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G = C7×Q64order 448 = 26·7

Direct product of C7 and Q64

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

C1C16 — C7×Q64
C1C2C4C8C16C112C7×Q32 — C7×Q64
C1C2C4C8C16 — C7×Q64
C1C14C28C56C112 — C7×Q64

Generators and relations for C7×Q64
 G = < a,b,c | a7=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >

8C4
8C4
4Q8
4Q8
8C28
8C28
2Q16
2Q16
4C7×Q8
4C7×Q8
2C7×Q16
2C7×Q16

Smallest permutation representation of C7×Q64
Regular action on 448 points
Generators in S448
(1 129 87 33 320 330 268)(2 130 88 34 289 331 269)(3 131 89 35 290 332 270)(4 132 90 36 291 333 271)(5 133 91 37 292 334 272)(6 134 92 38 293 335 273)(7 135 93 39 294 336 274)(8 136 94 40 295 337 275)(9 137 95 41 296 338 276)(10 138 96 42 297 339 277)(11 139 65 43 298 340 278)(12 140 66 44 299 341 279)(13 141 67 45 300 342 280)(14 142 68 46 301 343 281)(15 143 69 47 302 344 282)(16 144 70 48 303 345 283)(17 145 71 49 304 346 284)(18 146 72 50 305 347 285)(19 147 73 51 306 348 286)(20 148 74 52 307 349 287)(21 149 75 53 308 350 288)(22 150 76 54 309 351 257)(23 151 77 55 310 352 258)(24 152 78 56 311 321 259)(25 153 79 57 312 322 260)(26 154 80 58 313 323 261)(27 155 81 59 314 324 262)(28 156 82 60 315 325 263)(29 157 83 61 316 326 264)(30 158 84 62 317 327 265)(31 159 85 63 318 328 266)(32 160 86 64 319 329 267)(97 226 417 204 403 361 191)(98 227 418 205 404 362 192)(99 228 419 206 405 363 161)(100 229 420 207 406 364 162)(101 230 421 208 407 365 163)(102 231 422 209 408 366 164)(103 232 423 210 409 367 165)(104 233 424 211 410 368 166)(105 234 425 212 411 369 167)(106 235 426 213 412 370 168)(107 236 427 214 413 371 169)(108 237 428 215 414 372 170)(109 238 429 216 415 373 171)(110 239 430 217 416 374 172)(111 240 431 218 385 375 173)(112 241 432 219 386 376 174)(113 242 433 220 387 377 175)(114 243 434 221 388 378 176)(115 244 435 222 389 379 177)(116 245 436 223 390 380 178)(117 246 437 224 391 381 179)(118 247 438 193 392 382 180)(119 248 439 194 393 383 181)(120 249 440 195 394 384 182)(121 250 441 196 395 353 183)(122 251 442 197 396 354 184)(123 252 443 198 397 355 185)(124 253 444 199 398 356 186)(125 254 445 200 399 357 187)(126 255 446 201 400 358 188)(127 256 447 202 401 359 189)(128 225 448 203 402 360 190)
(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)
(1 436 17 420)(2 435 18 419)(3 434 19 418)(4 433 20 417)(5 432 21 448)(6 431 22 447)(7 430 23 446)(8 429 24 445)(9 428 25 444)(10 427 26 443)(11 426 27 442)(12 425 28 441)(13 424 29 440)(14 423 30 439)(15 422 31 438)(16 421 32 437)(33 380 49 364)(34 379 50 363)(35 378 51 362)(36 377 52 361)(37 376 53 360)(38 375 54 359)(39 374 55 358)(40 373 56 357)(41 372 57 356)(42 371 58 355)(43 370 59 354)(44 369 60 353)(45 368 61 384)(46 367 62 383)(47 366 63 382)(48 365 64 381)(65 412 81 396)(66 411 82 395)(67 410 83 394)(68 409 84 393)(69 408 85 392)(70 407 86 391)(71 406 87 390)(72 405 88 389)(73 404 89 388)(74 403 90 387)(75 402 91 386)(76 401 92 385)(77 400 93 416)(78 399 94 415)(79 398 95 414)(80 397 96 413)(97 333 113 349)(98 332 114 348)(99 331 115 347)(100 330 116 346)(101 329 117 345)(102 328 118 344)(103 327 119 343)(104 326 120 342)(105 325 121 341)(106 324 122 340)(107 323 123 339)(108 322 124 338)(109 321 125 337)(110 352 126 336)(111 351 127 335)(112 350 128 334)(129 223 145 207)(130 222 146 206)(131 221 147 205)(132 220 148 204)(133 219 149 203)(134 218 150 202)(135 217 151 201)(136 216 152 200)(137 215 153 199)(138 214 154 198)(139 213 155 197)(140 212 156 196)(141 211 157 195)(142 210 158 194)(143 209 159 193)(144 208 160 224)(161 289 177 305)(162 320 178 304)(163 319 179 303)(164 318 180 302)(165 317 181 301)(166 316 182 300)(167 315 183 299)(168 314 184 298)(169 313 185 297)(170 312 186 296)(171 311 187 295)(172 310 188 294)(173 309 189 293)(174 308 190 292)(175 307 191 291)(176 306 192 290)(225 272 241 288)(226 271 242 287)(227 270 243 286)(228 269 244 285)(229 268 245 284)(230 267 246 283)(231 266 247 282)(232 265 248 281)(233 264 249 280)(234 263 250 279)(235 262 251 278)(236 261 252 277)(237 260 253 276)(238 259 254 275)(239 258 255 274)(240 257 256 273)

G:=sub<Sym(448)| (1,129,87,33,320,330,268)(2,130,88,34,289,331,269)(3,131,89,35,290,332,270)(4,132,90,36,291,333,271)(5,133,91,37,292,334,272)(6,134,92,38,293,335,273)(7,135,93,39,294,336,274)(8,136,94,40,295,337,275)(9,137,95,41,296,338,276)(10,138,96,42,297,339,277)(11,139,65,43,298,340,278)(12,140,66,44,299,341,279)(13,141,67,45,300,342,280)(14,142,68,46,301,343,281)(15,143,69,47,302,344,282)(16,144,70,48,303,345,283)(17,145,71,49,304,346,284)(18,146,72,50,305,347,285)(19,147,73,51,306,348,286)(20,148,74,52,307,349,287)(21,149,75,53,308,350,288)(22,150,76,54,309,351,257)(23,151,77,55,310,352,258)(24,152,78,56,311,321,259)(25,153,79,57,312,322,260)(26,154,80,58,313,323,261)(27,155,81,59,314,324,262)(28,156,82,60,315,325,263)(29,157,83,61,316,326,264)(30,158,84,62,317,327,265)(31,159,85,63,318,328,266)(32,160,86,64,319,329,267)(97,226,417,204,403,361,191)(98,227,418,205,404,362,192)(99,228,419,206,405,363,161)(100,229,420,207,406,364,162)(101,230,421,208,407,365,163)(102,231,422,209,408,366,164)(103,232,423,210,409,367,165)(104,233,424,211,410,368,166)(105,234,425,212,411,369,167)(106,235,426,213,412,370,168)(107,236,427,214,413,371,169)(108,237,428,215,414,372,170)(109,238,429,216,415,373,171)(110,239,430,217,416,374,172)(111,240,431,218,385,375,173)(112,241,432,219,386,376,174)(113,242,433,220,387,377,175)(114,243,434,221,388,378,176)(115,244,435,222,389,379,177)(116,245,436,223,390,380,178)(117,246,437,224,391,381,179)(118,247,438,193,392,382,180)(119,248,439,194,393,383,181)(120,249,440,195,394,384,182)(121,250,441,196,395,353,183)(122,251,442,197,396,354,184)(123,252,443,198,397,355,185)(124,253,444,199,398,356,186)(125,254,445,200,399,357,187)(126,255,446,201,400,358,188)(127,256,447,202,401,359,189)(128,225,448,203,402,360,190), (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), (1,436,17,420)(2,435,18,419)(3,434,19,418)(4,433,20,417)(5,432,21,448)(6,431,22,447)(7,430,23,446)(8,429,24,445)(9,428,25,444)(10,427,26,443)(11,426,27,442)(12,425,28,441)(13,424,29,440)(14,423,30,439)(15,422,31,438)(16,421,32,437)(33,380,49,364)(34,379,50,363)(35,378,51,362)(36,377,52,361)(37,376,53,360)(38,375,54,359)(39,374,55,358)(40,373,56,357)(41,372,57,356)(42,371,58,355)(43,370,59,354)(44,369,60,353)(45,368,61,384)(46,367,62,383)(47,366,63,382)(48,365,64,381)(65,412,81,396)(66,411,82,395)(67,410,83,394)(68,409,84,393)(69,408,85,392)(70,407,86,391)(71,406,87,390)(72,405,88,389)(73,404,89,388)(74,403,90,387)(75,402,91,386)(76,401,92,385)(77,400,93,416)(78,399,94,415)(79,398,95,414)(80,397,96,413)(97,333,113,349)(98,332,114,348)(99,331,115,347)(100,330,116,346)(101,329,117,345)(102,328,118,344)(103,327,119,343)(104,326,120,342)(105,325,121,341)(106,324,122,340)(107,323,123,339)(108,322,124,338)(109,321,125,337)(110,352,126,336)(111,351,127,335)(112,350,128,334)(129,223,145,207)(130,222,146,206)(131,221,147,205)(132,220,148,204)(133,219,149,203)(134,218,150,202)(135,217,151,201)(136,216,152,200)(137,215,153,199)(138,214,154,198)(139,213,155,197)(140,212,156,196)(141,211,157,195)(142,210,158,194)(143,209,159,193)(144,208,160,224)(161,289,177,305)(162,320,178,304)(163,319,179,303)(164,318,180,302)(165,317,181,301)(166,316,182,300)(167,315,183,299)(168,314,184,298)(169,313,185,297)(170,312,186,296)(171,311,187,295)(172,310,188,294)(173,309,189,293)(174,308,190,292)(175,307,191,291)(176,306,192,290)(225,272,241,288)(226,271,242,287)(227,270,243,286)(228,269,244,285)(229,268,245,284)(230,267,246,283)(231,266,247,282)(232,265,248,281)(233,264,249,280)(234,263,250,279)(235,262,251,278)(236,261,252,277)(237,260,253,276)(238,259,254,275)(239,258,255,274)(240,257,256,273)>;

G:=Group( (1,129,87,33,320,330,268)(2,130,88,34,289,331,269)(3,131,89,35,290,332,270)(4,132,90,36,291,333,271)(5,133,91,37,292,334,272)(6,134,92,38,293,335,273)(7,135,93,39,294,336,274)(8,136,94,40,295,337,275)(9,137,95,41,296,338,276)(10,138,96,42,297,339,277)(11,139,65,43,298,340,278)(12,140,66,44,299,341,279)(13,141,67,45,300,342,280)(14,142,68,46,301,343,281)(15,143,69,47,302,344,282)(16,144,70,48,303,345,283)(17,145,71,49,304,346,284)(18,146,72,50,305,347,285)(19,147,73,51,306,348,286)(20,148,74,52,307,349,287)(21,149,75,53,308,350,288)(22,150,76,54,309,351,257)(23,151,77,55,310,352,258)(24,152,78,56,311,321,259)(25,153,79,57,312,322,260)(26,154,80,58,313,323,261)(27,155,81,59,314,324,262)(28,156,82,60,315,325,263)(29,157,83,61,316,326,264)(30,158,84,62,317,327,265)(31,159,85,63,318,328,266)(32,160,86,64,319,329,267)(97,226,417,204,403,361,191)(98,227,418,205,404,362,192)(99,228,419,206,405,363,161)(100,229,420,207,406,364,162)(101,230,421,208,407,365,163)(102,231,422,209,408,366,164)(103,232,423,210,409,367,165)(104,233,424,211,410,368,166)(105,234,425,212,411,369,167)(106,235,426,213,412,370,168)(107,236,427,214,413,371,169)(108,237,428,215,414,372,170)(109,238,429,216,415,373,171)(110,239,430,217,416,374,172)(111,240,431,218,385,375,173)(112,241,432,219,386,376,174)(113,242,433,220,387,377,175)(114,243,434,221,388,378,176)(115,244,435,222,389,379,177)(116,245,436,223,390,380,178)(117,246,437,224,391,381,179)(118,247,438,193,392,382,180)(119,248,439,194,393,383,181)(120,249,440,195,394,384,182)(121,250,441,196,395,353,183)(122,251,442,197,396,354,184)(123,252,443,198,397,355,185)(124,253,444,199,398,356,186)(125,254,445,200,399,357,187)(126,255,446,201,400,358,188)(127,256,447,202,401,359,189)(128,225,448,203,402,360,190), (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), (1,436,17,420)(2,435,18,419)(3,434,19,418)(4,433,20,417)(5,432,21,448)(6,431,22,447)(7,430,23,446)(8,429,24,445)(9,428,25,444)(10,427,26,443)(11,426,27,442)(12,425,28,441)(13,424,29,440)(14,423,30,439)(15,422,31,438)(16,421,32,437)(33,380,49,364)(34,379,50,363)(35,378,51,362)(36,377,52,361)(37,376,53,360)(38,375,54,359)(39,374,55,358)(40,373,56,357)(41,372,57,356)(42,371,58,355)(43,370,59,354)(44,369,60,353)(45,368,61,384)(46,367,62,383)(47,366,63,382)(48,365,64,381)(65,412,81,396)(66,411,82,395)(67,410,83,394)(68,409,84,393)(69,408,85,392)(70,407,86,391)(71,406,87,390)(72,405,88,389)(73,404,89,388)(74,403,90,387)(75,402,91,386)(76,401,92,385)(77,400,93,416)(78,399,94,415)(79,398,95,414)(80,397,96,413)(97,333,113,349)(98,332,114,348)(99,331,115,347)(100,330,116,346)(101,329,117,345)(102,328,118,344)(103,327,119,343)(104,326,120,342)(105,325,121,341)(106,324,122,340)(107,323,123,339)(108,322,124,338)(109,321,125,337)(110,352,126,336)(111,351,127,335)(112,350,128,334)(129,223,145,207)(130,222,146,206)(131,221,147,205)(132,220,148,204)(133,219,149,203)(134,218,150,202)(135,217,151,201)(136,216,152,200)(137,215,153,199)(138,214,154,198)(139,213,155,197)(140,212,156,196)(141,211,157,195)(142,210,158,194)(143,209,159,193)(144,208,160,224)(161,289,177,305)(162,320,178,304)(163,319,179,303)(164,318,180,302)(165,317,181,301)(166,316,182,300)(167,315,183,299)(168,314,184,298)(169,313,185,297)(170,312,186,296)(171,311,187,295)(172,310,188,294)(173,309,189,293)(174,308,190,292)(175,307,191,291)(176,306,192,290)(225,272,241,288)(226,271,242,287)(227,270,243,286)(228,269,244,285)(229,268,245,284)(230,267,246,283)(231,266,247,282)(232,265,248,281)(233,264,249,280)(234,263,250,279)(235,262,251,278)(236,261,252,277)(237,260,253,276)(238,259,254,275)(239,258,255,274)(240,257,256,273) );

G=PermutationGroup([(1,129,87,33,320,330,268),(2,130,88,34,289,331,269),(3,131,89,35,290,332,270),(4,132,90,36,291,333,271),(5,133,91,37,292,334,272),(6,134,92,38,293,335,273),(7,135,93,39,294,336,274),(8,136,94,40,295,337,275),(9,137,95,41,296,338,276),(10,138,96,42,297,339,277),(11,139,65,43,298,340,278),(12,140,66,44,299,341,279),(13,141,67,45,300,342,280),(14,142,68,46,301,343,281),(15,143,69,47,302,344,282),(16,144,70,48,303,345,283),(17,145,71,49,304,346,284),(18,146,72,50,305,347,285),(19,147,73,51,306,348,286),(20,148,74,52,307,349,287),(21,149,75,53,308,350,288),(22,150,76,54,309,351,257),(23,151,77,55,310,352,258),(24,152,78,56,311,321,259),(25,153,79,57,312,322,260),(26,154,80,58,313,323,261),(27,155,81,59,314,324,262),(28,156,82,60,315,325,263),(29,157,83,61,316,326,264),(30,158,84,62,317,327,265),(31,159,85,63,318,328,266),(32,160,86,64,319,329,267),(97,226,417,204,403,361,191),(98,227,418,205,404,362,192),(99,228,419,206,405,363,161),(100,229,420,207,406,364,162),(101,230,421,208,407,365,163),(102,231,422,209,408,366,164),(103,232,423,210,409,367,165),(104,233,424,211,410,368,166),(105,234,425,212,411,369,167),(106,235,426,213,412,370,168),(107,236,427,214,413,371,169),(108,237,428,215,414,372,170),(109,238,429,216,415,373,171),(110,239,430,217,416,374,172),(111,240,431,218,385,375,173),(112,241,432,219,386,376,174),(113,242,433,220,387,377,175),(114,243,434,221,388,378,176),(115,244,435,222,389,379,177),(116,245,436,223,390,380,178),(117,246,437,224,391,381,179),(118,247,438,193,392,382,180),(119,248,439,194,393,383,181),(120,249,440,195,394,384,182),(121,250,441,196,395,353,183),(122,251,442,197,396,354,184),(123,252,443,198,397,355,185),(124,253,444,199,398,356,186),(125,254,445,200,399,357,187),(126,255,446,201,400,358,188),(127,256,447,202,401,359,189),(128,225,448,203,402,360,190)], [(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)], [(1,436,17,420),(2,435,18,419),(3,434,19,418),(4,433,20,417),(5,432,21,448),(6,431,22,447),(7,430,23,446),(8,429,24,445),(9,428,25,444),(10,427,26,443),(11,426,27,442),(12,425,28,441),(13,424,29,440),(14,423,30,439),(15,422,31,438),(16,421,32,437),(33,380,49,364),(34,379,50,363),(35,378,51,362),(36,377,52,361),(37,376,53,360),(38,375,54,359),(39,374,55,358),(40,373,56,357),(41,372,57,356),(42,371,58,355),(43,370,59,354),(44,369,60,353),(45,368,61,384),(46,367,62,383),(47,366,63,382),(48,365,64,381),(65,412,81,396),(66,411,82,395),(67,410,83,394),(68,409,84,393),(69,408,85,392),(70,407,86,391),(71,406,87,390),(72,405,88,389),(73,404,89,388),(74,403,90,387),(75,402,91,386),(76,401,92,385),(77,400,93,416),(78,399,94,415),(79,398,95,414),(80,397,96,413),(97,333,113,349),(98,332,114,348),(99,331,115,347),(100,330,116,346),(101,329,117,345),(102,328,118,344),(103,327,119,343),(104,326,120,342),(105,325,121,341),(106,324,122,340),(107,323,123,339),(108,322,124,338),(109,321,125,337),(110,352,126,336),(111,351,127,335),(112,350,128,334),(129,223,145,207),(130,222,146,206),(131,221,147,205),(132,220,148,204),(133,219,149,203),(134,218,150,202),(135,217,151,201),(136,216,152,200),(137,215,153,199),(138,214,154,198),(139,213,155,197),(140,212,156,196),(141,211,157,195),(142,210,158,194),(143,209,159,193),(144,208,160,224),(161,289,177,305),(162,320,178,304),(163,319,179,303),(164,318,180,302),(165,317,181,301),(166,316,182,300),(167,315,183,299),(168,314,184,298),(169,313,185,297),(170,312,186,296),(171,311,187,295),(172,310,188,294),(173,309,189,293),(174,308,190,292),(175,307,191,291),(176,306,192,290),(225,272,241,288),(226,271,242,287),(227,270,243,286),(228,269,244,285),(229,268,245,284),(230,267,246,283),(231,266,247,282),(232,265,248,281),(233,264,249,280),(234,263,250,279),(235,262,251,278),(236,261,252,277),(237,260,253,276),(238,259,254,275),(239,258,255,274),(240,257,256,273)])

133 conjugacy classes

class 1  2 4A4B4C7A···7F8A8B14A···14F16A16B16C16D28A···28F28G···28R32A···32H56A···56L112A···112X224A···224AV
order124447···78814···141616161628···2828···2832···3256···56112···112224···224
size11216161···1221···122222···216···162···22···22···22···2

133 irreducible representations

dim11111122222222
type++++++-
imageC1C2C2C7C14C14D4D8D16C7×D4Q64C7×D8C7×D16C7×Q64
kernelC7×Q64C224C7×Q32Q64C32Q32C56C28C14C8C7C4C2C1
# reps112661212468122448

Matrix representation of C7×Q64 in GL2(𝔽449) generated by

3590
0359
,
252399
50252
,
440219
2199
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

Subgroup lattice of C7×Q64 in TeX

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