metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊2Q64, Q32.D7, C28.8D8, C56.12D4, C16.7D14, C14.11D16, C112.5C22, Dic56.2C2, C7⋊C32.C2, C4.4(D4⋊D7), (C7×Q32).1C2, C2.7(C7⋊D16), C8.12(C7⋊D4), SmallGroup(448,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊Q64
G = < a,b,c | a7=b32=1, c2=b16, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7⋊Q64
►Regular action on 448 points
Generators in S448
(1 139 371 212 326 33 314)(2 315 34 327 213 372 140)(3 141 373 214 328 35 316)(4 317 36 329 215 374 142)(5 143 375 216 330 37 318)(6 319 38 331 217 376 144)(7 145 377 218 332 39 320)(8 289 40 333 219 378 146)(9 147 379 220 334 41 290)(10 291 42 335 221 380 148)(11 149 381 222 336 43 292)(12 293 44 337 223 382 150)(13 151 383 224 338 45 294)(14 295 46 339 193 384 152)(15 153 353 194 340 47 296)(16 297 48 341 195 354 154)(17 155 355 196 342 49 298)(18 299 50 343 197 356 156)(19 157 357 198 344 51 300)(20 301 52 345 199 358 158)(21 159 359 200 346 53 302)(22 303 54 347 201 360 160)(23 129 361 202 348 55 304)(24 305 56 349 203 362 130)(25 131 363 204 350 57 306)(26 307 58 351 205 364 132)(27 133 365 206 352 59 308)(28 309 60 321 207 366 134)(29 135 367 208 322 61 310)(30 311 62 323 209 368 136)(31 137 369 210 324 63 312)(32 313 64 325 211 370 138)(65 426 232 124 184 283 398)(66 399 284 185 125 233 427)(67 428 234 126 186 285 400)(68 401 286 187 127 235 429)(69 430 236 128 188 287 402)(70 403 288 189 97 237 431)(71 432 238 98 190 257 404)(72 405 258 191 99 239 433)(73 434 240 100 192 259 406)(74 407 260 161 101 241 435)(75 436 242 102 162 261 408)(76 409 262 163 103 243 437)(77 438 244 104 164 263 410)(78 411 264 165 105 245 439)(79 440 246 106 166 265 412)(80 413 266 167 107 247 441)(81 442 248 108 168 267 414)(82 415 268 169 109 249 443)(83 444 250 110 170 269 416)(84 385 270 171 111 251 445)(85 446 252 112 172 271 386)(86 387 272 173 113 253 447)(87 448 254 114 174 273 388)(88 389 274 175 115 255 417)(89 418 256 116 176 275 390)(90 391 276 177 117 225 419)(91 420 226 118 178 277 392)(92 393 278 179 119 227 421)(93 422 228 120 180 279 394)(94 395 280 181 121 229 423)(95 424 230 122 182 281 396)(96 397 282 183 123 231 425)
(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)(417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)
(1 75 17 91)(2 74 18 90)(3 73 19 89)(4 72 20 88)(5 71 21 87)(6 70 22 86)(7 69 23 85)(8 68 24 84)(9 67 25 83)(10 66 26 82)(11 65 27 81)(12 96 28 80)(13 95 29 79)(14 94 30 78)(15 93 31 77)(16 92 32 76)(33 261 49 277)(34 260 50 276)(35 259 51 275)(36 258 52 274)(37 257 53 273)(38 288 54 272)(39 287 55 271)(40 286 56 270)(41 285 57 269)(42 284 58 268)(43 283 59 267)(44 282 60 266)(45 281 61 265)(46 280 62 264)(47 279 63 263)(48 278 64 262)(97 201 113 217)(98 200 114 216)(99 199 115 215)(100 198 116 214)(101 197 117 213)(102 196 118 212)(103 195 119 211)(104 194 120 210)(105 193 121 209)(106 224 122 208)(107 223 123 207)(108 222 124 206)(109 221 125 205)(110 220 126 204)(111 219 127 203)(112 218 128 202)(129 446 145 430)(130 445 146 429)(131 444 147 428)(132 443 148 427)(133 442 149 426)(134 441 150 425)(135 440 151 424)(136 439 152 423)(137 438 153 422)(138 437 154 421)(139 436 155 420)(140 435 156 419)(141 434 157 418)(142 433 158 417)(143 432 159 448)(144 431 160 447)(161 343 177 327)(162 342 178 326)(163 341 179 325)(164 340 180 324)(165 339 181 323)(166 338 182 322)(167 337 183 321)(168 336 184 352)(169 335 185 351)(170 334 186 350)(171 333 187 349)(172 332 188 348)(173 331 189 347)(174 330 190 346)(175 329 191 345)(176 328 192 344)(225 372 241 356)(226 371 242 355)(227 370 243 354)(228 369 244 353)(229 368 245 384)(230 367 246 383)(231 366 247 382)(232 365 248 381)(233 364 249 380)(234 363 250 379)(235 362 251 378)(236 361 252 377)(237 360 253 376)(238 359 254 375)(239 358 255 374)(240 357 256 373)(289 401 305 385)(290 400 306 416)(291 399 307 415)(292 398 308 414)(293 397 309 413)(294 396 310 412)(295 395 311 411)(296 394 312 410)(297 393 313 409)(298 392 314 408)(299 391 315 407)(300 390 316 406)(301 389 317 405)(302 388 318 404)(303 387 319 403)(304 386 320 402)
G:=sub<Sym(448)| (1,139,371,212,326,33,314)(2,315,34,327,213,372,140)(3,141,373,214,328,35,316)(4,317,36,329,215,374,142)(5,143,375,216,330,37,318)(6,319,38,331,217,376,144)(7,145,377,218,332,39,320)(8,289,40,333,219,378,146)(9,147,379,220,334,41,290)(10,291,42,335,221,380,148)(11,149,381,222,336,43,292)(12,293,44,337,223,382,150)(13,151,383,224,338,45,294)(14,295,46,339,193,384,152)(15,153,353,194,340,47,296)(16,297,48,341,195,354,154)(17,155,355,196,342,49,298)(18,299,50,343,197,356,156)(19,157,357,198,344,51,300)(20,301,52,345,199,358,158)(21,159,359,200,346,53,302)(22,303,54,347,201,360,160)(23,129,361,202,348,55,304)(24,305,56,349,203,362,130)(25,131,363,204,350,57,306)(26,307,58,351,205,364,132)(27,133,365,206,352,59,308)(28,309,60,321,207,366,134)(29,135,367,208,322,61,310)(30,311,62,323,209,368,136)(31,137,369,210,324,63,312)(32,313,64,325,211,370,138)(65,426,232,124,184,283,398)(66,399,284,185,125,233,427)(67,428,234,126,186,285,400)(68,401,286,187,127,235,429)(69,430,236,128,188,287,402)(70,403,288,189,97,237,431)(71,432,238,98,190,257,404)(72,405,258,191,99,239,433)(73,434,240,100,192,259,406)(74,407,260,161,101,241,435)(75,436,242,102,162,261,408)(76,409,262,163,103,243,437)(77,438,244,104,164,263,410)(78,411,264,165,105,245,439)(79,440,246,106,166,265,412)(80,413,266,167,107,247,441)(81,442,248,108,168,267,414)(82,415,268,169,109,249,443)(83,444,250,110,170,269,416)(84,385,270,171,111,251,445)(85,446,252,112,172,271,386)(86,387,272,173,113,253,447)(87,448,254,114,174,273,388)(88,389,274,175,115,255,417)(89,418,256,116,176,275,390)(90,391,276,177,117,225,419)(91,420,226,118,178,277,392)(92,393,278,179,119,227,421)(93,422,228,120,180,279,394)(94,395,280,181,121,229,423)(95,424,230,122,182,281,396)(96,397,282,183,123,231,425), (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)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,75,17,91)(2,74,18,90)(3,73,19,89)(4,72,20,88)(5,71,21,87)(6,70,22,86)(7,69,23,85)(8,68,24,84)(9,67,25,83)(10,66,26,82)(11,65,27,81)(12,96,28,80)(13,95,29,79)(14,94,30,78)(15,93,31,77)(16,92,32,76)(33,261,49,277)(34,260,50,276)(35,259,51,275)(36,258,52,274)(37,257,53,273)(38,288,54,272)(39,287,55,271)(40,286,56,270)(41,285,57,269)(42,284,58,268)(43,283,59,267)(44,282,60,266)(45,281,61,265)(46,280,62,264)(47,279,63,263)(48,278,64,262)(97,201,113,217)(98,200,114,216)(99,199,115,215)(100,198,116,214)(101,197,117,213)(102,196,118,212)(103,195,119,211)(104,194,120,210)(105,193,121,209)(106,224,122,208)(107,223,123,207)(108,222,124,206)(109,221,125,205)(110,220,126,204)(111,219,127,203)(112,218,128,202)(129,446,145,430)(130,445,146,429)(131,444,147,428)(132,443,148,427)(133,442,149,426)(134,441,150,425)(135,440,151,424)(136,439,152,423)(137,438,153,422)(138,437,154,421)(139,436,155,420)(140,435,156,419)(141,434,157,418)(142,433,158,417)(143,432,159,448)(144,431,160,447)(161,343,177,327)(162,342,178,326)(163,341,179,325)(164,340,180,324)(165,339,181,323)(166,338,182,322)(167,337,183,321)(168,336,184,352)(169,335,185,351)(170,334,186,350)(171,333,187,349)(172,332,188,348)(173,331,189,347)(174,330,190,346)(175,329,191,345)(176,328,192,344)(225,372,241,356)(226,371,242,355)(227,370,243,354)(228,369,244,353)(229,368,245,384)(230,367,246,383)(231,366,247,382)(232,365,248,381)(233,364,249,380)(234,363,250,379)(235,362,251,378)(236,361,252,377)(237,360,253,376)(238,359,254,375)(239,358,255,374)(240,357,256,373)(289,401,305,385)(290,400,306,416)(291,399,307,415)(292,398,308,414)(293,397,309,413)(294,396,310,412)(295,395,311,411)(296,394,312,410)(297,393,313,409)(298,392,314,408)(299,391,315,407)(300,390,316,406)(301,389,317,405)(302,388,318,404)(303,387,319,403)(304,386,320,402)>;
G:=Group( (1,139,371,212,326,33,314)(2,315,34,327,213,372,140)(3,141,373,214,328,35,316)(4,317,36,329,215,374,142)(5,143,375,216,330,37,318)(6,319,38,331,217,376,144)(7,145,377,218,332,39,320)(8,289,40,333,219,378,146)(9,147,379,220,334,41,290)(10,291,42,335,221,380,148)(11,149,381,222,336,43,292)(12,293,44,337,223,382,150)(13,151,383,224,338,45,294)(14,295,46,339,193,384,152)(15,153,353,194,340,47,296)(16,297,48,341,195,354,154)(17,155,355,196,342,49,298)(18,299,50,343,197,356,156)(19,157,357,198,344,51,300)(20,301,52,345,199,358,158)(21,159,359,200,346,53,302)(22,303,54,347,201,360,160)(23,129,361,202,348,55,304)(24,305,56,349,203,362,130)(25,131,363,204,350,57,306)(26,307,58,351,205,364,132)(27,133,365,206,352,59,308)(28,309,60,321,207,366,134)(29,135,367,208,322,61,310)(30,311,62,323,209,368,136)(31,137,369,210,324,63,312)(32,313,64,325,211,370,138)(65,426,232,124,184,283,398)(66,399,284,185,125,233,427)(67,428,234,126,186,285,400)(68,401,286,187,127,235,429)(69,430,236,128,188,287,402)(70,403,288,189,97,237,431)(71,432,238,98,190,257,404)(72,405,258,191,99,239,433)(73,434,240,100,192,259,406)(74,407,260,161,101,241,435)(75,436,242,102,162,261,408)(76,409,262,163,103,243,437)(77,438,244,104,164,263,410)(78,411,264,165,105,245,439)(79,440,246,106,166,265,412)(80,413,266,167,107,247,441)(81,442,248,108,168,267,414)(82,415,268,169,109,249,443)(83,444,250,110,170,269,416)(84,385,270,171,111,251,445)(85,446,252,112,172,271,386)(86,387,272,173,113,253,447)(87,448,254,114,174,273,388)(88,389,274,175,115,255,417)(89,418,256,116,176,275,390)(90,391,276,177,117,225,419)(91,420,226,118,178,277,392)(92,393,278,179,119,227,421)(93,422,228,120,180,279,394)(94,395,280,181,121,229,423)(95,424,230,122,182,281,396)(96,397,282,183,123,231,425), (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)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,75,17,91)(2,74,18,90)(3,73,19,89)(4,72,20,88)(5,71,21,87)(6,70,22,86)(7,69,23,85)(8,68,24,84)(9,67,25,83)(10,66,26,82)(11,65,27,81)(12,96,28,80)(13,95,29,79)(14,94,30,78)(15,93,31,77)(16,92,32,76)(33,261,49,277)(34,260,50,276)(35,259,51,275)(36,258,52,274)(37,257,53,273)(38,288,54,272)(39,287,55,271)(40,286,56,270)(41,285,57,269)(42,284,58,268)(43,283,59,267)(44,282,60,266)(45,281,61,265)(46,280,62,264)(47,279,63,263)(48,278,64,262)(97,201,113,217)(98,200,114,216)(99,199,115,215)(100,198,116,214)(101,197,117,213)(102,196,118,212)(103,195,119,211)(104,194,120,210)(105,193,121,209)(106,224,122,208)(107,223,123,207)(108,222,124,206)(109,221,125,205)(110,220,126,204)(111,219,127,203)(112,218,128,202)(129,446,145,430)(130,445,146,429)(131,444,147,428)(132,443,148,427)(133,442,149,426)(134,441,150,425)(135,440,151,424)(136,439,152,423)(137,438,153,422)(138,437,154,421)(139,436,155,420)(140,435,156,419)(141,434,157,418)(142,433,158,417)(143,432,159,448)(144,431,160,447)(161,343,177,327)(162,342,178,326)(163,341,179,325)(164,340,180,324)(165,339,181,323)(166,338,182,322)(167,337,183,321)(168,336,184,352)(169,335,185,351)(170,334,186,350)(171,333,187,349)(172,332,188,348)(173,331,189,347)(174,330,190,346)(175,329,191,345)(176,328,192,344)(225,372,241,356)(226,371,242,355)(227,370,243,354)(228,369,244,353)(229,368,245,384)(230,367,246,383)(231,366,247,382)(232,365,248,381)(233,364,249,380)(234,363,250,379)(235,362,251,378)(236,361,252,377)(237,360,253,376)(238,359,254,375)(239,358,255,374)(240,357,256,373)(289,401,305,385)(290,400,306,416)(291,399,307,415)(292,398,308,414)(293,397,309,413)(294,396,310,412)(295,395,311,411)(296,394,312,410)(297,393,313,409)(298,392,314,408)(299,391,315,407)(300,390,316,406)(301,389,317,405)(302,388,318,404)(303,387,319,403)(304,386,320,402) );
G=PermutationGroup([[(1,139,371,212,326,33,314),(2,315,34,327,213,372,140),(3,141,373,214,328,35,316),(4,317,36,329,215,374,142),(5,143,375,216,330,37,318),(6,319,38,331,217,376,144),(7,145,377,218,332,39,320),(8,289,40,333,219,378,146),(9,147,379,220,334,41,290),(10,291,42,335,221,380,148),(11,149,381,222,336,43,292),(12,293,44,337,223,382,150),(13,151,383,224,338,45,294),(14,295,46,339,193,384,152),(15,153,353,194,340,47,296),(16,297,48,341,195,354,154),(17,155,355,196,342,49,298),(18,299,50,343,197,356,156),(19,157,357,198,344,51,300),(20,301,52,345,199,358,158),(21,159,359,200,346,53,302),(22,303,54,347,201,360,160),(23,129,361,202,348,55,304),(24,305,56,349,203,362,130),(25,131,363,204,350,57,306),(26,307,58,351,205,364,132),(27,133,365,206,352,59,308),(28,309,60,321,207,366,134),(29,135,367,208,322,61,310),(30,311,62,323,209,368,136),(31,137,369,210,324,63,312),(32,313,64,325,211,370,138),(65,426,232,124,184,283,398),(66,399,284,185,125,233,427),(67,428,234,126,186,285,400),(68,401,286,187,127,235,429),(69,430,236,128,188,287,402),(70,403,288,189,97,237,431),(71,432,238,98,190,257,404),(72,405,258,191,99,239,433),(73,434,240,100,192,259,406),(74,407,260,161,101,241,435),(75,436,242,102,162,261,408),(76,409,262,163,103,243,437),(77,438,244,104,164,263,410),(78,411,264,165,105,245,439),(79,440,246,106,166,265,412),(80,413,266,167,107,247,441),(81,442,248,108,168,267,414),(82,415,268,169,109,249,443),(83,444,250,110,170,269,416),(84,385,270,171,111,251,445),(85,446,252,112,172,271,386),(86,387,272,173,113,253,447),(87,448,254,114,174,273,388),(88,389,274,175,115,255,417),(89,418,256,116,176,275,390),(90,391,276,177,117,225,419),(91,420,226,118,178,277,392),(92,393,278,179,119,227,421),(93,422,228,120,180,279,394),(94,395,280,181,121,229,423),(95,424,230,122,182,281,396),(96,397,282,183,123,231,425)], [(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),(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)], [(1,75,17,91),(2,74,18,90),(3,73,19,89),(4,72,20,88),(5,71,21,87),(6,70,22,86),(7,69,23,85),(8,68,24,84),(9,67,25,83),(10,66,26,82),(11,65,27,81),(12,96,28,80),(13,95,29,79),(14,94,30,78),(15,93,31,77),(16,92,32,76),(33,261,49,277),(34,260,50,276),(35,259,51,275),(36,258,52,274),(37,257,53,273),(38,288,54,272),(39,287,55,271),(40,286,56,270),(41,285,57,269),(42,284,58,268),(43,283,59,267),(44,282,60,266),(45,281,61,265),(46,280,62,264),(47,279,63,263),(48,278,64,262),(97,201,113,217),(98,200,114,216),(99,199,115,215),(100,198,116,214),(101,197,117,213),(102,196,118,212),(103,195,119,211),(104,194,120,210),(105,193,121,209),(106,224,122,208),(107,223,123,207),(108,222,124,206),(109,221,125,205),(110,220,126,204),(111,219,127,203),(112,218,128,202),(129,446,145,430),(130,445,146,429),(131,444,147,428),(132,443,148,427),(133,442,149,426),(134,441,150,425),(135,440,151,424),(136,439,152,423),(137,438,153,422),(138,437,154,421),(139,436,155,420),(140,435,156,419),(141,434,157,418),(142,433,158,417),(143,432,159,448),(144,431,160,447),(161,343,177,327),(162,342,178,326),(163,341,179,325),(164,340,180,324),(165,339,181,323),(166,338,182,322),(167,337,183,321),(168,336,184,352),(169,335,185,351),(170,334,186,350),(171,333,187,349),(172,332,188,348),(173,331,189,347),(174,330,190,346),(175,329,191,345),(176,328,192,344),(225,372,241,356),(226,371,242,355),(227,370,243,354),(228,369,244,353),(229,368,245,384),(230,367,246,383),(231,366,247,382),(232,365,248,381),(233,364,249,380),(234,363,250,379),(235,362,251,378),(236,361,252,377),(237,360,253,376),(238,359,254,375),(239,358,255,374),(240,357,256,373),(289,401,305,385),(290,400,306,416),(291,399,307,415),(292,398,308,414),(293,397,309,413),(294,396,310,412),(295,395,311,411),(296,394,312,410),(297,393,313,409),(298,392,314,408),(299,391,315,407),(300,390,316,406),(301,389,317,405),(302,388,318,404),(303,387,319,403),(304,386,320,402)]])
52 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 28D | ··· | 28I | 32A | ··· | 32H | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 16 | 112 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | ··· | 16 | 14 | ··· | 14 | 4 | ··· | 4 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | D16 | C7⋊D4 | Q64 | D4⋊D7 | C7⋊D16 | C7⋊Q64 |
| kernel | C7⋊Q64 | C7⋊C32 | Dic56 | C7×Q32 | C56 | Q32 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 3 | 6 | 12 |
Matrix representation of C7⋊Q64 ►in GL4(𝔽449) generated by
| 398 | 1 | 0 | 0 |
| 144 | 94 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 89 | 252 | 0 | 0 |
| 414 | 360 | 0 | 0 |
| 0 | 0 | 147 | 112 |
| 0 | 0 | 244 | 247 |
| 300 | 321 | 0 | 0 |
| 426 | 149 | 0 | 0 |
| 0 | 0 | 318 | 327 |
| 0 | 0 | 229 | 131 |
G:=sub<GL(4,GF(449))| [398,144,0,0,1,94,0,0,0,0,1,0,0,0,0,1],[89,414,0,0,252,360,0,0,0,0,147,244,0,0,112,247],[300,426,0,0,321,149,0,0,0,0,318,229,0,0,327,131] >;
C7⋊Q64 in GAP, Magma, Sage, TeX
C_7\rtimes Q_{64} % in TeX
G:=Group("C7:Q64"); // GroupNames label
G:=SmallGroup(448,79);
// by ID
G=gap.SmallGroup(448,79);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,85,232,254,135,142,675,346,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^7=b^32=1,c^2=b^16,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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