direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×Dic9, C18⋊C20, C90⋊4C4, C30.59D6, C10.16D18, C90.16C22, C30.10Dic3, C9⋊2(C2×C20), C45⋊11(C2×C4), (C2×C18).C10, C22.(C5×D9), C6.9(S3×C10), (C2×C30).8S3, (C2×C90).2C2, (C2×C10).2D9, C2.2(C10×D9), C3.(C10×Dic3), C18.4(C2×C10), C6.2(C5×Dic3), C15.4(C2×Dic3), (C2×C6).2(C5×S3), SmallGroup(360,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C10×Dic9 |
Generators and relations for C10×Dic9
G = < a,b,c | a10=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C10×Dic9
►Regular action on 360 points
Generators in S360
(1 134 80 109 62 101 51 177 26 155)(2 135 81 110 63 102 52 178 27 156)(3 136 82 111 64 103 53 179 28 157)(4 137 83 112 65 104 54 180 29 158)(5 138 84 113 66 105 37 163 30 159)(6 139 85 114 67 106 38 164 31 160)(7 140 86 115 68 107 39 165 32 161)(8 141 87 116 69 108 40 166 33 162)(9 142 88 117 70 91 41 167 34 145)(10 143 89 118 71 92 42 168 35 146)(11 144 90 119 72 93 43 169 36 147)(12 127 73 120 55 94 44 170 19 148)(13 128 74 121 56 95 45 171 20 149)(14 129 75 122 57 96 46 172 21 150)(15 130 76 123 58 97 47 173 22 151)(16 131 77 124 59 98 48 174 23 152)(17 132 78 125 60 99 49 175 24 153)(18 133 79 126 61 100 50 176 25 154)(181 314 267 294 250 277 223 348 213 327)(182 315 268 295 251 278 224 349 214 328)(183 316 269 296 252 279 225 350 215 329)(184 317 270 297 235 280 226 351 216 330)(185 318 253 298 236 281 227 352 199 331)(186 319 254 299 237 282 228 353 200 332)(187 320 255 300 238 283 229 354 201 333)(188 321 256 301 239 284 230 355 202 334)(189 322 257 302 240 285 231 356 203 335)(190 323 258 303 241 286 232 357 204 336)(191 324 259 304 242 287 233 358 205 337)(192 307 260 305 243 288 234 359 206 338)(193 308 261 306 244 271 217 360 207 339)(194 309 262 289 245 272 218 343 208 340)(195 310 263 290 246 273 219 344 209 341)(196 311 264 291 247 274 220 345 210 342)(197 312 265 292 248 275 221 346 211 325)(198 313 266 293 249 276 222 347 212 326)
(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)
(1 183 10 192)(2 182 11 191)(3 181 12 190)(4 198 13 189)(5 197 14 188)(6 196 15 187)(7 195 16 186)(8 194 17 185)(9 193 18 184)(19 204 28 213)(20 203 29 212)(21 202 30 211)(22 201 31 210)(23 200 32 209)(24 199 33 208)(25 216 34 207)(26 215 35 206)(27 214 36 205)(37 221 46 230)(38 220 47 229)(39 219 48 228)(40 218 49 227)(41 217 50 226)(42 234 51 225)(43 233 52 224)(44 232 53 223)(45 231 54 222)(55 241 64 250)(56 240 65 249)(57 239 66 248)(58 238 67 247)(59 237 68 246)(60 236 69 245)(61 235 70 244)(62 252 71 243)(63 251 72 242)(73 258 82 267)(74 257 83 266)(75 256 84 265)(76 255 85 264)(77 254 86 263)(78 253 87 262)(79 270 88 261)(80 269 89 260)(81 268 90 259)(91 271 100 280)(92 288 101 279)(93 287 102 278)(94 286 103 277)(95 285 104 276)(96 284 105 275)(97 283 106 274)(98 282 107 273)(99 281 108 272)(109 296 118 305)(110 295 119 304)(111 294 120 303)(112 293 121 302)(113 292 122 301)(114 291 123 300)(115 290 124 299)(116 289 125 298)(117 306 126 297)(127 323 136 314)(128 322 137 313)(129 321 138 312)(130 320 139 311)(131 319 140 310)(132 318 141 309)(133 317 142 308)(134 316 143 307)(135 315 144 324)(145 339 154 330)(146 338 155 329)(147 337 156 328)(148 336 157 327)(149 335 158 326)(150 334 159 325)(151 333 160 342)(152 332 161 341)(153 331 162 340)(163 346 172 355)(164 345 173 354)(165 344 174 353)(166 343 175 352)(167 360 176 351)(168 359 177 350)(169 358 178 349)(170 357 179 348)(171 356 180 347)
G:=sub<Sym(360)| (1,134,80,109,62,101,51,177,26,155)(2,135,81,110,63,102,52,178,27,156)(3,136,82,111,64,103,53,179,28,157)(4,137,83,112,65,104,54,180,29,158)(5,138,84,113,66,105,37,163,30,159)(6,139,85,114,67,106,38,164,31,160)(7,140,86,115,68,107,39,165,32,161)(8,141,87,116,69,108,40,166,33,162)(9,142,88,117,70,91,41,167,34,145)(10,143,89,118,71,92,42,168,35,146)(11,144,90,119,72,93,43,169,36,147)(12,127,73,120,55,94,44,170,19,148)(13,128,74,121,56,95,45,171,20,149)(14,129,75,122,57,96,46,172,21,150)(15,130,76,123,58,97,47,173,22,151)(16,131,77,124,59,98,48,174,23,152)(17,132,78,125,60,99,49,175,24,153)(18,133,79,126,61,100,50,176,25,154)(181,314,267,294,250,277,223,348,213,327)(182,315,268,295,251,278,224,349,214,328)(183,316,269,296,252,279,225,350,215,329)(184,317,270,297,235,280,226,351,216,330)(185,318,253,298,236,281,227,352,199,331)(186,319,254,299,237,282,228,353,200,332)(187,320,255,300,238,283,229,354,201,333)(188,321,256,301,239,284,230,355,202,334)(189,322,257,302,240,285,231,356,203,335)(190,323,258,303,241,286,232,357,204,336)(191,324,259,304,242,287,233,358,205,337)(192,307,260,305,243,288,234,359,206,338)(193,308,261,306,244,271,217,360,207,339)(194,309,262,289,245,272,218,343,208,340)(195,310,263,290,246,273,219,344,209,341)(196,311,264,291,247,274,220,345,210,342)(197,312,265,292,248,275,221,346,211,325)(198,313,266,293,249,276,222,347,212,326), (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), (1,183,10,192)(2,182,11,191)(3,181,12,190)(4,198,13,189)(5,197,14,188)(6,196,15,187)(7,195,16,186)(8,194,17,185)(9,193,18,184)(19,204,28,213)(20,203,29,212)(21,202,30,211)(22,201,31,210)(23,200,32,209)(24,199,33,208)(25,216,34,207)(26,215,35,206)(27,214,36,205)(37,221,46,230)(38,220,47,229)(39,219,48,228)(40,218,49,227)(41,217,50,226)(42,234,51,225)(43,233,52,224)(44,232,53,223)(45,231,54,222)(55,241,64,250)(56,240,65,249)(57,239,66,248)(58,238,67,247)(59,237,68,246)(60,236,69,245)(61,235,70,244)(62,252,71,243)(63,251,72,242)(73,258,82,267)(74,257,83,266)(75,256,84,265)(76,255,85,264)(77,254,86,263)(78,253,87,262)(79,270,88,261)(80,269,89,260)(81,268,90,259)(91,271,100,280)(92,288,101,279)(93,287,102,278)(94,286,103,277)(95,285,104,276)(96,284,105,275)(97,283,106,274)(98,282,107,273)(99,281,108,272)(109,296,118,305)(110,295,119,304)(111,294,120,303)(112,293,121,302)(113,292,122,301)(114,291,123,300)(115,290,124,299)(116,289,125,298)(117,306,126,297)(127,323,136,314)(128,322,137,313)(129,321,138,312)(130,320,139,311)(131,319,140,310)(132,318,141,309)(133,317,142,308)(134,316,143,307)(135,315,144,324)(145,339,154,330)(146,338,155,329)(147,337,156,328)(148,336,157,327)(149,335,158,326)(150,334,159,325)(151,333,160,342)(152,332,161,341)(153,331,162,340)(163,346,172,355)(164,345,173,354)(165,344,174,353)(166,343,175,352)(167,360,176,351)(168,359,177,350)(169,358,178,349)(170,357,179,348)(171,356,180,347)>;
G:=Group( (1,134,80,109,62,101,51,177,26,155)(2,135,81,110,63,102,52,178,27,156)(3,136,82,111,64,103,53,179,28,157)(4,137,83,112,65,104,54,180,29,158)(5,138,84,113,66,105,37,163,30,159)(6,139,85,114,67,106,38,164,31,160)(7,140,86,115,68,107,39,165,32,161)(8,141,87,116,69,108,40,166,33,162)(9,142,88,117,70,91,41,167,34,145)(10,143,89,118,71,92,42,168,35,146)(11,144,90,119,72,93,43,169,36,147)(12,127,73,120,55,94,44,170,19,148)(13,128,74,121,56,95,45,171,20,149)(14,129,75,122,57,96,46,172,21,150)(15,130,76,123,58,97,47,173,22,151)(16,131,77,124,59,98,48,174,23,152)(17,132,78,125,60,99,49,175,24,153)(18,133,79,126,61,100,50,176,25,154)(181,314,267,294,250,277,223,348,213,327)(182,315,268,295,251,278,224,349,214,328)(183,316,269,296,252,279,225,350,215,329)(184,317,270,297,235,280,226,351,216,330)(185,318,253,298,236,281,227,352,199,331)(186,319,254,299,237,282,228,353,200,332)(187,320,255,300,238,283,229,354,201,333)(188,321,256,301,239,284,230,355,202,334)(189,322,257,302,240,285,231,356,203,335)(190,323,258,303,241,286,232,357,204,336)(191,324,259,304,242,287,233,358,205,337)(192,307,260,305,243,288,234,359,206,338)(193,308,261,306,244,271,217,360,207,339)(194,309,262,289,245,272,218,343,208,340)(195,310,263,290,246,273,219,344,209,341)(196,311,264,291,247,274,220,345,210,342)(197,312,265,292,248,275,221,346,211,325)(198,313,266,293,249,276,222,347,212,326), (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), (1,183,10,192)(2,182,11,191)(3,181,12,190)(4,198,13,189)(5,197,14,188)(6,196,15,187)(7,195,16,186)(8,194,17,185)(9,193,18,184)(19,204,28,213)(20,203,29,212)(21,202,30,211)(22,201,31,210)(23,200,32,209)(24,199,33,208)(25,216,34,207)(26,215,35,206)(27,214,36,205)(37,221,46,230)(38,220,47,229)(39,219,48,228)(40,218,49,227)(41,217,50,226)(42,234,51,225)(43,233,52,224)(44,232,53,223)(45,231,54,222)(55,241,64,250)(56,240,65,249)(57,239,66,248)(58,238,67,247)(59,237,68,246)(60,236,69,245)(61,235,70,244)(62,252,71,243)(63,251,72,242)(73,258,82,267)(74,257,83,266)(75,256,84,265)(76,255,85,264)(77,254,86,263)(78,253,87,262)(79,270,88,261)(80,269,89,260)(81,268,90,259)(91,271,100,280)(92,288,101,279)(93,287,102,278)(94,286,103,277)(95,285,104,276)(96,284,105,275)(97,283,106,274)(98,282,107,273)(99,281,108,272)(109,296,118,305)(110,295,119,304)(111,294,120,303)(112,293,121,302)(113,292,122,301)(114,291,123,300)(115,290,124,299)(116,289,125,298)(117,306,126,297)(127,323,136,314)(128,322,137,313)(129,321,138,312)(130,320,139,311)(131,319,140,310)(132,318,141,309)(133,317,142,308)(134,316,143,307)(135,315,144,324)(145,339,154,330)(146,338,155,329)(147,337,156,328)(148,336,157,327)(149,335,158,326)(150,334,159,325)(151,333,160,342)(152,332,161,341)(153,331,162,340)(163,346,172,355)(164,345,173,354)(165,344,174,353)(166,343,175,352)(167,360,176,351)(168,359,177,350)(169,358,178,349)(170,357,179,348)(171,356,180,347) );
G=PermutationGroup([[(1,134,80,109,62,101,51,177,26,155),(2,135,81,110,63,102,52,178,27,156),(3,136,82,111,64,103,53,179,28,157),(4,137,83,112,65,104,54,180,29,158),(5,138,84,113,66,105,37,163,30,159),(6,139,85,114,67,106,38,164,31,160),(7,140,86,115,68,107,39,165,32,161),(8,141,87,116,69,108,40,166,33,162),(9,142,88,117,70,91,41,167,34,145),(10,143,89,118,71,92,42,168,35,146),(11,144,90,119,72,93,43,169,36,147),(12,127,73,120,55,94,44,170,19,148),(13,128,74,121,56,95,45,171,20,149),(14,129,75,122,57,96,46,172,21,150),(15,130,76,123,58,97,47,173,22,151),(16,131,77,124,59,98,48,174,23,152),(17,132,78,125,60,99,49,175,24,153),(18,133,79,126,61,100,50,176,25,154),(181,314,267,294,250,277,223,348,213,327),(182,315,268,295,251,278,224,349,214,328),(183,316,269,296,252,279,225,350,215,329),(184,317,270,297,235,280,226,351,216,330),(185,318,253,298,236,281,227,352,199,331),(186,319,254,299,237,282,228,353,200,332),(187,320,255,300,238,283,229,354,201,333),(188,321,256,301,239,284,230,355,202,334),(189,322,257,302,240,285,231,356,203,335),(190,323,258,303,241,286,232,357,204,336),(191,324,259,304,242,287,233,358,205,337),(192,307,260,305,243,288,234,359,206,338),(193,308,261,306,244,271,217,360,207,339),(194,309,262,289,245,272,218,343,208,340),(195,310,263,290,246,273,219,344,209,341),(196,311,264,291,247,274,220,345,210,342),(197,312,265,292,248,275,221,346,211,325),(198,313,266,293,249,276,222,347,212,326)], [(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)], [(1,183,10,192),(2,182,11,191),(3,181,12,190),(4,198,13,189),(5,197,14,188),(6,196,15,187),(7,195,16,186),(8,194,17,185),(9,193,18,184),(19,204,28,213),(20,203,29,212),(21,202,30,211),(22,201,31,210),(23,200,32,209),(24,199,33,208),(25,216,34,207),(26,215,35,206),(27,214,36,205),(37,221,46,230),(38,220,47,229),(39,219,48,228),(40,218,49,227),(41,217,50,226),(42,234,51,225),(43,233,52,224),(44,232,53,223),(45,231,54,222),(55,241,64,250),(56,240,65,249),(57,239,66,248),(58,238,67,247),(59,237,68,246),(60,236,69,245),(61,235,70,244),(62,252,71,243),(63,251,72,242),(73,258,82,267),(74,257,83,266),(75,256,84,265),(76,255,85,264),(77,254,86,263),(78,253,87,262),(79,270,88,261),(80,269,89,260),(81,268,90,259),(91,271,100,280),(92,288,101,279),(93,287,102,278),(94,286,103,277),(95,285,104,276),(96,284,105,275),(97,283,106,274),(98,282,107,273),(99,281,108,272),(109,296,118,305),(110,295,119,304),(111,294,120,303),(112,293,121,302),(113,292,122,301),(114,291,123,300),(115,290,124,299),(116,289,125,298),(117,306,126,297),(127,323,136,314),(128,322,137,313),(129,321,138,312),(130,320,139,311),(131,319,140,310),(132,318,141,309),(133,317,142,308),(134,316,143,307),(135,315,144,324),(145,339,154,330),(146,338,155,329),(147,337,156,328),(148,336,157,327),(149,335,158,326),(150,334,159,325),(151,333,160,342),(152,332,161,341),(153,331,162,340),(163,346,172,355),(164,345,173,354),(165,344,174,353),(166,343,175,352),(167,360,176,351),(168,359,177,350),(169,358,178,349),(170,357,179,348),(171,356,180,347)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 9A | 9B | 9C | 10A | ··· | 10L | 15A | 15B | 15C | 15D | 18A | ··· | 18I | 20A | ··· | 20P | 30A | ··· | 30L | 45A | ··· | 45L | 90A | ··· | 90AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 20 | ··· | 20 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | Dic3 | D6 | D9 | C5×S3 | Dic9 | D18 | C5×Dic3 | S3×C10 | C5×D9 | C5×Dic9 | C10×D9 |
| kernel | C10×Dic9 | C5×Dic9 | C2×C90 | C90 | C2×Dic9 | Dic9 | C2×C18 | C18 | C2×C30 | C30 | C30 | C2×C10 | C2×C6 | C10 | C10 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 2 | 1 | 3 | 4 | 6 | 3 | 8 | 4 | 12 | 24 | 12 |
Matrix representation of C10×Dic9 ►in GL4(𝔽181) generated by
| 59 | 0 | 0 | 0 |
| 0 | 180 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 127 | 4 |
| 0 | 0 | 177 | 131 |
| 180 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 15 |
| 0 | 0 | 48 | 148 |
G:=sub<GL(4,GF(181))| [59,0,0,0,0,180,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,127,177,0,0,4,131],[180,0,0,0,0,1,0,0,0,0,33,48,0,0,15,148] >;
C10×Dic9 in GAP, Magma, Sage, TeX
C_{10}\times {\rm Dic}_9 % in TeX
G:=Group("C10xDic9"); // GroupNames label
G:=SmallGroup(360,23);
// by ID
G=gap.SmallGroup(360,23);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,6004,208,8645]);
// Polycyclic
G:=Group<a,b,c|a^10=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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