direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×C5⋊C8, C5⋊C72, C45⋊2C8, C10.C36, C15.C24, C90.2C4, C18.2F5, C30.2C12, Dic5.2C18, C2.(C9×F5), C6.2(C3×F5), (C9×Dic5).4C2, (C3×Dic5).6C6, C3.(C3×C5⋊C8), (C3×C5⋊C8).C3, SmallGroup(360,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C9×C5⋊C8 |
Generators and relations for C9×C5⋊C8
G = < a,b,c | a9=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C9×C5⋊C8
►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)
(1 155 101 253 190)(2 156 102 254 191)(3 157 103 255 192)(4 158 104 256 193)(5 159 105 257 194)(6 160 106 258 195)(7 161 107 259 196)(8 162 108 260 197)(9 154 100 261 198)(10 297 64 351 234)(11 289 65 343 226)(12 290 66 344 227)(13 291 67 345 228)(14 292 68 346 229)(15 293 69 347 230)(16 294 70 348 231)(17 295 71 349 232)(18 296 72 350 233)(19 285 135 73 339)(20 286 127 74 340)(21 287 128 75 341)(22 288 129 76 342)(23 280 130 77 334)(24 281 131 78 335)(25 282 132 79 336)(26 283 133 80 337)(27 284 134 81 338)(28 84 115 244 150)(29 85 116 245 151)(30 86 117 246 152)(31 87 109 247 153)(32 88 110 248 145)(33 89 111 249 146)(34 90 112 250 147)(35 82 113 251 148)(36 83 114 252 149)(37 307 200 263 49)(38 308 201 264 50)(39 309 202 265 51)(40 310 203 266 52)(41 311 204 267 53)(42 312 205 268 54)(43 313 206 269 46)(44 314 207 270 47)(45 315 199 262 48)(55 136 360 225 306)(56 137 352 217 298)(57 138 353 218 299)(58 139 354 219 300)(59 140 355 220 301)(60 141 356 221 302)(61 142 357 222 303)(62 143 358 223 304)(63 144 359 224 305)(91 323 171 175 182)(92 324 163 176 183)(93 316 164 177 184)(94 317 165 178 185)(95 318 166 179 186)(96 319 167 180 187)(97 320 168 172 188)(98 321 169 173 189)(99 322 170 174 181)(118 278 242 332 215)(119 279 243 333 216)(120 271 235 325 208)(121 272 236 326 209)(122 273 237 327 210)(123 274 238 328 211)(124 275 239 329 212)(125 276 240 330 213)(126 277 241 331 214)
(1 220 91 123 246 80 263 348)(2 221 92 124 247 81 264 349)(3 222 93 125 248 73 265 350)(4 223 94 126 249 74 266 351)(5 224 95 118 250 75 267 343)(6 225 96 119 251 76 268 344)(7 217 97 120 252 77 269 345)(8 218 98 121 244 78 270 346)(9 219 99 122 245 79 262 347)(10 193 143 317 241 111 286 52)(11 194 144 318 242 112 287 53)(12 195 136 319 243 113 288 54)(13 196 137 320 235 114 280 46)(14 197 138 321 236 115 281 47)(15 198 139 322 237 116 282 48)(16 190 140 323 238 117 283 49)(17 191 141 324 239 109 284 50)(18 192 142 316 240 110 285 51)(19 202 296 157 61 184 330 145)(20 203 297 158 62 185 331 146)(21 204 289 159 63 186 332 147)(22 205 290 160 55 187 333 148)(23 206 291 161 56 188 325 149)(24 207 292 162 57 189 326 150)(25 199 293 154 58 181 327 151)(26 200 294 155 59 182 328 152)(27 201 295 156 60 183 329 153)(28 131 314 229 108 353 173 272)(29 132 315 230 100 354 174 273)(30 133 307 231 101 355 175 274)(31 134 308 232 102 356 176 275)(32 135 309 233 103 357 177 276)(33 127 310 234 104 358 178 277)(34 128 311 226 105 359 179 278)(35 129 312 227 106 360 180 279)(36 130 313 228 107 352 172 271)(37 70 253 301 171 211 86 337)(38 71 254 302 163 212 87 338)(39 72 255 303 164 213 88 339)(40 64 256 304 165 214 89 340)(41 65 257 305 166 215 90 341)(42 66 258 306 167 216 82 342)(43 67 259 298 168 208 83 334)(44 68 260 299 169 209 84 335)(45 69 261 300 170 210 85 336)
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), (1,155,101,253,190)(2,156,102,254,191)(3,157,103,255,192)(4,158,104,256,193)(5,159,105,257,194)(6,160,106,258,195)(7,161,107,259,196)(8,162,108,260,197)(9,154,100,261,198)(10,297,64,351,234)(11,289,65,343,226)(12,290,66,344,227)(13,291,67,345,228)(14,292,68,346,229)(15,293,69,347,230)(16,294,70,348,231)(17,295,71,349,232)(18,296,72,350,233)(19,285,135,73,339)(20,286,127,74,340)(21,287,128,75,341)(22,288,129,76,342)(23,280,130,77,334)(24,281,131,78,335)(25,282,132,79,336)(26,283,133,80,337)(27,284,134,81,338)(28,84,115,244,150)(29,85,116,245,151)(30,86,117,246,152)(31,87,109,247,153)(32,88,110,248,145)(33,89,111,249,146)(34,90,112,250,147)(35,82,113,251,148)(36,83,114,252,149)(37,307,200,263,49)(38,308,201,264,50)(39,309,202,265,51)(40,310,203,266,52)(41,311,204,267,53)(42,312,205,268,54)(43,313,206,269,46)(44,314,207,270,47)(45,315,199,262,48)(55,136,360,225,306)(56,137,352,217,298)(57,138,353,218,299)(58,139,354,219,300)(59,140,355,220,301)(60,141,356,221,302)(61,142,357,222,303)(62,143,358,223,304)(63,144,359,224,305)(91,323,171,175,182)(92,324,163,176,183)(93,316,164,177,184)(94,317,165,178,185)(95,318,166,179,186)(96,319,167,180,187)(97,320,168,172,188)(98,321,169,173,189)(99,322,170,174,181)(118,278,242,332,215)(119,279,243,333,216)(120,271,235,325,208)(121,272,236,326,209)(122,273,237,327,210)(123,274,238,328,211)(124,275,239,329,212)(125,276,240,330,213)(126,277,241,331,214), (1,220,91,123,246,80,263,348)(2,221,92,124,247,81,264,349)(3,222,93,125,248,73,265,350)(4,223,94,126,249,74,266,351)(5,224,95,118,250,75,267,343)(6,225,96,119,251,76,268,344)(7,217,97,120,252,77,269,345)(8,218,98,121,244,78,270,346)(9,219,99,122,245,79,262,347)(10,193,143,317,241,111,286,52)(11,194,144,318,242,112,287,53)(12,195,136,319,243,113,288,54)(13,196,137,320,235,114,280,46)(14,197,138,321,236,115,281,47)(15,198,139,322,237,116,282,48)(16,190,140,323,238,117,283,49)(17,191,141,324,239,109,284,50)(18,192,142,316,240,110,285,51)(19,202,296,157,61,184,330,145)(20,203,297,158,62,185,331,146)(21,204,289,159,63,186,332,147)(22,205,290,160,55,187,333,148)(23,206,291,161,56,188,325,149)(24,207,292,162,57,189,326,150)(25,199,293,154,58,181,327,151)(26,200,294,155,59,182,328,152)(27,201,295,156,60,183,329,153)(28,131,314,229,108,353,173,272)(29,132,315,230,100,354,174,273)(30,133,307,231,101,355,175,274)(31,134,308,232,102,356,176,275)(32,135,309,233,103,357,177,276)(33,127,310,234,104,358,178,277)(34,128,311,226,105,359,179,278)(35,129,312,227,106,360,180,279)(36,130,313,228,107,352,172,271)(37,70,253,301,171,211,86,337)(38,71,254,302,163,212,87,338)(39,72,255,303,164,213,88,339)(40,64,256,304,165,214,89,340)(41,65,257,305,166,215,90,341)(42,66,258,306,167,216,82,342)(43,67,259,298,168,208,83,334)(44,68,260,299,169,209,84,335)(45,69,261,300,170,210,85,336)>;
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), (1,155,101,253,190)(2,156,102,254,191)(3,157,103,255,192)(4,158,104,256,193)(5,159,105,257,194)(6,160,106,258,195)(7,161,107,259,196)(8,162,108,260,197)(9,154,100,261,198)(10,297,64,351,234)(11,289,65,343,226)(12,290,66,344,227)(13,291,67,345,228)(14,292,68,346,229)(15,293,69,347,230)(16,294,70,348,231)(17,295,71,349,232)(18,296,72,350,233)(19,285,135,73,339)(20,286,127,74,340)(21,287,128,75,341)(22,288,129,76,342)(23,280,130,77,334)(24,281,131,78,335)(25,282,132,79,336)(26,283,133,80,337)(27,284,134,81,338)(28,84,115,244,150)(29,85,116,245,151)(30,86,117,246,152)(31,87,109,247,153)(32,88,110,248,145)(33,89,111,249,146)(34,90,112,250,147)(35,82,113,251,148)(36,83,114,252,149)(37,307,200,263,49)(38,308,201,264,50)(39,309,202,265,51)(40,310,203,266,52)(41,311,204,267,53)(42,312,205,268,54)(43,313,206,269,46)(44,314,207,270,47)(45,315,199,262,48)(55,136,360,225,306)(56,137,352,217,298)(57,138,353,218,299)(58,139,354,219,300)(59,140,355,220,301)(60,141,356,221,302)(61,142,357,222,303)(62,143,358,223,304)(63,144,359,224,305)(91,323,171,175,182)(92,324,163,176,183)(93,316,164,177,184)(94,317,165,178,185)(95,318,166,179,186)(96,319,167,180,187)(97,320,168,172,188)(98,321,169,173,189)(99,322,170,174,181)(118,278,242,332,215)(119,279,243,333,216)(120,271,235,325,208)(121,272,236,326,209)(122,273,237,327,210)(123,274,238,328,211)(124,275,239,329,212)(125,276,240,330,213)(126,277,241,331,214), (1,220,91,123,246,80,263,348)(2,221,92,124,247,81,264,349)(3,222,93,125,248,73,265,350)(4,223,94,126,249,74,266,351)(5,224,95,118,250,75,267,343)(6,225,96,119,251,76,268,344)(7,217,97,120,252,77,269,345)(8,218,98,121,244,78,270,346)(9,219,99,122,245,79,262,347)(10,193,143,317,241,111,286,52)(11,194,144,318,242,112,287,53)(12,195,136,319,243,113,288,54)(13,196,137,320,235,114,280,46)(14,197,138,321,236,115,281,47)(15,198,139,322,237,116,282,48)(16,190,140,323,238,117,283,49)(17,191,141,324,239,109,284,50)(18,192,142,316,240,110,285,51)(19,202,296,157,61,184,330,145)(20,203,297,158,62,185,331,146)(21,204,289,159,63,186,332,147)(22,205,290,160,55,187,333,148)(23,206,291,161,56,188,325,149)(24,207,292,162,57,189,326,150)(25,199,293,154,58,181,327,151)(26,200,294,155,59,182,328,152)(27,201,295,156,60,183,329,153)(28,131,314,229,108,353,173,272)(29,132,315,230,100,354,174,273)(30,133,307,231,101,355,175,274)(31,134,308,232,102,356,176,275)(32,135,309,233,103,357,177,276)(33,127,310,234,104,358,178,277)(34,128,311,226,105,359,179,278)(35,129,312,227,106,360,180,279)(36,130,313,228,107,352,172,271)(37,70,253,301,171,211,86,337)(38,71,254,302,163,212,87,338)(39,72,255,303,164,213,88,339)(40,64,256,304,165,214,89,340)(41,65,257,305,166,215,90,341)(42,66,258,306,167,216,82,342)(43,67,259,298,168,208,83,334)(44,68,260,299,169,209,84,335)(45,69,261,300,170,210,85,336) );
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)], [(1,155,101,253,190),(2,156,102,254,191),(3,157,103,255,192),(4,158,104,256,193),(5,159,105,257,194),(6,160,106,258,195),(7,161,107,259,196),(8,162,108,260,197),(9,154,100,261,198),(10,297,64,351,234),(11,289,65,343,226),(12,290,66,344,227),(13,291,67,345,228),(14,292,68,346,229),(15,293,69,347,230),(16,294,70,348,231),(17,295,71,349,232),(18,296,72,350,233),(19,285,135,73,339),(20,286,127,74,340),(21,287,128,75,341),(22,288,129,76,342),(23,280,130,77,334),(24,281,131,78,335),(25,282,132,79,336),(26,283,133,80,337),(27,284,134,81,338),(28,84,115,244,150),(29,85,116,245,151),(30,86,117,246,152),(31,87,109,247,153),(32,88,110,248,145),(33,89,111,249,146),(34,90,112,250,147),(35,82,113,251,148),(36,83,114,252,149),(37,307,200,263,49),(38,308,201,264,50),(39,309,202,265,51),(40,310,203,266,52),(41,311,204,267,53),(42,312,205,268,54),(43,313,206,269,46),(44,314,207,270,47),(45,315,199,262,48),(55,136,360,225,306),(56,137,352,217,298),(57,138,353,218,299),(58,139,354,219,300),(59,140,355,220,301),(60,141,356,221,302),(61,142,357,222,303),(62,143,358,223,304),(63,144,359,224,305),(91,323,171,175,182),(92,324,163,176,183),(93,316,164,177,184),(94,317,165,178,185),(95,318,166,179,186),(96,319,167,180,187),(97,320,168,172,188),(98,321,169,173,189),(99,322,170,174,181),(118,278,242,332,215),(119,279,243,333,216),(120,271,235,325,208),(121,272,236,326,209),(122,273,237,327,210),(123,274,238,328,211),(124,275,239,329,212),(125,276,240,330,213),(126,277,241,331,214)], [(1,220,91,123,246,80,263,348),(2,221,92,124,247,81,264,349),(3,222,93,125,248,73,265,350),(4,223,94,126,249,74,266,351),(5,224,95,118,250,75,267,343),(6,225,96,119,251,76,268,344),(7,217,97,120,252,77,269,345),(8,218,98,121,244,78,270,346),(9,219,99,122,245,79,262,347),(10,193,143,317,241,111,286,52),(11,194,144,318,242,112,287,53),(12,195,136,319,243,113,288,54),(13,196,137,320,235,114,280,46),(14,197,138,321,236,115,281,47),(15,198,139,322,237,116,282,48),(16,190,140,323,238,117,283,49),(17,191,141,324,239,109,284,50),(18,192,142,316,240,110,285,51),(19,202,296,157,61,184,330,145),(20,203,297,158,62,185,331,146),(21,204,289,159,63,186,332,147),(22,205,290,160,55,187,333,148),(23,206,291,161,56,188,325,149),(24,207,292,162,57,189,326,150),(25,199,293,154,58,181,327,151),(26,200,294,155,59,182,328,152),(27,201,295,156,60,183,329,153),(28,131,314,229,108,353,173,272),(29,132,315,230,100,354,174,273),(30,133,307,231,101,355,175,274),(31,134,308,232,102,356,176,275),(32,135,309,233,103,357,177,276),(33,127,310,234,104,358,178,277),(34,128,311,226,105,359,179,278),(35,129,312,227,106,360,180,279),(36,130,313,228,107,352,172,271),(37,70,253,301,171,211,86,337),(38,71,254,302,163,212,87,338),(39,72,255,303,164,213,88,339),(40,64,256,304,165,214,89,340),(41,65,257,305,166,215,90,341),(42,66,258,306,167,216,82,342),(43,67,259,298,168,208,83,334),(44,68,260,299,169,209,84,335),(45,69,261,300,170,210,85,336)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 18A | ··· | 18F | 24A | ··· | 24H | 30A | 30B | 36A | ··· | 36L | 45A | ··· | 45F | 72A | ··· | 72X | 90A | ··· | 90F |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 18 | ··· | 18 | 24 | ··· | 24 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 72 | ··· | 72 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 4 | 5 | 5 | 5 | 5 | 4 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 5 | ··· | 5 | 4 | ··· | 4 | 5 | ··· | 5 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C9 | C12 | C18 | C24 | C36 | C72 | F5 | C5⋊C8 | C3×F5 | C3×C5⋊C8 | C9×F5 | C9×C5⋊C8 |
| kernel | C9×C5⋊C8 | C9×Dic5 | C3×C5⋊C8 | C90 | C3×Dic5 | C45 | C5⋊C8 | C30 | Dic5 | C15 | C10 | C5 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 4 | 6 | 8 | 12 | 24 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of C9×C5⋊C8 ►in GL5(𝔽1801)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 144 | 0 | 0 | 0 |
| 0 | 0 | 144 | 0 | 0 |
| 0 | 0 | 0 | 144 | 0 |
| 0 | 0 | 0 | 0 | 144 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1800 |
| 0 | 1 | 0 | 0 | 1800 |
| 0 | 0 | 1 | 0 | 1800 |
| 0 | 0 | 0 | 1 | 1800 |
| 464 | 0 | 0 | 0 | 0 |
| 0 | 882 | 736 | 877 | 1785 |
| 0 | 1759 | 720 | 199 | 866 |
| 0 | 1081 | 1602 | 935 | 1743 |
| 0 | 16 | 678 | 919 | 1065 |
G:=sub<GL(5,GF(1801))| [1,0,0,0,0,0,144,0,0,0,0,0,144,0,0,0,0,0,144,0,0,0,0,0,144],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1800,1800,1800,1800],[464,0,0,0,0,0,882,1759,1081,16,0,736,720,1602,678,0,877,199,935,919,0,1785,866,1743,1065] >;
C9×C5⋊C8 in GAP, Magma, Sage, TeX
C_9\times C_5\rtimes C_8
% in TeX
G:=Group("C9xC5:C8"); // GroupNames label
G:=SmallGroup(360,5);
// by ID
G=gap.SmallGroup(360,5);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-2,-5,36,79,122,5189,1745]);
// Polycyclic
G:=Group<a,b,c|a^9=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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