,27,264)(13,278,58,263,43,248,28,293)(14,247,59,292,44,277,29,262)(15,276,60,261,45,246,30,291)(61,346,106,331,91,316,76,301)(62,315,107,360,92,345,77,330)(63,344,108,329,93,314,78,359)(64,313,109,358,94,343,79,328)(65,342,110,327,95,312,80,357)(66,311,111,356,96,341,81,326)(67,340,112,325,97,310,82,355)(68,309,113,354,98,339,83,324)(69,338,114,323,99,308,84,353)(70,307,115,352,100,337,85,322)(71,336,116,321,101,306,86,351)(72,305,117,350,102,335,87,320)(73,334,118,319,103,304,88,349)(74,303,119,348,104,333,89,318)(75,332,120,317,105,302,90,347)(121,228,166,213,151,198,136,183)(122,197,167,182,152,227,137,212)(123,226,168,211,153,196,138,181)(124,195,169,240,154,225,139,210)(125,224,170,209,155,194,140,239)(126,193,171,238,156,223,141,208)(127,222,172,207,157,192,142,237)(128,191,173,236,158,221,143,206)(129,220,174,205,159,190,144,235)(130,189,175,234,160,219,145,204)(131,218,176,203,161,188,146,233)(132,187,177,232,162,217,147,202)(133,216,178,201,163,186,148,231)(134,185,179,230,164,215,149,200)(135,214,180,199,165,184,150,229)>;
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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(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), (1,228,332)(2,229,333)(3,230,334)(4,231,335)(5,232,336)(6,233,337)(7,234,338)(8,235,339)(9,236,340)(10,237,341)(11,238,342)(12,239,343)(13,240,344)(14,181,345)(15,182,346)(16,183,347)(17,184,348)(18,185,349)(19,186,350)(20,187,351)(21,188,352)(22,189,353)(23,190,354)(24,191,355)(25,192,356)(26,193,357)(27,194,358)(28,195,359)(29,196,360)(30,197,301)(31,198,302)(32,199,303)(33,200,304)(34,201,305)(35,202,306)(36,203,307)(37,204,308)(38,205,309)(39,206,310)(40,207,311)(41,208,312)(42,209,313)(43,210,314)(44,211,315)(45,212,316)(46,213,317)(47,214,318)(48,215,319)(49,216,320)(50,217,321)(51,218,322)(52,219,323)(53,220,324)(54,221,325)(55,222,326)(56,223,327)(57,224,328)(58,225,329)(59,226,330)(60,227,331)(61,167,291)(62,168,292)(63,169,293)(64,170,294)(65,171,295)(66,172,296)(67,173,297)(68,174,298)(69,175,299)(70,176,300)(71,177,241)(72,178,242)(73,179,243)(74,180,244)(75,121,245)(76,122,246)(77,123,247)(78,124,248)(79,125,249)(80,126,250)(81,127,251)(82,128,252)(83,129,253)(84,130,254)(85,131,255)(86,132,256)(87,133,257)(88,134,258)(89,135,259)(90,136,260)(91,137,261)(92,138,262)(93,139,263)(94,140,264)(95,141,265)(96,142,266)(97,143,267)(98,144,268)(99,145,269)(100,146,270)(101,147,271)(102,148,272)(103,149,273)(104,150,274)(105,151,275)(106,152,276)(107,153,277)(108,154,278)(109,155,279)(110,156,280)(111,157,281)(112,158,282)(113,159,283)(114,160,284)(115,161,285)(116,162,286)(117,163,287)(118,164,288)(119,165,289)(120,166,290), (1,290,46,275,31,260,16,245)(2,259,47,244,32,289,17,274)(3,288,48,273,33,258,18,243)(4,257,49,242,34,287,19,272)(5,286,50,271,35,256,20,241)(6,255,51,300,36,285,21,270)(7,284,52,269,37,254,22,299)(8,253,53,298,38,283,23,268)(9,282,54,267,39,252,24,297)(10,251,55,296,40,281,25,266)(11,280,56,265,41,250,26,295)(12,249,57,294,42,279,27,264)(13,278,58,263,43,248,28,293)(14,247,59,292,44,277,29,262)(15,276,60,261,45,246,30,291)(61,346,106,331,91,316,76,301)(62,315,107,360,92,345,77,330)(63,344,108,329,93,314,78,359)(64,313,109,358,94,343,79,328)(65,342,110,327,95,312,80,357)(66,311,111,356,96,341,81,326)(67,340,112,325,97,310,82,355)(68,309,113,354,98,339,83,324)(69,338,114,323,99,308,84,353)(70,307,115,352,100,337,85,322)(71,336,116,321,101,306,86,351)(72,305,117,350,102,335,87,320)(73,334,118,319,103,304,88,349)(74,303,119,348,104,333,89,318)(75,332,120,317,105,302,90,347)(121,228,166,213,151,198,136,183)(122,197,167,182,152,227,137,212)(123,226,168,211,153,196,138,181)(124,195,169,240,154,225,139,210)(125,224,170,209,155,194,140,239)(126,193,171,238,156,223,141,208)(127,222,172,207,157,192,142,237)(128,191,173,236,158,221,143,206)(129,220,174,205,159,190,144,235)(130,189,175,234,160,219,145,204)(131,218,176,203,161,188,146,233)(132,187,177,232,162,217,147,202)(133,216,178,201,163,186,148,231)(134,185,179,230,164,215,149,200)(135,214,180,199,165,184,150,229) );
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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(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)], [(1,228,332),(2,229,333),(3,230,334),(4,231,335),(5,232,336),(6,233,337),(7,234,338),(8,235,339),(9,236,340),(10,237,341),(11,238,342),(12,239,343),(13,240,344),(14,181,345),(15,182,346),(16,183,347),(17,184,348),(18,185,349),(19,186,350),(20,187,351),(21,188,352),(22,189,353),(23,190,354),(24,191,355),(25,192,356),(26,193,357),(27,194,358),(28,195,359),(29,196,360),(30,197,301),(31,198,302),(32,199,303),(33,200,304),(34,201,305),(35,202,306),(36,203,307),(37,204,308),(38,205,309),(39,206,310),(40,207,311),(41,208,312),(42,209,313),(43,210,314),(44,211,315),(45,212,316),(46,213,317),(47,214,318),(48,215,319),(49,216,320),(50,217,321),(51,218,322),(52,219,323),(53,220,324),(54,221,325),(55,222,326),(56,223,327),(57,224,328),(58,225,329),(59,226,330),(60,227,331),(61,167,291),(62,168,292),(63,169,293),(64,170,294),(65,171,295),(66,172,296),(67,173,297),(68,174,298),(69,175,299),(70,176,300),(71,177,241),(72,178,242),(73,179,243),(74,180,244),(75,121,245),(76,122,246),(77,123,247),(78,124,248),(79,125,249),(80,126,250),(81,127,251),(82,128,252),(83,129,253),(84,130,254),(85,131,255),(86,132,256),(87,133,257),(88,134,258),(89,135,259),(90,136,260),(91,137,261),(92,138,262),(93,139,263),(94,140,264),(95,141,265),(96,142,266),(97,143,267),(98,144,268),(99,145,269),(100,146,270),(101,147,271),(102,148,272),(103,149,273),(104,150,274),(105,151,275),(106,152,276),(107,153,277),(108,154,278),(109,155,279),(110,156,280),(111,157,281),(112,158,282),(113,159,283),(114,160,284),(115,161,285),(116,162,286),(117,163,287),(118,164,288),(119,165,289),(120,166,290)], [(1,290,46,275,31,260,16,245),(2,259,47,244,32,289,17,274),(3,288,48,273,33,258,18,243),(4,257,49,242,34,287,19,272),(5,286,50,271,35,256,20,241),(6,255,51,300,36,285,21,270),(7,284,52,269,37,254,22,299),(8,253,53,298,38,283,23,268),(9,282,54,267,39,252,24,297),(10,251,55,296,40,281,25,266),(11,280,56,265,41,250,26,295),(12,249,57,294,42,279,27,264),(13,278,58,263,43,248,28,293),(14,247,59,292,44,277,29,262),(15,276,60,261,45,246,30,291),(61,346,106,331,91,316,76,301),(62,315,107,360,92,345,77,330),(63,344,108,329,93,314,78,359),(64,313,109,358,94,343,79,328),(65,342,110,327,95,312,80,357),(66,311,111,356,96,341,81,326),(67,340,112,325,97,310,82,355),(68,309,113,354,98,339,83,324),(69,338,114,323,99,308,84,353),(70,307,115,352,100,337,85,322),(71,336,116,321,101,306,86,351),(72,305,117,350,102,335,87,320),(73,334,118,319,103,304,88,349),(74,303,119,348,104,333,89,318),(75,332,120,317,105,302,90,347),(121,228,166,213,151,198,136,183),(122,197,167,182,152,227,137,212),(123,226,168,211,153,196,138,181),(124,195,169,240,154,225,139,210),(125,224,170,209,155,194,140,239),(126,193,171,238,156,223,141,208),(127,222,172,207,157,192,142,237),(128,191,173,236,158,221,143,206),(129,220,174,205,159,190,144,235),(130,189,175,234,160,219,145,204),(131,218,176,203,161,188,146,233),(132,187,177,232,162,217,147,202),(133,216,178,201,163,186,148,231),(134,185,179,230,164,215,149,200),(135,214,180,199,165,184,150,229)]])
96 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 12A | ··· | 12H | 15A | ··· | 15P | 20A | 20B | 20C | 20D | 30A | ··· | 30P | 60A | ··· | 60AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 45 | 45 | 45 | 45 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | D5 | Dic3 | Dic5 | C3⋊C8 | D15 | C5⋊2C8 | Dic15 | C15⋊3C8 |
| kernel | C60.S3 | C3×C60 | C3×C30 | C3×C15 | C60 | C3×C12 | C30 | C3×C6 | C15 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C60.S3 ►in GL5(𝔽241)
| 177 | 0 | 0 | 0 | 0 |
| 0 | 189 | 189 | 0 | 0 |
| 0 | 52 | 1 | 0 | 0 |
| 0 | 0 | 0 | 173 | 16 |
| 0 | 0 | 0 | 225 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 177 | 30 | 0 | 0 |
| 0 | 211 | 63 | 0 | 0 |
| 0 | 0 | 0 | 63 | 211 |
| 0 | 0 | 0 | 30 | 177 |
| 211 | 0 | 0 | 0 | 0 |
| 0 | 221 | 63 | 0 | 0 |
| 0 | 139 | 20 | 0 | 0 |
| 0 | 0 | 0 | 163 | 169 |
| 0 | 0 | 0 | 128 | 78 |
G:=sub<GL(5,GF(241))| [177,0,0,0,0,0,189,52,0,0,0,189,1,0,0,0,0,0,173,225,0,0,0,16,64],[1,0,0,0,0,0,177,211,0,0,0,30,63,0,0,0,0,0,63,30,0,0,0,211,177],[211,0,0,0,0,0,221,139,0,0,0,63,20,0,0,0,0,0,163,128,0,0,0,169,78] >;
C60.S3 in GAP, Magma, Sage, TeX
C_{60}.S_3 % in TeX
G:=Group("C60.S3"); // GroupNames label
G:=SmallGroup(360,37);
// by ID
G=gap.SmallGroup(360,37);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,12,31,387,1444,10373]);
// Polycyclic
G:=Group<a,b,c|a^60=b^3=1,c^2=a^45,a*b=b*a,c*a*c^-1=a^29,c*b*c^-1=b^-1>;
// generators/relations
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