direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C7×Q64, C32.C14, Q32.C14, C224.2C2, C28.41D8, C56.70D4, C14.17D16, C112.21C22, C8.7(C7×D4), C4.3(C7×D8), C2.5(C7×D16), C16.4(C2×C14), (C7×Q32).2C2, SmallGroup(448,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q64
G = < a,b,c | a7=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Q64
►Regular action on 448 points
Generators in S448
(1 391 149 269 61 314 216)(2 392 150 270 62 315 217)(3 393 151 271 63 316 218)(4 394 152 272 64 317 219)(5 395 153 273 33 318 220)(6 396 154 274 34 319 221)(7 397 155 275 35 320 222)(8 398 156 276 36 289 223)(9 399 157 277 37 290 224)(10 400 158 278 38 291 193)(11 401 159 279 39 292 194)(12 402 160 280 40 293 195)(13 403 129 281 41 294 196)(14 404 130 282 42 295 197)(15 405 131 283 43 296 198)(16 406 132 284 44 297 199)(17 407 133 285 45 298 200)(18 408 134 286 46 299 201)(19 409 135 287 47 300 202)(20 410 136 288 48 301 203)(21 411 137 257 49 302 204)(22 412 138 258 50 303 205)(23 413 139 259 51 304 206)(24 414 140 260 52 305 207)(25 415 141 261 53 306 208)(26 416 142 262 54 307 209)(27 385 143 263 55 308 210)(28 386 144 264 56 309 211)(29 387 145 265 57 310 212)(30 388 146 266 58 311 213)(31 389 147 267 59 312 214)(32 390 148 268 60 313 215)(65 367 114 438 245 163 347)(66 368 115 439 246 164 348)(67 369 116 440 247 165 349)(68 370 117 441 248 166 350)(69 371 118 442 249 167 351)(70 372 119 443 250 168 352)(71 373 120 444 251 169 321)(72 374 121 445 252 170 322)(73 375 122 446 253 171 323)(74 376 123 447 254 172 324)(75 377 124 448 255 173 325)(76 378 125 417 256 174 326)(77 379 126 418 225 175 327)(78 380 127 419 226 176 328)(79 381 128 420 227 177 329)(80 382 97 421 228 178 330)(81 383 98 422 229 179 331)(82 384 99 423 230 180 332)(83 353 100 424 231 181 333)(84 354 101 425 232 182 334)(85 355 102 426 233 183 335)(86 356 103 427 234 184 336)(87 357 104 428 235 185 337)(88 358 105 429 236 186 338)(89 359 106 430 237 187 339)(90 360 107 431 238 188 340)(91 361 108 432 239 189 341)(92 362 109 433 240 190 342)(93 363 110 434 241 191 343)(94 364 111 435 242 192 344)(95 365 112 436 243 161 345)(96 366 113 437 244 162 346)
(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)
(1 432 17 448)(2 431 18 447)(3 430 19 446)(4 429 20 445)(5 428 21 444)(6 427 22 443)(7 426 23 442)(8 425 24 441)(9 424 25 440)(10 423 26 439)(11 422 27 438)(12 421 28 437)(13 420 29 436)(14 419 30 435)(15 418 31 434)(16 417 32 433)(33 87 49 71)(34 86 50 70)(35 85 51 69)(36 84 52 68)(37 83 53 67)(38 82 54 66)(39 81 55 65)(40 80 56 96)(41 79 57 95)(42 78 58 94)(43 77 59 93)(44 76 60 92)(45 75 61 91)(46 74 62 90)(47 73 63 89)(48 72 64 88)(97 211 113 195)(98 210 114 194)(99 209 115 193)(100 208 116 224)(101 207 117 223)(102 206 118 222)(103 205 119 221)(104 204 120 220)(105 203 121 219)(106 202 122 218)(107 201 123 217)(108 200 124 216)(109 199 125 215)(110 198 126 214)(111 197 127 213)(112 196 128 212)(129 177 145 161)(130 176 146 192)(131 175 147 191)(132 174 148 190)(133 173 149 189)(134 172 150 188)(135 171 151 187)(136 170 152 186)(137 169 153 185)(138 168 154 184)(139 167 155 183)(140 166 156 182)(141 165 157 181)(142 164 158 180)(143 163 159 179)(144 162 160 178)(225 389 241 405)(226 388 242 404)(227 387 243 403)(228 386 244 402)(229 385 245 401)(230 416 246 400)(231 415 247 399)(232 414 248 398)(233 413 249 397)(234 412 250 396)(235 411 251 395)(236 410 252 394)(237 409 253 393)(238 408 254 392)(239 407 255 391)(240 406 256 390)(257 321 273 337)(258 352 274 336)(259 351 275 335)(260 350 276 334)(261 349 277 333)(262 348 278 332)(263 347 279 331)(264 346 280 330)(265 345 281 329)(266 344 282 328)(267 343 283 327)(268 342 284 326)(269 341 285 325)(270 340 286 324)(271 339 287 323)(272 338 288 322)(289 354 305 370)(290 353 306 369)(291 384 307 368)(292 383 308 367)(293 382 309 366)(294 381 310 365)(295 380 311 364)(296 379 312 363)(297 378 313 362)(298 377 314 361)(299 376 315 360)(300 375 316 359)(301 374 317 358)(302 373 318 357)(303 372 319 356)(304 371 320 355)
G:=sub<Sym(448)| (1,391,149,269,61,314,216)(2,392,150,270,62,315,217)(3,393,151,271,63,316,218)(4,394,152,272,64,317,219)(5,395,153,273,33,318,220)(6,396,154,274,34,319,221)(7,397,155,275,35,320,222)(8,398,156,276,36,289,223)(9,399,157,277,37,290,224)(10,400,158,278,38,291,193)(11,401,159,279,39,292,194)(12,402,160,280,40,293,195)(13,403,129,281,41,294,196)(14,404,130,282,42,295,197)(15,405,131,283,43,296,198)(16,406,132,284,44,297,199)(17,407,133,285,45,298,200)(18,408,134,286,46,299,201)(19,409,135,287,47,300,202)(20,410,136,288,48,301,203)(21,411,137,257,49,302,204)(22,412,138,258,50,303,205)(23,413,139,259,51,304,206)(24,414,140,260,52,305,207)(25,415,141,261,53,306,208)(26,416,142,262,54,307,209)(27,385,143,263,55,308,210)(28,386,144,264,56,309,211)(29,387,145,265,57,310,212)(30,388,146,266,58,311,213)(31,389,147,267,59,312,214)(32,390,148,268,60,313,215)(65,367,114,438,245,163,347)(66,368,115,439,246,164,348)(67,369,116,440,247,165,349)(68,370,117,441,248,166,350)(69,371,118,442,249,167,351)(70,372,119,443,250,168,352)(71,373,120,444,251,169,321)(72,374,121,445,252,170,322)(73,375,122,446,253,171,323)(74,376,123,447,254,172,324)(75,377,124,448,255,173,325)(76,378,125,417,256,174,326)(77,379,126,418,225,175,327)(78,380,127,419,226,176,328)(79,381,128,420,227,177,329)(80,382,97,421,228,178,330)(81,383,98,422,229,179,331)(82,384,99,423,230,180,332)(83,353,100,424,231,181,333)(84,354,101,425,232,182,334)(85,355,102,426,233,183,335)(86,356,103,427,234,184,336)(87,357,104,428,235,185,337)(88,358,105,429,236,186,338)(89,359,106,430,237,187,339)(90,360,107,431,238,188,340)(91,361,108,432,239,189,341)(92,362,109,433,240,190,342)(93,363,110,434,241,191,343)(94,364,111,435,242,192,344)(95,365,112,436,243,161,345)(96,366,113,437,244,162,346), (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), (1,432,17,448)(2,431,18,447)(3,430,19,446)(4,429,20,445)(5,428,21,444)(6,427,22,443)(7,426,23,442)(8,425,24,441)(9,424,25,440)(10,423,26,439)(11,422,27,438)(12,421,28,437)(13,420,29,436)(14,419,30,435)(15,418,31,434)(16,417,32,433)(33,87,49,71)(34,86,50,70)(35,85,51,69)(36,84,52,68)(37,83,53,67)(38,82,54,66)(39,81,55,65)(40,80,56,96)(41,79,57,95)(42,78,58,94)(43,77,59,93)(44,76,60,92)(45,75,61,91)(46,74,62,90)(47,73,63,89)(48,72,64,88)(97,211,113,195)(98,210,114,194)(99,209,115,193)(100,208,116,224)(101,207,117,223)(102,206,118,222)(103,205,119,221)(104,204,120,220)(105,203,121,219)(106,202,122,218)(107,201,123,217)(108,200,124,216)(109,199,125,215)(110,198,126,214)(111,197,127,213)(112,196,128,212)(129,177,145,161)(130,176,146,192)(131,175,147,191)(132,174,148,190)(133,173,149,189)(134,172,150,188)(135,171,151,187)(136,170,152,186)(137,169,153,185)(138,168,154,184)(139,167,155,183)(140,166,156,182)(141,165,157,181)(142,164,158,180)(143,163,159,179)(144,162,160,178)(225,389,241,405)(226,388,242,404)(227,387,243,403)(228,386,244,402)(229,385,245,401)(230,416,246,400)(231,415,247,399)(232,414,248,398)(233,413,249,397)(234,412,250,396)(235,411,251,395)(236,410,252,394)(237,409,253,393)(238,408,254,392)(239,407,255,391)(240,406,256,390)(257,321,273,337)(258,352,274,336)(259,351,275,335)(260,350,276,334)(261,349,277,333)(262,348,278,332)(263,347,279,331)(264,346,280,330)(265,345,281,329)(266,344,282,328)(267,343,283,327)(268,342,284,326)(269,341,285,325)(270,340,286,324)(271,339,287,323)(272,338,288,322)(289,354,305,370)(290,353,306,369)(291,384,307,368)(292,383,308,367)(293,382,309,366)(294,381,310,365)(295,380,311,364)(296,379,312,363)(297,378,313,362)(298,377,314,361)(299,376,315,360)(300,375,316,359)(301,374,317,358)(302,373,318,357)(303,372,319,356)(304,371,320,355)>;
G:=Group( (1,391,149,269,61,314,216)(2,392,150,270,62,315,217)(3,393,151,271,63,316,218)(4,394,152,272,64,317,219)(5,395,153,273,33,318,220)(6,396,154,274,34,319,221)(7,397,155,275,35,320,222)(8,398,156,276,36,289,223)(9,399,157,277,37,290,224)(10,400,158,278,38,291,193)(11,401,159,279,39,292,194)(12,402,160,280,40,293,195)(13,403,129,281,41,294,196)(14,404,130,282,42,295,197)(15,405,131,283,43,296,198)(16,406,132,284,44,297,199)(17,407,133,285,45,298,200)(18,408,134,286,46,299,201)(19,409,135,287,47,300,202)(20,410,136,288,48,301,203)(21,411,137,257,49,302,204)(22,412,138,258,50,303,205)(23,413,139,259,51,304,206)(24,414,140,260,52,305,207)(25,415,141,261,53,306,208)(26,416,142,262,54,307,209)(27,385,143,263,55,308,210)(28,386,144,264,56,309,211)(29,387,145,265,57,310,212)(30,388,146,266,58,311,213)(31,389,147,267,59,312,214)(32,390,148,268,60,313,215)(65,367,114,438,245,163,347)(66,368,115,439,246,164,348)(67,369,116,440,247,165,349)(68,370,117,441,248,166,350)(69,371,118,442,249,167,351)(70,372,119,443,250,168,352)(71,373,120,444,251,169,321)(72,374,121,445,252,170,322)(73,375,122,446,253,171,323)(74,376,123,447,254,172,324)(75,377,124,448,255,173,325)(76,378,125,417,256,174,326)(77,379,126,418,225,175,327)(78,380,127,419,226,176,328)(79,381,128,420,227,177,329)(80,382,97,421,228,178,330)(81,383,98,422,229,179,331)(82,384,99,423,230,180,332)(83,353,100,424,231,181,333)(84,354,101,425,232,182,334)(85,355,102,426,233,183,335)(86,356,103,427,234,184,336)(87,357,104,428,235,185,337)(88,358,105,429,236,186,338)(89,359,106,430,237,187,339)(90,360,107,431,238,188,340)(91,361,108,432,239,189,341)(92,362,109,433,240,190,342)(93,363,110,434,241,191,343)(94,364,111,435,242,192,344)(95,365,112,436,243,161,345)(96,366,113,437,244,162,346), (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), (1,432,17,448)(2,431,18,447)(3,430,19,446)(4,429,20,445)(5,428,21,444)(6,427,22,443)(7,426,23,442)(8,425,24,441)(9,424,25,440)(10,423,26,439)(11,422,27,438)(12,421,28,437)(13,420,29,436)(14,419,30,435)(15,418,31,434)(16,417,32,433)(33,87,49,71)(34,86,50,70)(35,85,51,69)(36,84,52,68)(37,83,53,67)(38,82,54,66)(39,81,55,65)(40,80,56,96)(41,79,57,95)(42,78,58,94)(43,77,59,93)(44,76,60,92)(45,75,61,91)(46,74,62,90)(47,73,63,89)(48,72,64,88)(97,211,113,195)(98,210,114,194)(99,209,115,193)(100,208,116,224)(101,207,117,223)(102,206,118,222)(103,205,119,221)(104,204,120,220)(105,203,121,219)(106,202,122,218)(107,201,123,217)(108,200,124,216)(109,199,125,215)(110,198,126,214)(111,197,127,213)(112,196,128,212)(129,177,145,161)(130,176,146,192)(131,175,147,191)(132,174,148,190)(133,173,149,189)(134,172,150,188)(135,171,151,187)(136,170,152,186)(137,169,153,185)(138,168,154,184)(139,167,155,183)(140,166,156,182)(141,165,157,181)(142,164,158,180)(143,163,159,179)(144,162,160,178)(225,389,241,405)(226,388,242,404)(227,387,243,403)(228,386,244,402)(229,385,245,401)(230,416,246,400)(231,415,247,399)(232,414,248,398)(233,413,249,397)(234,412,250,396)(235,411,251,395)(236,410,252,394)(237,409,253,393)(238,408,254,392)(239,407,255,391)(240,406,256,390)(257,321,273,337)(258,352,274,336)(259,351,275,335)(260,350,276,334)(261,349,277,333)(262,348,278,332)(263,347,279,331)(264,346,280,330)(265,345,281,329)(266,344,282,328)(267,343,283,327)(268,342,284,326)(269,341,285,325)(270,340,286,324)(271,339,287,323)(272,338,288,322)(289,354,305,370)(290,353,306,369)(291,384,307,368)(292,383,308,367)(293,382,309,366)(294,381,310,365)(295,380,311,364)(296,379,312,363)(297,378,313,362)(298,377,314,361)(299,376,315,360)(300,375,316,359)(301,374,317,358)(302,373,318,357)(303,372,319,356)(304,371,320,355) );
G=PermutationGroup([[(1,391,149,269,61,314,216),(2,392,150,270,62,315,217),(3,393,151,271,63,316,218),(4,394,152,272,64,317,219),(5,395,153,273,33,318,220),(6,396,154,274,34,319,221),(7,397,155,275,35,320,222),(8,398,156,276,36,289,223),(9,399,157,277,37,290,224),(10,400,158,278,38,291,193),(11,401,159,279,39,292,194),(12,402,160,280,40,293,195),(13,403,129,281,41,294,196),(14,404,130,282,42,295,197),(15,405,131,283,43,296,198),(16,406,132,284,44,297,199),(17,407,133,285,45,298,200),(18,408,134,286,46,299,201),(19,409,135,287,47,300,202),(20,410,136,288,48,301,203),(21,411,137,257,49,302,204),(22,412,138,258,50,303,205),(23,413,139,259,51,304,206),(24,414,140,260,52,305,207),(25,415,141,261,53,306,208),(26,416,142,262,54,307,209),(27,385,143,263,55,308,210),(28,386,144,264,56,309,211),(29,387,145,265,57,310,212),(30,388,146,266,58,311,213),(31,389,147,267,59,312,214),(32,390,148,268,60,313,215),(65,367,114,438,245,163,347),(66,368,115,439,246,164,348),(67,369,116,440,247,165,349),(68,370,117,441,248,166,350),(69,371,118,442,249,167,351),(70,372,119,443,250,168,352),(71,373,120,444,251,169,321),(72,374,121,445,252,170,322),(73,375,122,446,253,171,323),(74,376,123,447,254,172,324),(75,377,124,448,255,173,325),(76,378,125,417,256,174,326),(77,379,126,418,225,175,327),(78,380,127,419,226,176,328),(79,381,128,420,227,177,329),(80,382,97,421,228,178,330),(81,383,98,422,229,179,331),(82,384,99,423,230,180,332),(83,353,100,424,231,181,333),(84,354,101,425,232,182,334),(85,355,102,426,233,183,335),(86,356,103,427,234,184,336),(87,357,104,428,235,185,337),(88,358,105,429,236,186,338),(89,359,106,430,237,187,339),(90,360,107,431,238,188,340),(91,361,108,432,239,189,341),(92,362,109,433,240,190,342),(93,363,110,434,241,191,343),(94,364,111,435,242,192,344),(95,365,112,436,243,161,345),(96,366,113,437,244,162,346)], [(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)], [(1,432,17,448),(2,431,18,447),(3,430,19,446),(4,429,20,445),(5,428,21,444),(6,427,22,443),(7,426,23,442),(8,425,24,441),(9,424,25,440),(10,423,26,439),(11,422,27,438),(12,421,28,437),(13,420,29,436),(14,419,30,435),(15,418,31,434),(16,417,32,433),(33,87,49,71),(34,86,50,70),(35,85,51,69),(36,84,52,68),(37,83,53,67),(38,82,54,66),(39,81,55,65),(40,80,56,96),(41,79,57,95),(42,78,58,94),(43,77,59,93),(44,76,60,92),(45,75,61,91),(46,74,62,90),(47,73,63,89),(48,72,64,88),(97,211,113,195),(98,210,114,194),(99,209,115,193),(100,208,116,224),(101,207,117,223),(102,206,118,222),(103,205,119,221),(104,204,120,220),(105,203,121,219),(106,202,122,218),(107,201,123,217),(108,200,124,216),(109,199,125,215),(110,198,126,214),(111,197,127,213),(112,196,128,212),(129,177,145,161),(130,176,146,192),(131,175,147,191),(132,174,148,190),(133,173,149,189),(134,172,150,188),(135,171,151,187),(136,170,152,186),(137,169,153,185),(138,168,154,184),(139,167,155,183),(140,166,156,182),(141,165,157,181),(142,164,158,180),(143,163,159,179),(144,162,160,178),(225,389,241,405),(226,388,242,404),(227,387,243,403),(228,386,244,402),(229,385,245,401),(230,416,246,400),(231,415,247,399),(232,414,248,398),(233,413,249,397),(234,412,250,396),(235,411,251,395),(236,410,252,394),(237,409,253,393),(238,408,254,392),(239,407,255,391),(240,406,256,390),(257,321,273,337),(258,352,274,336),(259,351,275,335),(260,350,276,334),(261,349,277,333),(262,348,278,332),(263,347,279,331),(264,346,280,330),(265,345,281,329),(266,344,282,328),(267,343,283,327),(268,342,284,326),(269,341,285,325),(270,340,286,324),(271,339,287,323),(272,338,288,322),(289,354,305,370),(290,353,306,369),(291,384,307,368),(292,383,308,367),(293,382,309,366),(294,381,310,365),(295,380,311,364),(296,379,312,363),(297,378,313,362),(298,377,314,361),(299,376,315,360),(300,375,316,359),(301,374,317,358),(302,373,318,357),(303,372,319,356),(304,371,320,355)]])
133 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | ··· | 28R | 32A | ··· | 32H | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 2 | 16 | 16 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 16 | ··· | 16 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | D8 | D16 | C7×D4 | Q64 | C7×D8 | C7×D16 | C7×Q64 |
| kernel | C7×Q64 | C224 | C7×Q32 | Q64 | C32 | Q32 | C56 | C28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 4 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of C7×Q64 ►in GL2(𝔽449) generated by
| 359 | 0 |
| 0 | 359 |
| 252 | 399 |
| 50 | 252 |
| 440 | 219 |
| 219 | 9 |
G:=sub<GL(2,GF(449))| [359,0,0,359],[252,50,399,252],[440,219,219,9] >;
C7×Q64 in GAP, Magma, Sage, TeX
C_7\times Q_{64} % in TeX
G:=Group("C7xQ64"); // GroupNames label
G:=SmallGroup(448,177);
// by ID
G=gap.SmallGroup(448,177);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,1568,421,1576,2355,1186,192,5884,2951,242,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c|a^7=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export