metabelian, supersoluble, monomial, A-group
Aliases: C72⋊2C9, C7⋊1(C7⋊C9), C21.3(C7⋊C3), (C7×C21).2C3, C3.(C72⋊C3), SmallGroup(441,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C72⋊C9 |
Generators and relations for C72⋊C9
G = < a,b,c | a7=b7=c9=1, ab=ba, cac-1=a4, cbc-1=b4 >
49C9
Smallest permutation representation of C72⋊C9
►Regular action on 441 points
Generators in S441
(1 87 102 35 245 121 281)(2 246 88 122 103 282 36)(3 104 247 283 89 28 123)(4 90 105 29 248 124 284)(5 249 82 125 106 285 30)(6 107 250 286 83 31 126)(7 84 108 32 251 118 287)(8 252 85 119 100 288 33)(9 101 244 280 86 34 120)(10 179 395 262 196 71 241)(11 197 180 72 396 242 263)(12 388 198 243 172 264 64)(13 173 389 265 190 65 235)(14 191 174 66 390 236 266)(15 391 192 237 175 267 67)(16 176 392 268 193 68 238)(17 194 177 69 393 239 269)(18 394 195 240 178 270 70)(19 189 114 274 358 75 147)(20 359 181 76 115 148 275)(21 116 360 149 182 276 77)(22 183 117 277 352 78 150)(23 353 184 79 109 151 278)(24 110 354 152 185 279 80)(25 186 111 271 355 81 153)(26 356 187 73 112 145 272)(27 113 357 146 188 273 74)(37 361 314 441 128 397 93)(38 129 362 398 315 94 433)(39 307 130 95 363 434 399)(40 364 308 435 131 400 96)(41 132 365 401 309 97 436)(42 310 133 98 366 437 402)(43 367 311 438 134 403 99)(44 135 368 404 312 91 439)(45 313 127 92 369 440 405)(46 322 140 212 372 299 409)(47 373 323 300 141 410 213)(48 142 374 411 324 214 301)(49 316 143 215 375 302 412)(50 376 317 303 144 413 216)(51 136 377 414 318 208 304)(52 319 137 209 378 305 406)(53 370 320 306 138 407 210)(54 139 371 408 321 211 298)(55 296 430 338 233 163 257)(56 234 297 164 431 258 339)(57 432 226 259 289 340 165)(58 290 424 341 227 166 260)(59 228 291 167 425 261 342)(60 426 229 253 292 334 168)(61 293 427 335 230 169 254)(62 231 294 170 428 255 336)(63 429 232 256 295 337 171)(154 421 200 329 343 219 385)(155 344 422 220 201 386 330)(156 202 345 387 423 331 221)(157 415 203 332 346 222 379)(158 347 416 223 204 380 333)(159 205 348 381 417 325 224)(160 418 206 326 349 225 382)(161 350 419 217 207 383 327)(162 199 351 384 420 328 218)
(1 176 219 310 147 336 409)(2 148 177 337 220 410 311)(3 221 149 411 178 312 338)(4 179 222 313 150 339 412)(5 151 180 340 223 413 314)(6 224 152 414 172 315 341)(7 173 225 307 153 342 406)(8 145 174 334 217 407 308)(9 218 146 408 175 309 335)(10 346 45 78 258 302 284)(11 259 347 303 37 285 79)(12 38 260 286 348 80 304)(13 349 39 81 261 305 287)(14 253 350 306 40 288 73)(15 41 254 280 351 74 298)(16 343 42 75 255 299 281)(17 256 344 300 43 282 76)(18 44 257 283 345 77 301)(19 62 46 87 392 385 133)(20 393 63 386 47 134 88)(21 48 394 135 55 89 387)(22 56 49 90 395 379 127)(23 396 57 380 50 128 82)(24 51 388 129 58 83 381)(25 59 52 84 389 382 130)(26 390 60 383 53 131 85)(27 54 391 132 61 86 384)(28 423 116 142 195 368 296)(29 196 415 369 117 297 143)(30 109 197 289 416 144 361)(31 417 110 136 198 362 290)(32 190 418 363 111 291 137)(33 112 191 292 419 138 364)(34 420 113 139 192 365 293)(35 193 421 366 114 294 140)(36 115 194 295 422 141 367)(64 433 166 250 205 279 208)(65 206 434 271 167 209 251)(66 168 207 210 435 252 272)(67 436 169 244 199 273 211)(68 200 437 274 170 212 245)(69 171 201 213 438 246 275)(70 439 163 247 202 276 214)(71 203 440 277 164 215 248)(72 165 204 216 441 249 278)(91 233 104 156 182 324 270)(92 183 234 316 105 262 157)(93 106 184 263 226 158 317)(94 227 107 159 185 318 264)(95 186 228 319 108 265 160)(96 100 187 266 229 161 320)(97 230 101 162 188 321 267)(98 189 231 322 102 268 154)(99 103 181 269 232 155 323)(118 235 326 399 355 425 378)(119 356 236 426 327 370 400)(120 328 357 371 237 401 427)(121 238 329 402 358 428 372)(122 359 239 429 330 373 403)(123 331 360 374 240 404 430)(124 241 332 405 352 431 375)(125 353 242 432 333 376 397)(126 325 354 377 243 398 424)
(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)(406 407 408 409 410 411 412 413 414)(415 416 417 418 419 420 421 422 423)(424 425 426 427 428 429 430 431 432)(433 434 435 436 437 438 439 440 441)
G:=sub<Sym(441)| (1,87,102,35,245,121,281)(2,246,88,122,103,282,36)(3,104,247,283,89,28,123)(4,90,105,29,248,124,284)(5,249,82,125,106,285,30)(6,107,250,286,83,31,126)(7,84,108,32,251,118,287)(8,252,85,119,100,288,33)(9,101,244,280,86,34,120)(10,179,395,262,196,71,241)(11,197,180,72,396,242,263)(12,388,198,243,172,264,64)(13,173,389,265,190,65,235)(14,191,174,66,390,236,266)(15,391,192,237,175,267,67)(16,176,392,268,193,68,238)(17,194,177,69,393,239,269)(18,394,195,240,178,270,70)(19,189,114,274,358,75,147)(20,359,181,76,115,148,275)(21,116,360,149,182,276,77)(22,183,117,277,352,78,150)(23,353,184,79,109,151,278)(24,110,354,152,185,279,80)(25,186,111,271,355,81,153)(26,356,187,73,112,145,272)(27,113,357,146,188,273,74)(37,361,314,441,128,397,93)(38,129,362,398,315,94,433)(39,307,130,95,363,434,399)(40,364,308,435,131,400,96)(41,132,365,401,309,97,436)(42,310,133,98,366,437,402)(43,367,311,438,134,403,99)(44,135,368,404,312,91,439)(45,313,127,92,369,440,405)(46,322,140,212,372,299,409)(47,373,323,300,141,410,213)(48,142,374,411,324,214,301)(49,316,143,215,375,302,412)(50,376,317,303,144,413,216)(51,136,377,414,318,208,304)(52,319,137,209,378,305,406)(53,370,320,306,138,407,210)(54,139,371,408,321,211,298)(55,296,430,338,233,163,257)(56,234,297,164,431,258,339)(57,432,226,259,289,340,165)(58,290,424,341,227,166,260)(59,228,291,167,425,261,342)(60,426,229,253,292,334,168)(61,293,427,335,230,169,254)(62,231,294,170,428,255,336)(63,429,232,256,295,337,171)(154,421,200,329,343,219,385)(155,344,422,220,201,386,330)(156,202,345,387,423,331,221)(157,415,203,332,346,222,379)(158,347,416,223,204,380,333)(159,205,348,381,417,325,224)(160,418,206,326,349,225,382)(161,350,419,217,207,383,327)(162,199,351,384,420,328,218), (1,176,219,310,147,336,409)(2,148,177,337,220,410,311)(3,221,149,411,178,312,338)(4,179,222,313,150,339,412)(5,151,180,340,223,413,314)(6,224,152,414,172,315,341)(7,173,225,307,153,342,406)(8,145,174,334,217,407,308)(9,218,146,408,175,309,335)(10,346,45,78,258,302,284)(11,259,347,303,37,285,79)(12,38,260,286,348,80,304)(13,349,39,81,261,305,287)(14,253,350,306,40,288,73)(15,41,254,280,351,74,298)(16,343,42,75,255,299,281)(17,256,344,300,43,282,76)(18,44,257,283,345,77,301)(19,62,46,87,392,385,133)(20,393,63,386,47,134,88)(21,48,394,135,55,89,387)(22,56,49,90,395,379,127)(23,396,57,380,50,128,82)(24,51,388,129,58,83,381)(25,59,52,84,389,382,130)(26,390,60,383,53,131,85)(27,54,391,132,61,86,384)(28,423,116,142,195,368,296)(29,196,415,369,117,297,143)(30,109,197,289,416,144,361)(31,417,110,136,198,362,290)(32,190,418,363,111,291,137)(33,112,191,292,419,138,364)(34,420,113,139,192,365,293)(35,193,421,366,114,294,140)(36,115,194,295,422,141,367)(64,433,166,250,205,279,208)(65,206,434,271,167,209,251)(66,168,207,210,435,252,272)(67,436,169,244,199,273,211)(68,200,437,274,170,212,245)(69,171,201,213,438,246,275)(70,439,163,247,202,276,214)(71,203,440,277,164,215,248)(72,165,204,216,441,249,278)(91,233,104,156,182,324,270)(92,183,234,316,105,262,157)(93,106,184,263,226,158,317)(94,227,107,159,185,318,264)(95,186,228,319,108,265,160)(96,100,187,266,229,161,320)(97,230,101,162,188,321,267)(98,189,231,322,102,268,154)(99,103,181,269,232,155,323)(118,235,326,399,355,425,378)(119,356,236,426,327,370,400)(120,328,357,371,237,401,427)(121,238,329,402,358,428,372)(122,359,239,429,330,373,403)(123,331,360,374,240,404,430)(124,241,332,405,352,431,375)(125,353,242,432,333,376,397)(126,325,354,377,243,398,424), (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)(406,407,408,409,410,411,412,413,414)(415,416,417,418,419,420,421,422,423)(424,425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440,441)>;
G:=Group( (1,87,102,35,245,121,281)(2,246,88,122,103,282,36)(3,104,247,283,89,28,123)(4,90,105,29,248,124,284)(5,249,82,125,106,285,30)(6,107,250,286,83,31,126)(7,84,108,32,251,118,287)(8,252,85,119,100,288,33)(9,101,244,280,86,34,120)(10,179,395,262,196,71,241)(11,197,180,72,396,242,263)(12,388,198,243,172,264,64)(13,173,389,265,190,65,235)(14,191,174,66,390,236,266)(15,391,192,237,175,267,67)(16,176,392,268,193,68,238)(17,194,177,69,393,239,269)(18,394,195,240,178,270,70)(19,189,114,274,358,75,147)(20,359,181,76,115,148,275)(21,116,360,149,182,276,77)(22,183,117,277,352,78,150)(23,353,184,79,109,151,278)(24,110,354,152,185,279,80)(25,186,111,271,355,81,153)(26,356,187,73,112,145,272)(27,113,357,146,188,273,74)(37,361,314,441,128,397,93)(38,129,362,398,315,94,433)(39,307,130,95,363,434,399)(40,364,308,435,131,400,96)(41,132,365,401,309,97,436)(42,310,133,98,366,437,402)(43,367,311,438,134,403,99)(44,135,368,404,312,91,439)(45,313,127,92,369,440,405)(46,322,140,212,372,299,409)(47,373,323,300,141,410,213)(48,142,374,411,324,214,301)(49,316,143,215,375,302,412)(50,376,317,303,144,413,216)(51,136,377,414,318,208,304)(52,319,137,209,378,305,406)(53,370,320,306,138,407,210)(54,139,371,408,321,211,298)(55,296,430,338,233,163,257)(56,234,297,164,431,258,339)(57,432,226,259,289,340,165)(58,290,424,341,227,166,260)(59,228,291,167,425,261,342)(60,426,229,253,292,334,168)(61,293,427,335,230,169,254)(62,231,294,170,428,255,336)(63,429,232,256,295,337,171)(154,421,200,329,343,219,385)(155,344,422,220,201,386,330)(156,202,345,387,423,331,221)(157,415,203,332,346,222,379)(158,347,416,223,204,380,333)(159,205,348,381,417,325,224)(160,418,206,326,349,225,382)(161,350,419,217,207,383,327)(162,199,351,384,420,328,218), (1,176,219,310,147,336,409)(2,148,177,337,220,410,311)(3,221,149,411,178,312,338)(4,179,222,313,150,339,412)(5,151,180,340,223,413,314)(6,224,152,414,172,315,341)(7,173,225,307,153,342,406)(8,145,174,334,217,407,308)(9,218,146,408,175,309,335)(10,346,45,78,258,302,284)(11,259,347,303,37,285,79)(12,38,260,286,348,80,304)(13,349,39,81,261,305,287)(14,253,350,306,40,288,73)(15,41,254,280,351,74,298)(16,343,42,75,255,299,281)(17,256,344,300,43,282,76)(18,44,257,283,345,77,301)(19,62,46,87,392,385,133)(20,393,63,386,47,134,88)(21,48,394,135,55,89,387)(22,56,49,90,395,379,127)(23,396,57,380,50,128,82)(24,51,388,129,58,83,381)(25,59,52,84,389,382,130)(26,390,60,383,53,131,85)(27,54,391,132,61,86,384)(28,423,116,142,195,368,296)(29,196,415,369,117,297,143)(30,109,197,289,416,144,361)(31,417,110,136,198,362,290)(32,190,418,363,111,291,137)(33,112,191,292,419,138,364)(34,420,113,139,192,365,293)(35,193,421,366,114,294,140)(36,115,194,295,422,141,367)(64,433,166,250,205,279,208)(65,206,434,271,167,209,251)(66,168,207,210,435,252,272)(67,436,169,244,199,273,211)(68,200,437,274,170,212,245)(69,171,201,213,438,246,275)(70,439,163,247,202,276,214)(71,203,440,277,164,215,248)(72,165,204,216,441,249,278)(91,233,104,156,182,324,270)(92,183,234,316,105,262,157)(93,106,184,263,226,158,317)(94,227,107,159,185,318,264)(95,186,228,319,108,265,160)(96,100,187,266,229,161,320)(97,230,101,162,188,321,267)(98,189,231,322,102,268,154)(99,103,181,269,232,155,323)(118,235,326,399,355,425,378)(119,356,236,426,327,370,400)(120,328,357,371,237,401,427)(121,238,329,402,358,428,372)(122,359,239,429,330,373,403)(123,331,360,374,240,404,430)(124,241,332,405,352,431,375)(125,353,242,432,333,376,397)(126,325,354,377,243,398,424), (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)(406,407,408,409,410,411,412,413,414)(415,416,417,418,419,420,421,422,423)(424,425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440,441) );
G=PermutationGroup([[(1,87,102,35,245,121,281),(2,246,88,122,103,282,36),(3,104,247,283,89,28,123),(4,90,105,29,248,124,284),(5,249,82,125,106,285,30),(6,107,250,286,83,31,126),(7,84,108,32,251,118,287),(8,252,85,119,100,288,33),(9,101,244,280,86,34,120),(10,179,395,262,196,71,241),(11,197,180,72,396,242,263),(12,388,198,243,172,264,64),(13,173,389,265,190,65,235),(14,191,174,66,390,236,266),(15,391,192,237,175,267,67),(16,176,392,268,193,68,238),(17,194,177,69,393,239,269),(18,394,195,240,178,270,70),(19,189,114,274,358,75,147),(20,359,181,76,115,148,275),(21,116,360,149,182,276,77),(22,183,117,277,352,78,150),(23,353,184,79,109,151,278),(24,110,354,152,185,279,80),(25,186,111,271,355,81,153),(26,356,187,73,112,145,272),(27,113,357,146,188,273,74),(37,361,314,441,128,397,93),(38,129,362,398,315,94,433),(39,307,130,95,363,434,399),(40,364,308,435,131,400,96),(41,132,365,401,309,97,436),(42,310,133,98,366,437,402),(43,367,311,438,134,403,99),(44,135,368,404,312,91,439),(45,313,127,92,369,440,405),(46,322,140,212,372,299,409),(47,373,323,300,141,410,213),(48,142,374,411,324,214,301),(49,316,143,215,375,302,412),(50,376,317,303,144,413,216),(51,136,377,414,318,208,304),(52,319,137,209,378,305,406),(53,370,320,306,138,407,210),(54,139,371,408,321,211,298),(55,296,430,338,233,163,257),(56,234,297,164,431,258,339),(57,432,226,259,289,340,165),(58,290,424,341,227,166,260),(59,228,291,167,425,261,342),(60,426,229,253,292,334,168),(61,293,427,335,230,169,254),(62,231,294,170,428,255,336),(63,429,232,256,295,337,171),(154,421,200,329,343,219,385),(155,344,422,220,201,386,330),(156,202,345,387,423,331,221),(157,415,203,332,346,222,379),(158,347,416,223,204,380,333),(159,205,348,381,417,325,224),(160,418,206,326,349,225,382),(161,350,419,217,207,383,327),(162,199,351,384,420,328,218)], [(1,176,219,310,147,336,409),(2,148,177,337,220,410,311),(3,221,149,411,178,312,338),(4,179,222,313,150,339,412),(5,151,180,340,223,413,314),(6,224,152,414,172,315,341),(7,173,225,307,153,342,406),(8,145,174,334,217,407,308),(9,218,146,408,175,309,335),(10,346,45,78,258,302,284),(11,259,347,303,37,285,79),(12,38,260,286,348,80,304),(13,349,39,81,261,305,287),(14,253,350,306,40,288,73),(15,41,254,280,351,74,298),(16,343,42,75,255,299,281),(17,256,344,300,43,282,76),(18,44,257,283,345,77,301),(19,62,46,87,392,385,133),(20,393,63,386,47,134,88),(21,48,394,135,55,89,387),(22,56,49,90,395,379,127),(23,396,57,380,50,128,82),(24,51,388,129,58,83,381),(25,59,52,84,389,382,130),(26,390,60,383,53,131,85),(27,54,391,132,61,86,384),(28,423,116,142,195,368,296),(29,196,415,369,117,297,143),(30,109,197,289,416,144,361),(31,417,110,136,198,362,290),(32,190,418,363,111,291,137),(33,112,191,292,419,138,364),(34,420,113,139,192,365,293),(35,193,421,366,114,294,140),(36,115,194,295,422,141,367),(64,433,166,250,205,279,208),(65,206,434,271,167,209,251),(66,168,207,210,435,252,272),(67,436,169,244,199,273,211),(68,200,437,274,170,212,245),(69,171,201,213,438,246,275),(70,439,163,247,202,276,214),(71,203,440,277,164,215,248),(72,165,204,216,441,249,278),(91,233,104,156,182,324,270),(92,183,234,316,105,262,157),(93,106,184,263,226,158,317),(94,227,107,159,185,318,264),(95,186,228,319,108,265,160),(96,100,187,266,229,161,320),(97,230,101,162,188,321,267),(98,189,231,322,102,268,154),(99,103,181,269,232,155,323),(118,235,326,399,355,425,378),(119,356,236,426,327,370,400),(120,328,357,371,237,401,427),(121,238,329,402,358,428,372),(122,359,239,429,330,373,403),(123,331,360,374,240,404,430),(124,241,332,405,352,431,375),(125,353,242,432,333,376,397),(126,325,354,377,243,398,424)], [(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),(406,407,408,409,410,411,412,413,414),(415,416,417,418,419,420,421,422,423),(424,425,426,427,428,429,430,431,432),(433,434,435,436,437,438,439,440,441)]])
57 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7P | 9A | ··· | 9F | 21A | ··· | 21AF |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 9 | ··· | 9 | 21 | ··· | 21 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 49 | ··· | 49 | 3 | ··· | 3 |
57 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C9 | C7⋊C3 | C7⋊C9 |
| kernel | C72⋊C9 | C7×C21 | C72 | C21 | C7 |
| # reps | 1 | 2 | 6 | 16 | 32 |
Matrix representation of C72⋊C9 ►in GL6(𝔽127)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 95 | 1 |
| 0 | 0 | 0 | 27 | 73 | 54 |
| 0 | 0 | 0 | 29 | 81 | 76 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 95 | 1 | 0 |
| 0 | 0 | 0 | 119 | 0 | 1 |
| 0 | 0 | 0 | 25 | 81 | 54 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 107 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 113 | 2 |
| 0 | 0 | 0 | 0 | 15 | 38 |
| 0 | 0 | 0 | 17 | 79 | 40 |
G:=sub<GL(6,GF(127))| [64,0,0,0,0,0,0,32,0,0,0,0,0,0,8,0,0,0,0,0,0,0,27,29,0,0,0,95,73,81,0,0,0,1,54,76],[32,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,95,119,25,0,0,0,1,0,81,0,0,0,0,1,54],[0,0,107,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,17,0,0,0,113,15,79,0,0,0,2,38,40] >;
C72⋊C9 in GAP, Magma, Sage, TeX
C_7^2\rtimes C_9
% in TeX
G:=Group("C7^2:C9"); // GroupNames label
G:=SmallGroup(441,6);
// by ID
G=gap.SmallGroup(441,6);
# by ID
G:=PCGroup([4,-3,-3,-7,-7,12,218,2019]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^9=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^4>;
// generators/relations
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