direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C5×C9⋊C9, C9⋊C45, C45⋊C9, C15.23- 1+2, (C3×C9).1C15, C3.2(C3×C45), C15.2(C3×C9), (C3×C45).1C3, (C3×C15).7C32, C32.8(C3×C15), C3.2(C5×3- 1+2), SmallGroup(405,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C9⋊C9
G = < a,b,c | a5=b9=c9=1, ab=ba, ac=ca, cbc-1=b7 >
Smallest permutation representation of C5×C9⋊C9
►Regular action on 405 points
Generators in S405
(1 124 77 33 115)(2 125 78 34 116)(3 126 79 35 117)(4 118 80 36 109)(5 119 81 28 110)(6 120 73 29 111)(7 121 74 30 112)(8 122 75 31 113)(9 123 76 32 114)(10 72 27 101 62)(11 64 19 102 63)(12 65 20 103 55)(13 66 21 104 56)(14 67 22 105 57)(15 68 23 106 58)(16 69 24 107 59)(17 70 25 108 60)(18 71 26 100 61)(37 186 139 128 90)(38 187 140 129 82)(39 188 141 130 83)(40 189 142 131 84)(41 181 143 132 85)(42 182 144 133 86)(43 183 136 134 87)(44 184 137 135 88)(45 185 138 127 89)(46 397 353 312 99)(47 398 354 313 91)(48 399 355 314 92)(49 400 356 315 93)(50 401 357 307 94)(51 402 358 308 95)(52 403 359 309 96)(53 404 360 310 97)(54 405 352 311 98)(145 233 154 242 192)(146 234 155 243 193)(147 226 156 235 194)(148 227 157 236 195)(149 228 158 237 196)(150 229 159 238 197)(151 230 160 239 198)(152 231 161 240 190)(153 232 162 241 191)(163 251 201 260 210)(164 252 202 261 211)(165 244 203 253 212)(166 245 204 254 213)(167 246 205 255 214)(168 247 206 256 215)(169 248 207 257 216)(170 249 199 258 208)(171 250 200 259 209)(172 324 274 269 219)(173 316 275 270 220)(174 317 276 262 221)(175 318 277 263 222)(176 319 278 264 223)(177 320 279 265 224)(178 321 271 266 225)(179 322 272 267 217)(180 323 273 268 218)(280 365 289 374 330)(281 366 290 375 331)(282 367 291 376 332)(283 368 292 377 333)(284 369 293 378 325)(285 361 294 370 326)(286 362 295 371 327)(287 363 296 372 328)(288 364 297 373 329)(298 383 339 392 348)(299 384 340 393 349)(300 385 341 394 350)(301 386 342 395 351)(302 387 334 396 343)(303 379 335 388 344)(304 380 336 389 345)(305 381 337 390 346)(306 382 338 391 347)
(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)(361 362 363 364 365 366 367 368 369)(370 371 372 373 374 375 376 377 378)(379 380 381 382 383 384 385 386 387)(388 389 390 391 392 393 394 395 396)(397 398 399 400 401 402 403 404 405)
(1 366 231 16 307 175 43 335 206)(2 361 229 17 311 173 44 339 204)(3 365 227 18 315 180 45 334 202)(4 369 234 10 310 178 37 338 200)(5 364 232 11 314 176 38 342 207)(6 368 230 12 309 174 39 337 205)(7 363 228 13 313 172 40 341 203)(8 367 226 14 308 179 41 336 201)(9 362 233 15 312 177 42 340 199)(19 48 278 140 351 216 81 373 241)(20 52 276 141 346 214 73 377 239)(21 47 274 142 350 212 74 372 237)(22 51 272 143 345 210 75 376 235)(23 46 279 144 349 208 76 371 242)(24 50 277 136 344 215 77 375 240)(25 54 275 137 348 213 78 370 238)(26 49 273 138 343 211 79 374 236)(27 53 271 139 347 209 80 378 243)(28 329 191 102 399 264 129 301 169)(29 333 198 103 403 262 130 305 167)(30 328 196 104 398 269 131 300 165)(31 332 194 105 402 267 132 304 163)(32 327 192 106 397 265 133 299 170)(33 331 190 107 401 263 134 303 168)(34 326 197 108 405 270 135 298 166)(35 330 195 100 400 268 127 302 164)(36 325 193 101 404 266 128 306 171)(55 359 221 83 381 246 111 283 151)(56 354 219 84 385 244 112 287 149)(57 358 217 85 380 251 113 282 147)(58 353 224 86 384 249 114 286 145)(59 357 222 87 379 247 115 281 152)(60 352 220 88 383 245 116 285 150)(61 356 218 89 387 252 117 280 148)(62 360 225 90 382 250 109 284 146)(63 355 223 82 386 248 110 288 153)(64 92 319 187 395 257 119 297 162)(65 96 317 188 390 255 120 292 160)(66 91 324 189 394 253 121 296 158)(67 95 322 181 389 260 122 291 156)(68 99 320 182 393 258 123 295 154)(69 94 318 183 388 256 124 290 161)(70 98 316 184 392 254 125 294 159)(71 93 323 185 396 261 126 289 157)(72 97 321 186 391 259 118 293 155)
G:=sub<Sym(405)| (1,124,77,33,115)(2,125,78,34,116)(3,126,79,35,117)(4,118,80,36,109)(5,119,81,28,110)(6,120,73,29,111)(7,121,74,30,112)(8,122,75,31,113)(9,123,76,32,114)(10,72,27,101,62)(11,64,19,102,63)(12,65,20,103,55)(13,66,21,104,56)(14,67,22,105,57)(15,68,23,106,58)(16,69,24,107,59)(17,70,25,108,60)(18,71,26,100,61)(37,186,139,128,90)(38,187,140,129,82)(39,188,141,130,83)(40,189,142,131,84)(41,181,143,132,85)(42,182,144,133,86)(43,183,136,134,87)(44,184,137,135,88)(45,185,138,127,89)(46,397,353,312,99)(47,398,354,313,91)(48,399,355,314,92)(49,400,356,315,93)(50,401,357,307,94)(51,402,358,308,95)(52,403,359,309,96)(53,404,360,310,97)(54,405,352,311,98)(145,233,154,242,192)(146,234,155,243,193)(147,226,156,235,194)(148,227,157,236,195)(149,228,158,237,196)(150,229,159,238,197)(151,230,160,239,198)(152,231,161,240,190)(153,232,162,241,191)(163,251,201,260,210)(164,252,202,261,211)(165,244,203,253,212)(166,245,204,254,213)(167,246,205,255,214)(168,247,206,256,215)(169,248,207,257,216)(170,249,199,258,208)(171,250,200,259,209)(172,324,274,269,219)(173,316,275,270,220)(174,317,276,262,221)(175,318,277,263,222)(176,319,278,264,223)(177,320,279,265,224)(178,321,271,266,225)(179,322,272,267,217)(180,323,273,268,218)(280,365,289,374,330)(281,366,290,375,331)(282,367,291,376,332)(283,368,292,377,333)(284,369,293,378,325)(285,361,294,370,326)(286,362,295,371,327)(287,363,296,372,328)(288,364,297,373,329)(298,383,339,392,348)(299,384,340,393,349)(300,385,341,394,350)(301,386,342,395,351)(302,387,334,396,343)(303,379,335,388,344)(304,380,336,389,345)(305,381,337,390,346)(306,382,338,391,347), (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)(361,362,363,364,365,366,367,368,369)(370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387)(388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405), (1,366,231,16,307,175,43,335,206)(2,361,229,17,311,173,44,339,204)(3,365,227,18,315,180,45,334,202)(4,369,234,10,310,178,37,338,200)(5,364,232,11,314,176,38,342,207)(6,368,230,12,309,174,39,337,205)(7,363,228,13,313,172,40,341,203)(8,367,226,14,308,179,41,336,201)(9,362,233,15,312,177,42,340,199)(19,48,278,140,351,216,81,373,241)(20,52,276,141,346,214,73,377,239)(21,47,274,142,350,212,74,372,237)(22,51,272,143,345,210,75,376,235)(23,46,279,144,349,208,76,371,242)(24,50,277,136,344,215,77,375,240)(25,54,275,137,348,213,78,370,238)(26,49,273,138,343,211,79,374,236)(27,53,271,139,347,209,80,378,243)(28,329,191,102,399,264,129,301,169)(29,333,198,103,403,262,130,305,167)(30,328,196,104,398,269,131,300,165)(31,332,194,105,402,267,132,304,163)(32,327,192,106,397,265,133,299,170)(33,331,190,107,401,263,134,303,168)(34,326,197,108,405,270,135,298,166)(35,330,195,100,400,268,127,302,164)(36,325,193,101,404,266,128,306,171)(55,359,221,83,381,246,111,283,151)(56,354,219,84,385,244,112,287,149)(57,358,217,85,380,251,113,282,147)(58,353,224,86,384,249,114,286,145)(59,357,222,87,379,247,115,281,152)(60,352,220,88,383,245,116,285,150)(61,356,218,89,387,252,117,280,148)(62,360,225,90,382,250,109,284,146)(63,355,223,82,386,248,110,288,153)(64,92,319,187,395,257,119,297,162)(65,96,317,188,390,255,120,292,160)(66,91,324,189,394,253,121,296,158)(67,95,322,181,389,260,122,291,156)(68,99,320,182,393,258,123,295,154)(69,94,318,183,388,256,124,290,161)(70,98,316,184,392,254,125,294,159)(71,93,323,185,396,261,126,289,157)(72,97,321,186,391,259,118,293,155)>;
G:=Group( (1,124,77,33,115)(2,125,78,34,116)(3,126,79,35,117)(4,118,80,36,109)(5,119,81,28,110)(6,120,73,29,111)(7,121,74,30,112)(8,122,75,31,113)(9,123,76,32,114)(10,72,27,101,62)(11,64,19,102,63)(12,65,20,103,55)(13,66,21,104,56)(14,67,22,105,57)(15,68,23,106,58)(16,69,24,107,59)(17,70,25,108,60)(18,71,26,100,61)(37,186,139,128,90)(38,187,140,129,82)(39,188,141,130,83)(40,189,142,131,84)(41,181,143,132,85)(42,182,144,133,86)(43,183,136,134,87)(44,184,137,135,88)(45,185,138,127,89)(46,397,353,312,99)(47,398,354,313,91)(48,399,355,314,92)(49,400,356,315,93)(50,401,357,307,94)(51,402,358,308,95)(52,403,359,309,96)(53,404,360,310,97)(54,405,352,311,98)(145,233,154,242,192)(146,234,155,243,193)(147,226,156,235,194)(148,227,157,236,195)(149,228,158,237,196)(150,229,159,238,197)(151,230,160,239,198)(152,231,161,240,190)(153,232,162,241,191)(163,251,201,260,210)(164,252,202,261,211)(165,244,203,253,212)(166,245,204,254,213)(167,246,205,255,214)(168,247,206,256,215)(169,248,207,257,216)(170,249,199,258,208)(171,250,200,259,209)(172,324,274,269,219)(173,316,275,270,220)(174,317,276,262,221)(175,318,277,263,222)(176,319,278,264,223)(177,320,279,265,224)(178,321,271,266,225)(179,322,272,267,217)(180,323,273,268,218)(280,365,289,374,330)(281,366,290,375,331)(282,367,291,376,332)(283,368,292,377,333)(284,369,293,378,325)(285,361,294,370,326)(286,362,295,371,327)(287,363,296,372,328)(288,364,297,373,329)(298,383,339,392,348)(299,384,340,393,349)(300,385,341,394,350)(301,386,342,395,351)(302,387,334,396,343)(303,379,335,388,344)(304,380,336,389,345)(305,381,337,390,346)(306,382,338,391,347), (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)(361,362,363,364,365,366,367,368,369)(370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387)(388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405), (1,366,231,16,307,175,43,335,206)(2,361,229,17,311,173,44,339,204)(3,365,227,18,315,180,45,334,202)(4,369,234,10,310,178,37,338,200)(5,364,232,11,314,176,38,342,207)(6,368,230,12,309,174,39,337,205)(7,363,228,13,313,172,40,341,203)(8,367,226,14,308,179,41,336,201)(9,362,233,15,312,177,42,340,199)(19,48,278,140,351,216,81,373,241)(20,52,276,141,346,214,73,377,239)(21,47,274,142,350,212,74,372,237)(22,51,272,143,345,210,75,376,235)(23,46,279,144,349,208,76,371,242)(24,50,277,136,344,215,77,375,240)(25,54,275,137,348,213,78,370,238)(26,49,273,138,343,211,79,374,236)(27,53,271,139,347,209,80,378,243)(28,329,191,102,399,264,129,301,169)(29,333,198,103,403,262,130,305,167)(30,328,196,104,398,269,131,300,165)(31,332,194,105,402,267,132,304,163)(32,327,192,106,397,265,133,299,170)(33,331,190,107,401,263,134,303,168)(34,326,197,108,405,270,135,298,166)(35,330,195,100,400,268,127,302,164)(36,325,193,101,404,266,128,306,171)(55,359,221,83,381,246,111,283,151)(56,354,219,84,385,244,112,287,149)(57,358,217,85,380,251,113,282,147)(58,353,224,86,384,249,114,286,145)(59,357,222,87,379,247,115,281,152)(60,352,220,88,383,245,116,285,150)(61,356,218,89,387,252,117,280,148)(62,360,225,90,382,250,109,284,146)(63,355,223,82,386,248,110,288,153)(64,92,319,187,395,257,119,297,162)(65,96,317,188,390,255,120,292,160)(66,91,324,189,394,253,121,296,158)(67,95,322,181,389,260,122,291,156)(68,99,320,182,393,258,123,295,154)(69,94,318,183,388,256,124,290,161)(70,98,316,184,392,254,125,294,159)(71,93,323,185,396,261,126,289,157)(72,97,321,186,391,259,118,293,155) );
G=PermutationGroup([[(1,124,77,33,115),(2,125,78,34,116),(3,126,79,35,117),(4,118,80,36,109),(5,119,81,28,110),(6,120,73,29,111),(7,121,74,30,112),(8,122,75,31,113),(9,123,76,32,114),(10,72,27,101,62),(11,64,19,102,63),(12,65,20,103,55),(13,66,21,104,56),(14,67,22,105,57),(15,68,23,106,58),(16,69,24,107,59),(17,70,25,108,60),(18,71,26,100,61),(37,186,139,128,90),(38,187,140,129,82),(39,188,141,130,83),(40,189,142,131,84),(41,181,143,132,85),(42,182,144,133,86),(43,183,136,134,87),(44,184,137,135,88),(45,185,138,127,89),(46,397,353,312,99),(47,398,354,313,91),(48,399,355,314,92),(49,400,356,315,93),(50,401,357,307,94),(51,402,358,308,95),(52,403,359,309,96),(53,404,360,310,97),(54,405,352,311,98),(145,233,154,242,192),(146,234,155,243,193),(147,226,156,235,194),(148,227,157,236,195),(149,228,158,237,196),(150,229,159,238,197),(151,230,160,239,198),(152,231,161,240,190),(153,232,162,241,191),(163,251,201,260,210),(164,252,202,261,211),(165,244,203,253,212),(166,245,204,254,213),(167,246,205,255,214),(168,247,206,256,215),(169,248,207,257,216),(170,249,199,258,208),(171,250,200,259,209),(172,324,274,269,219),(173,316,275,270,220),(174,317,276,262,221),(175,318,277,263,222),(176,319,278,264,223),(177,320,279,265,224),(178,321,271,266,225),(179,322,272,267,217),(180,323,273,268,218),(280,365,289,374,330),(281,366,290,375,331),(282,367,291,376,332),(283,368,292,377,333),(284,369,293,378,325),(285,361,294,370,326),(286,362,295,371,327),(287,363,296,372,328),(288,364,297,373,329),(298,383,339,392,348),(299,384,340,393,349),(300,385,341,394,350),(301,386,342,395,351),(302,387,334,396,343),(303,379,335,388,344),(304,380,336,389,345),(305,381,337,390,346),(306,382,338,391,347)], [(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),(361,362,363,364,365,366,367,368,369),(370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387),(388,389,390,391,392,393,394,395,396),(397,398,399,400,401,402,403,404,405)], [(1,366,231,16,307,175,43,335,206),(2,361,229,17,311,173,44,339,204),(3,365,227,18,315,180,45,334,202),(4,369,234,10,310,178,37,338,200),(5,364,232,11,314,176,38,342,207),(6,368,230,12,309,174,39,337,205),(7,363,228,13,313,172,40,341,203),(8,367,226,14,308,179,41,336,201),(9,362,233,15,312,177,42,340,199),(19,48,278,140,351,216,81,373,241),(20,52,276,141,346,214,73,377,239),(21,47,274,142,350,212,74,372,237),(22,51,272,143,345,210,75,376,235),(23,46,279,144,349,208,76,371,242),(24,50,277,136,344,215,77,375,240),(25,54,275,137,348,213,78,370,238),(26,49,273,138,343,211,79,374,236),(27,53,271,139,347,209,80,378,243),(28,329,191,102,399,264,129,301,169),(29,333,198,103,403,262,130,305,167),(30,328,196,104,398,269,131,300,165),(31,332,194,105,402,267,132,304,163),(32,327,192,106,397,265,133,299,170),(33,331,190,107,401,263,134,303,168),(34,326,197,108,405,270,135,298,166),(35,330,195,100,400,268,127,302,164),(36,325,193,101,404,266,128,306,171),(55,359,221,83,381,246,111,283,151),(56,354,219,84,385,244,112,287,149),(57,358,217,85,380,251,113,282,147),(58,353,224,86,384,249,114,286,145),(59,357,222,87,379,247,115,281,152),(60,352,220,88,383,245,116,285,150),(61,356,218,89,387,252,117,280,148),(62,360,225,90,382,250,109,284,146),(63,355,223,82,386,248,110,288,153),(64,92,319,187,395,257,119,297,162),(65,96,317,188,390,255,120,292,160),(66,91,324,189,394,253,121,296,158),(67,95,322,181,389,260,122,291,156),(68,99,320,182,393,258,123,295,154),(69,94,318,183,388,256,124,290,161),(70,98,316,184,392,254,125,294,159),(71,93,323,185,396,261,126,289,157),(72,97,321,186,391,259,118,293,155)]])
165 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 9A | ··· | 9X | 15A | ··· | 15AF | 45A | ··· | 45CR |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C5 | C9 | C15 | C45 | 3- 1+2 | C5×3- 1+2 |
| kernel | C5×C9⋊C9 | C3×C45 | C9⋊C9 | C45 | C3×C9 | C9 | C15 | C3 |
| # reps | 1 | 8 | 4 | 18 | 32 | 72 | 6 | 24 |
Matrix representation of C5×C9⋊C9 ►in GL4(𝔽181) generated by
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 48 | 0 | 0 | 0 |
| 0 | 0 | 48 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 132 | 0 | 0 |
| 62 | 0 | 0 | 0 |
| 0 | 171 | 144 | 80 |
| 0 | 80 | 128 | 3 |
| 0 | 3 | 62 | 63 |
G:=sub<GL(4,GF(181))| [1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[48,0,0,0,0,0,0,132,0,48,0,0,0,0,48,0],[62,0,0,0,0,171,80,3,0,144,128,62,0,80,3,63] >;
C5×C9⋊C9 in GAP, Magma, Sage, TeX
C_5\times C_9\rtimes C_9
% in TeX
G:=Group("C5xC9:C9"); // GroupNames label
G:=SmallGroup(405,4);
// by ID
G=gap.SmallGroup(405,4);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,675,481,156]);
// Polycyclic
G:=Group<a,b,c|a^5=b^9=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations
Export