direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊2C8, C8⋊2C56, C56⋊6C8, C28.56SD16, C28.31M4(2), C4⋊C8.4C14, C4.6(C2×C56), C28.46(C2×C8), (C4×C56).29C2, (C2×C8).10C28, (C2×C56).13C4, (C4×C8).11C14, C14.11(C4⋊C8), (C2×C28).47Q8, (C2×C28).530D4, C4.15(C7×SD16), C14.8(C4.Q8), C4.4(C7×M4(2)), C42.64(C2×C14), C14.7(C8.C4), (C4×C28).348C22, C2.3(C7×C4⋊C8), (C7×C4⋊C8).10C2, (C2×C4).8(C7×Q8), C2.1(C7×C4.Q8), (C2×C4).61(C2×C28), C2.1(C7×C8.C4), (C2×C4).139(C7×D4), C22.13(C7×C4⋊C4), (C2×C14).56(C4⋊C4), (C2×C28).322(C2×C4), SmallGroup(448,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊2C8
G = < a,b,c | a7=b8=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C7×C8⋊2C8
►Regular action on 448 points
Generators in S448
(1 30 181 60 165 44 149)(2 31 182 61 166 45 150)(3 32 183 62 167 46 151)(4 25 184 63 168 47 152)(5 26 177 64 161 48 145)(6 27 178 57 162 41 146)(7 28 179 58 163 42 147)(8 29 180 59 164 43 148)(9 325 436 309 420 293 122)(10 326 437 310 421 294 123)(11 327 438 311 422 295 124)(12 328 439 312 423 296 125)(13 321 440 305 424 289 126)(14 322 433 306 417 290 127)(15 323 434 307 418 291 128)(16 324 435 308 419 292 121)(17 131 169 56 153 40 137)(18 132 170 49 154 33 138)(19 133 171 50 155 34 139)(20 134 172 51 156 35 140)(21 135 173 52 157 36 141)(22 136 174 53 158 37 142)(23 129 175 54 159 38 143)(24 130 176 55 160 39 144)(65 233 218 97 202 81 186)(66 234 219 98 203 82 187)(67 235 220 99 204 83 188)(68 236 221 100 205 84 189)(69 237 222 101 206 85 190)(70 238 223 102 207 86 191)(71 239 224 103 208 87 192)(72 240 217 104 201 88 185)(73 225 344 105 210 89 194)(74 226 337 106 211 90 195)(75 227 338 107 212 91 196)(76 228 339 108 213 92 197)(77 229 340 109 214 93 198)(78 230 341 110 215 94 199)(79 231 342 111 216 95 200)(80 232 343 112 209 96 193)(113 445 332 429 316 413 300)(114 446 333 430 317 414 301)(115 447 334 431 318 415 302)(116 448 335 432 319 416 303)(117 441 336 425 320 409 304)(118 442 329 426 313 410 297)(119 443 330 427 314 411 298)(120 444 331 428 315 412 299)(241 360 408 273 392 257 376)(242 353 401 274 385 258 369)(243 354 402 275 386 259 370)(244 355 403 276 387 260 371)(245 356 404 277 388 261 372)(246 357 405 278 389 262 373)(247 358 406 279 390 263 374)(248 359 407 280 391 264 375)(249 346 394 281 378 265 362)(250 347 395 282 379 266 363)(251 348 396 283 380 267 364)(252 349 397 284 381 268 365)(253 350 398 285 382 269 366)(254 351 399 286 383 270 367)(255 352 400 287 384 271 368)(256 345 393 288 377 272 361)
(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)
(1 360 72 302 131 292 78 348)(2 355 65 297 132 295 79 351)(3 358 66 300 133 290 80 346)(4 353 67 303 134 293 73 349)(5 356 68 298 135 296 74 352)(6 359 69 301 136 291 75 347)(7 354 70 304 129 294 76 350)(8 357 71 299 130 289 77 345)(9 344 284 184 274 220 448 51)(10 339 285 179 275 223 441 54)(11 342 286 182 276 218 442 49)(12 337 287 177 277 221 443 52)(13 340 288 180 278 224 444 55)(14 343 281 183 279 219 445 50)(15 338 282 178 280 222 446 53)(16 341 283 181 273 217 447 56)(17 419 199 251 149 241 185 415)(18 422 200 254 150 244 186 410)(19 417 193 249 151 247 187 413)(20 420 194 252 152 242 188 416)(21 423 195 255 145 245 189 411)(22 418 196 250 146 248 190 414)(23 421 197 253 147 243 191 409)(24 424 198 256 148 246 192 412)(25 401 235 116 172 122 225 397)(26 404 236 119 173 125 226 400)(27 407 237 114 174 128 227 395)(28 402 238 117 175 123 228 398)(29 405 239 120 176 126 229 393)(30 408 240 115 169 121 230 396)(31 403 233 118 170 124 231 399)(32 406 234 113 171 127 232 394)(33 438 216 270 166 260 202 426)(34 433 209 265 167 263 203 429)(35 436 210 268 168 258 204 432)(36 439 211 271 161 261 205 427)(37 434 212 266 162 264 206 430)(38 437 213 269 163 259 207 425)(39 440 214 272 164 262 208 428)(40 435 215 267 165 257 201 431)(41 375 85 317 142 307 91 363)(42 370 86 320 143 310 92 366)(43 373 87 315 144 305 93 361)(44 376 88 318 137 308 94 364)(45 371 81 313 138 311 95 367)(46 374 82 316 139 306 96 362)(47 369 83 319 140 309 89 365)(48 372 84 314 141 312 90 368)(57 391 101 333 158 323 107 379)(58 386 102 336 159 326 108 382)(59 389 103 331 160 321 109 377)(60 392 104 334 153 324 110 380)(61 387 97 329 154 327 111 383)(62 390 98 332 155 322 112 378)(63 385 99 335 156 325 105 381)(64 388 100 330 157 328 106 384)
G:=sub<Sym(448)| (1,30,181,60,165,44,149)(2,31,182,61,166,45,150)(3,32,183,62,167,46,151)(4,25,184,63,168,47,152)(5,26,177,64,161,48,145)(6,27,178,57,162,41,146)(7,28,179,58,163,42,147)(8,29,180,59,164,43,148)(9,325,436,309,420,293,122)(10,326,437,310,421,294,123)(11,327,438,311,422,295,124)(12,328,439,312,423,296,125)(13,321,440,305,424,289,126)(14,322,433,306,417,290,127)(15,323,434,307,418,291,128)(16,324,435,308,419,292,121)(17,131,169,56,153,40,137)(18,132,170,49,154,33,138)(19,133,171,50,155,34,139)(20,134,172,51,156,35,140)(21,135,173,52,157,36,141)(22,136,174,53,158,37,142)(23,129,175,54,159,38,143)(24,130,176,55,160,39,144)(65,233,218,97,202,81,186)(66,234,219,98,203,82,187)(67,235,220,99,204,83,188)(68,236,221,100,205,84,189)(69,237,222,101,206,85,190)(70,238,223,102,207,86,191)(71,239,224,103,208,87,192)(72,240,217,104,201,88,185)(73,225,344,105,210,89,194)(74,226,337,106,211,90,195)(75,227,338,107,212,91,196)(76,228,339,108,213,92,197)(77,229,340,109,214,93,198)(78,230,341,110,215,94,199)(79,231,342,111,216,95,200)(80,232,343,112,209,96,193)(113,445,332,429,316,413,300)(114,446,333,430,317,414,301)(115,447,334,431,318,415,302)(116,448,335,432,319,416,303)(117,441,336,425,320,409,304)(118,442,329,426,313,410,297)(119,443,330,427,314,411,298)(120,444,331,428,315,412,299)(241,360,408,273,392,257,376)(242,353,401,274,385,258,369)(243,354,402,275,386,259,370)(244,355,403,276,387,260,371)(245,356,404,277,388,261,372)(246,357,405,278,389,262,373)(247,358,406,279,390,263,374)(248,359,407,280,391,264,375)(249,346,394,281,378,265,362)(250,347,395,282,379,266,363)(251,348,396,283,380,267,364)(252,349,397,284,381,268,365)(253,350,398,285,382,269,366)(254,351,399,286,383,270,367)(255,352,400,287,384,271,368)(256,345,393,288,377,272,361), (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), (1,360,72,302,131,292,78,348)(2,355,65,297,132,295,79,351)(3,358,66,300,133,290,80,346)(4,353,67,303,134,293,73,349)(5,356,68,298,135,296,74,352)(6,359,69,301,136,291,75,347)(7,354,70,304,129,294,76,350)(8,357,71,299,130,289,77,345)(9,344,284,184,274,220,448,51)(10,339,285,179,275,223,441,54)(11,342,286,182,276,218,442,49)(12,337,287,177,277,221,443,52)(13,340,288,180,278,224,444,55)(14,343,281,183,279,219,445,50)(15,338,282,178,280,222,446,53)(16,341,283,181,273,217,447,56)(17,419,199,251,149,241,185,415)(18,422,200,254,150,244,186,410)(19,417,193,249,151,247,187,413)(20,420,194,252,152,242,188,416)(21,423,195,255,145,245,189,411)(22,418,196,250,146,248,190,414)(23,421,197,253,147,243,191,409)(24,424,198,256,148,246,192,412)(25,401,235,116,172,122,225,397)(26,404,236,119,173,125,226,400)(27,407,237,114,174,128,227,395)(28,402,238,117,175,123,228,398)(29,405,239,120,176,126,229,393)(30,408,240,115,169,121,230,396)(31,403,233,118,170,124,231,399)(32,406,234,113,171,127,232,394)(33,438,216,270,166,260,202,426)(34,433,209,265,167,263,203,429)(35,436,210,268,168,258,204,432)(36,439,211,271,161,261,205,427)(37,434,212,266,162,264,206,430)(38,437,213,269,163,259,207,425)(39,440,214,272,164,262,208,428)(40,435,215,267,165,257,201,431)(41,375,85,317,142,307,91,363)(42,370,86,320,143,310,92,366)(43,373,87,315,144,305,93,361)(44,376,88,318,137,308,94,364)(45,371,81,313,138,311,95,367)(46,374,82,316,139,306,96,362)(47,369,83,319,140,309,89,365)(48,372,84,314,141,312,90,368)(57,391,101,333,158,323,107,379)(58,386,102,336,159,326,108,382)(59,389,103,331,160,321,109,377)(60,392,104,334,153,324,110,380)(61,387,97,329,154,327,111,383)(62,390,98,332,155,322,112,378)(63,385,99,335,156,325,105,381)(64,388,100,330,157,328,106,384)>;
G:=Group( (1,30,181,60,165,44,149)(2,31,182,61,166,45,150)(3,32,183,62,167,46,151)(4,25,184,63,168,47,152)(5,26,177,64,161,48,145)(6,27,178,57,162,41,146)(7,28,179,58,163,42,147)(8,29,180,59,164,43,148)(9,325,436,309,420,293,122)(10,326,437,310,421,294,123)(11,327,438,311,422,295,124)(12,328,439,312,423,296,125)(13,321,440,305,424,289,126)(14,322,433,306,417,290,127)(15,323,434,307,418,291,128)(16,324,435,308,419,292,121)(17,131,169,56,153,40,137)(18,132,170,49,154,33,138)(19,133,171,50,155,34,139)(20,134,172,51,156,35,140)(21,135,173,52,157,36,141)(22,136,174,53,158,37,142)(23,129,175,54,159,38,143)(24,130,176,55,160,39,144)(65,233,218,97,202,81,186)(66,234,219,98,203,82,187)(67,235,220,99,204,83,188)(68,236,221,100,205,84,189)(69,237,222,101,206,85,190)(70,238,223,102,207,86,191)(71,239,224,103,208,87,192)(72,240,217,104,201,88,185)(73,225,344,105,210,89,194)(74,226,337,106,211,90,195)(75,227,338,107,212,91,196)(76,228,339,108,213,92,197)(77,229,340,109,214,93,198)(78,230,341,110,215,94,199)(79,231,342,111,216,95,200)(80,232,343,112,209,96,193)(113,445,332,429,316,413,300)(114,446,333,430,317,414,301)(115,447,334,431,318,415,302)(116,448,335,432,319,416,303)(117,441,336,425,320,409,304)(118,442,329,426,313,410,297)(119,443,330,427,314,411,298)(120,444,331,428,315,412,299)(241,360,408,273,392,257,376)(242,353,401,274,385,258,369)(243,354,402,275,386,259,370)(244,355,403,276,387,260,371)(245,356,404,277,388,261,372)(246,357,405,278,389,262,373)(247,358,406,279,390,263,374)(248,359,407,280,391,264,375)(249,346,394,281,378,265,362)(250,347,395,282,379,266,363)(251,348,396,283,380,267,364)(252,349,397,284,381,268,365)(253,350,398,285,382,269,366)(254,351,399,286,383,270,367)(255,352,400,287,384,271,368)(256,345,393,288,377,272,361), (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), (1,360,72,302,131,292,78,348)(2,355,65,297,132,295,79,351)(3,358,66,300,133,290,80,346)(4,353,67,303,134,293,73,349)(5,356,68,298,135,296,74,352)(6,359,69,301,136,291,75,347)(7,354,70,304,129,294,76,350)(8,357,71,299,130,289,77,345)(9,344,284,184,274,220,448,51)(10,339,285,179,275,223,441,54)(11,342,286,182,276,218,442,49)(12,337,287,177,277,221,443,52)(13,340,288,180,278,224,444,55)(14,343,281,183,279,219,445,50)(15,338,282,178,280,222,446,53)(16,341,283,181,273,217,447,56)(17,419,199,251,149,241,185,415)(18,422,200,254,150,244,186,410)(19,417,193,249,151,247,187,413)(20,420,194,252,152,242,188,416)(21,423,195,255,145,245,189,411)(22,418,196,250,146,248,190,414)(23,421,197,253,147,243,191,409)(24,424,198,256,148,246,192,412)(25,401,235,116,172,122,225,397)(26,404,236,119,173,125,226,400)(27,407,237,114,174,128,227,395)(28,402,238,117,175,123,228,398)(29,405,239,120,176,126,229,393)(30,408,240,115,169,121,230,396)(31,403,233,118,170,124,231,399)(32,406,234,113,171,127,232,394)(33,438,216,270,166,260,202,426)(34,433,209,265,167,263,203,429)(35,436,210,268,168,258,204,432)(36,439,211,271,161,261,205,427)(37,434,212,266,162,264,206,430)(38,437,213,269,163,259,207,425)(39,440,214,272,164,262,208,428)(40,435,215,267,165,257,201,431)(41,375,85,317,142,307,91,363)(42,370,86,320,143,310,92,366)(43,373,87,315,144,305,93,361)(44,376,88,318,137,308,94,364)(45,371,81,313,138,311,95,367)(46,374,82,316,139,306,96,362)(47,369,83,319,140,309,89,365)(48,372,84,314,141,312,90,368)(57,391,101,333,158,323,107,379)(58,386,102,336,159,326,108,382)(59,389,103,331,160,321,109,377)(60,392,104,334,153,324,110,380)(61,387,97,329,154,327,111,383)(62,390,98,332,155,322,112,378)(63,385,99,335,156,325,105,381)(64,388,100,330,157,328,106,384) );
G=PermutationGroup([[(1,30,181,60,165,44,149),(2,31,182,61,166,45,150),(3,32,183,62,167,46,151),(4,25,184,63,168,47,152),(5,26,177,64,161,48,145),(6,27,178,57,162,41,146),(7,28,179,58,163,42,147),(8,29,180,59,164,43,148),(9,325,436,309,420,293,122),(10,326,437,310,421,294,123),(11,327,438,311,422,295,124),(12,328,439,312,423,296,125),(13,321,440,305,424,289,126),(14,322,433,306,417,290,127),(15,323,434,307,418,291,128),(16,324,435,308,419,292,121),(17,131,169,56,153,40,137),(18,132,170,49,154,33,138),(19,133,171,50,155,34,139),(20,134,172,51,156,35,140),(21,135,173,52,157,36,141),(22,136,174,53,158,37,142),(23,129,175,54,159,38,143),(24,130,176,55,160,39,144),(65,233,218,97,202,81,186),(66,234,219,98,203,82,187),(67,235,220,99,204,83,188),(68,236,221,100,205,84,189),(69,237,222,101,206,85,190),(70,238,223,102,207,86,191),(71,239,224,103,208,87,192),(72,240,217,104,201,88,185),(73,225,344,105,210,89,194),(74,226,337,106,211,90,195),(75,227,338,107,212,91,196),(76,228,339,108,213,92,197),(77,229,340,109,214,93,198),(78,230,341,110,215,94,199),(79,231,342,111,216,95,200),(80,232,343,112,209,96,193),(113,445,332,429,316,413,300),(114,446,333,430,317,414,301),(115,447,334,431,318,415,302),(116,448,335,432,319,416,303),(117,441,336,425,320,409,304),(118,442,329,426,313,410,297),(119,443,330,427,314,411,298),(120,444,331,428,315,412,299),(241,360,408,273,392,257,376),(242,353,401,274,385,258,369),(243,354,402,275,386,259,370),(244,355,403,276,387,260,371),(245,356,404,277,388,261,372),(246,357,405,278,389,262,373),(247,358,406,279,390,263,374),(248,359,407,280,391,264,375),(249,346,394,281,378,265,362),(250,347,395,282,379,266,363),(251,348,396,283,380,267,364),(252,349,397,284,381,268,365),(253,350,398,285,382,269,366),(254,351,399,286,383,270,367),(255,352,400,287,384,271,368),(256,345,393,288,377,272,361)], [(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)], [(1,360,72,302,131,292,78,348),(2,355,65,297,132,295,79,351),(3,358,66,300,133,290,80,346),(4,353,67,303,134,293,73,349),(5,356,68,298,135,296,74,352),(6,359,69,301,136,291,75,347),(7,354,70,304,129,294,76,350),(8,357,71,299,130,289,77,345),(9,344,284,184,274,220,448,51),(10,339,285,179,275,223,441,54),(11,342,286,182,276,218,442,49),(12,337,287,177,277,221,443,52),(13,340,288,180,278,224,444,55),(14,343,281,183,279,219,445,50),(15,338,282,178,280,222,446,53),(16,341,283,181,273,217,447,56),(17,419,199,251,149,241,185,415),(18,422,200,254,150,244,186,410),(19,417,193,249,151,247,187,413),(20,420,194,252,152,242,188,416),(21,423,195,255,145,245,189,411),(22,418,196,250,146,248,190,414),(23,421,197,253,147,243,191,409),(24,424,198,256,148,246,192,412),(25,401,235,116,172,122,225,397),(26,404,236,119,173,125,226,400),(27,407,237,114,174,128,227,395),(28,402,238,117,175,123,228,398),(29,405,239,120,176,126,229,393),(30,408,240,115,169,121,230,396),(31,403,233,118,170,124,231,399),(32,406,234,113,171,127,232,394),(33,438,216,270,166,260,202,426),(34,433,209,265,167,263,203,429),(35,436,210,268,168,258,204,432),(36,439,211,271,161,261,205,427),(37,434,212,266,162,264,206,430),(38,437,213,269,163,259,207,425),(39,440,214,272,164,262,208,428),(40,435,215,267,165,257,201,431),(41,375,85,317,142,307,91,363),(42,370,86,320,143,310,92,366),(43,373,87,315,144,305,93,361),(44,376,88,318,137,308,94,364),(45,371,81,313,138,311,95,367),(46,374,82,316,139,306,96,362),(47,369,83,319,140,309,89,365),(48,372,84,314,141,312,90,368),(57,391,101,333,158,323,107,379),(58,386,102,336,159,326,108,382),(59,389,103,331,160,321,109,377),(60,392,104,334,153,324,110,380),(61,387,97,329,154,327,111,383),(62,390,98,332,155,322,112,378),(63,385,99,335,156,325,105,381),(64,388,100,330,157,328,106,384)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AV | 56A | ··· | 56AV | 56AW | ··· | 56CR |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C8 | C14 | C14 | C28 | C56 | D4 | Q8 | M4(2) | SD16 | C8.C4 | C7×D4 | C7×Q8 | C7×M4(2) | C7×SD16 | C7×C8.C4 |
| kernel | C7×C8⋊2C8 | C4×C56 | C7×C4⋊C8 | C2×C56 | C8⋊2C8 | C56 | C4×C8 | C4⋊C8 | C2×C8 | C8 | C2×C28 | C2×C28 | C28 | C28 | C14 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 6 | 8 | 6 | 12 | 24 | 48 | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C7×C8⋊2C8 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 112 | 2 | 0 | 0 |
| 112 | 1 | 0 | 0 |
| 0 | 0 | 100 | 13 |
| 0 | 0 | 100 | 100 |
| 99 | 14 | 0 | 0 |
| 106 | 14 | 0 | 0 |
| 0 | 0 | 58 | 89 |
| 0 | 0 | 89 | 55 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,28,0,0,0,0,28],[112,112,0,0,2,1,0,0,0,0,100,100,0,0,13,100],[99,106,0,0,14,14,0,0,0,0,58,89,0,0,89,55] >;
C7×C8⋊2C8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_2C_8
% in TeX
G:=Group("C7xC8:2C8"); // GroupNames label
G:=SmallGroup(448,138);
// by ID
G=gap.SmallGroup(448,138);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,204,3923,136,172]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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