direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C8⋊C8, C56⋊7C8, C8⋊3C56, C28.36M4(2), (C2×C8).4C28, C2.1(C4×C56), (C2×C56).30C4, (C4×C56).31C2, (C4×C8).13C14, C14.10(C4×C8), C28.51(C2×C8), C4.11(C2×C56), C22.6(C4×C28), C14.7(C8⋊C4), C4.9(C7×M4(2)), C42.91(C2×C14), (C2×C14).28C42, (C4×C28).377C22, C2.1(C7×C8⋊C4), (C2×C4).80(C2×C28), (C2×C28).342(C2×C4), SmallGroup(448,126)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊C8
G = < a,b,c | a7=b8=c8=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 74 in 66 conjugacy classes, 58 normal (14 characteristic)
C1, C2, C2, C4, C22, C7, C8, C8, C2×C4, C2×C4, C14, C14, C42, C2×C8, C28, C2×C14, C4×C8, C4×C8, C56, C56, C2×C28, C2×C28, C8⋊C8, C4×C28, C2×C56, C4×C56, C4×C56, C7×C8⋊C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C14, C42, C2×C8, M4(2), C28, C2×C14, C4×C8, C8⋊C4, C56, C2×C28, C8⋊C8, C4×C28, C2×C56, C7×M4(2), C4×C56, C7×C8⋊C4, C7×C8⋊C8
►Regular action on 448 points
Generators in S448
(1 397 388 381 372 365 356)(2 398 389 382 373 366 357)(3 399 390 383 374 367 358)(4 400 391 384 375 368 359)(5 393 392 377 376 361 360)(6 394 385 378 369 362 353)(7 395 386 379 370 363 354)(8 396 387 380 371 364 355)(9 445 436 429 420 413 404)(10 446 437 430 421 414 405)(11 447 438 431 422 415 406)(12 448 439 432 423 416 407)(13 441 440 425 424 409 408)(14 442 433 426 417 410 401)(15 443 434 427 418 411 402)(16 444 435 428 419 412 403)(17 65 64 49 48 33 32)(18 66 57 50 41 34 25)(19 67 58 51 42 35 26)(20 68 59 52 43 36 27)(21 69 60 53 44 37 28)(22 70 61 54 45 38 29)(23 71 62 55 46 39 30)(24 72 63 56 47 40 31)(73 121 114 105 98 89 82)(74 122 115 106 99 90 83)(75 123 116 107 100 91 84)(76 124 117 108 101 92 85)(77 125 118 109 102 93 86)(78 126 119 110 103 94 87)(79 127 120 111 104 95 88)(80 128 113 112 97 96 81)(129 177 176 161 160 145 144)(130 178 169 162 153 146 137)(131 179 170 163 154 147 138)(132 180 171 164 155 148 139)(133 181 172 165 156 149 140)(134 182 173 166 157 150 141)(135 183 174 167 158 151 142)(136 184 175 168 159 152 143)(185 237 230 217 210 201 194)(186 238 231 218 211 202 195)(187 239 232 219 212 203 196)(188 240 225 220 213 204 197)(189 233 226 221 214 205 198)(190 234 227 222 215 206 199)(191 235 228 223 216 207 200)(192 236 229 224 209 208 193)(241 251 299 273 283 257 267)(242 252 300 274 284 258 268)(243 253 301 275 285 259 269)(244 254 302 276 286 260 270)(245 255 303 277 287 261 271)(246 256 304 278 288 262 272)(247 249 297 279 281 263 265)(248 250 298 280 282 264 266)(289 337 347 321 331 305 315)(290 338 348 322 332 306 316)(291 339 349 323 333 307 317)(292 340 350 324 334 308 318)(293 341 351 325 335 309 319)(294 342 352 326 336 310 320)(295 343 345 327 329 311 313)(296 344 346 328 330 312 314)
(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)
(1 291 253 186 131 73 18 404)(2 296 254 191 132 78 19 401)(3 293 255 188 133 75 20 406)(4 290 256 185 134 80 21 403)(5 295 249 190 135 77 22 408)(6 292 250 187 136 74 23 405)(7 289 251 192 129 79 24 402)(8 294 252 189 130 76 17 407)(9 397 339 301 238 179 121 66)(10 394 340 298 239 184 122 71)(11 399 341 303 240 181 123 68)(12 396 342 300 233 178 124 65)(13 393 343 297 234 183 125 70)(14 398 344 302 235 180 126 67)(15 395 337 299 236 177 127 72)(16 400 338 304 237 182 128 69)(25 413 356 317 243 195 138 82)(26 410 357 314 244 200 139 87)(27 415 358 319 245 197 140 84)(28 412 359 316 246 194 141 81)(29 409 360 313 247 199 142 86)(30 414 353 318 248 196 143 83)(31 411 354 315 241 193 144 88)(32 416 355 320 242 198 137 85)(33 423 364 310 268 205 146 92)(34 420 365 307 269 202 147 89)(35 417 366 312 270 207 148 94)(36 422 367 309 271 204 149 91)(37 419 368 306 272 201 150 96)(38 424 361 311 265 206 151 93)(39 421 362 308 266 203 152 90)(40 418 363 305 267 208 145 95)(41 429 372 333 259 211 154 98)(42 426 373 330 260 216 155 103)(43 431 374 335 261 213 156 100)(44 428 375 332 262 210 157 97)(45 425 376 329 263 215 158 102)(46 430 369 334 264 212 159 99)(47 427 370 331 257 209 160 104)(48 432 371 336 258 214 153 101)(49 439 380 326 284 221 162 108)(50 436 381 323 285 218 163 105)(51 433 382 328 286 223 164 110)(52 438 383 325 287 220 165 107)(53 435 384 322 288 217 166 112)(54 440 377 327 281 222 167 109)(55 437 378 324 282 219 168 106)(56 434 379 321 283 224 161 111)(57 445 388 349 275 231 170 114)(58 442 389 346 276 228 171 119)(59 447 390 351 277 225 172 116)(60 444 391 348 278 230 173 113)(61 441 392 345 279 227 174 118)(62 446 385 350 280 232 175 115)(63 443 386 347 273 229 176 120)(64 448 387 352 274 226 169 117)
G:=sub<Sym(448)| (1,397,388,381,372,365,356)(2,398,389,382,373,366,357)(3,399,390,383,374,367,358)(4,400,391,384,375,368,359)(5,393,392,377,376,361,360)(6,394,385,378,369,362,353)(7,395,386,379,370,363,354)(8,396,387,380,371,364,355)(9,445,436,429,420,413,404)(10,446,437,430,421,414,405)(11,447,438,431,422,415,406)(12,448,439,432,423,416,407)(13,441,440,425,424,409,408)(14,442,433,426,417,410,401)(15,443,434,427,418,411,402)(16,444,435,428,419,412,403)(17,65,64,49,48,33,32)(18,66,57,50,41,34,25)(19,67,58,51,42,35,26)(20,68,59,52,43,36,27)(21,69,60,53,44,37,28)(22,70,61,54,45,38,29)(23,71,62,55,46,39,30)(24,72,63,56,47,40,31)(73,121,114,105,98,89,82)(74,122,115,106,99,90,83)(75,123,116,107,100,91,84)(76,124,117,108,101,92,85)(77,125,118,109,102,93,86)(78,126,119,110,103,94,87)(79,127,120,111,104,95,88)(80,128,113,112,97,96,81)(129,177,176,161,160,145,144)(130,178,169,162,153,146,137)(131,179,170,163,154,147,138)(132,180,171,164,155,148,139)(133,181,172,165,156,149,140)(134,182,173,166,157,150,141)(135,183,174,167,158,151,142)(136,184,175,168,159,152,143)(185,237,230,217,210,201,194)(186,238,231,218,211,202,195)(187,239,232,219,212,203,196)(188,240,225,220,213,204,197)(189,233,226,221,214,205,198)(190,234,227,222,215,206,199)(191,235,228,223,216,207,200)(192,236,229,224,209,208,193)(241,251,299,273,283,257,267)(242,252,300,274,284,258,268)(243,253,301,275,285,259,269)(244,254,302,276,286,260,270)(245,255,303,277,287,261,271)(246,256,304,278,288,262,272)(247,249,297,279,281,263,265)(248,250,298,280,282,264,266)(289,337,347,321,331,305,315)(290,338,348,322,332,306,316)(291,339,349,323,333,307,317)(292,340,350,324,334,308,318)(293,341,351,325,335,309,319)(294,342,352,326,336,310,320)(295,343,345,327,329,311,313)(296,344,346,328,330,312,314), (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), (1,291,253,186,131,73,18,404)(2,296,254,191,132,78,19,401)(3,293,255,188,133,75,20,406)(4,290,256,185,134,80,21,403)(5,295,249,190,135,77,22,408)(6,292,250,187,136,74,23,405)(7,289,251,192,129,79,24,402)(8,294,252,189,130,76,17,407)(9,397,339,301,238,179,121,66)(10,394,340,298,239,184,122,71)(11,399,341,303,240,181,123,68)(12,396,342,300,233,178,124,65)(13,393,343,297,234,183,125,70)(14,398,344,302,235,180,126,67)(15,395,337,299,236,177,127,72)(16,400,338,304,237,182,128,69)(25,413,356,317,243,195,138,82)(26,410,357,314,244,200,139,87)(27,415,358,319,245,197,140,84)(28,412,359,316,246,194,141,81)(29,409,360,313,247,199,142,86)(30,414,353,318,248,196,143,83)(31,411,354,315,241,193,144,88)(32,416,355,320,242,198,137,85)(33,423,364,310,268,205,146,92)(34,420,365,307,269,202,147,89)(35,417,366,312,270,207,148,94)(36,422,367,309,271,204,149,91)(37,419,368,306,272,201,150,96)(38,424,361,311,265,206,151,93)(39,421,362,308,266,203,152,90)(40,418,363,305,267,208,145,95)(41,429,372,333,259,211,154,98)(42,426,373,330,260,216,155,103)(43,431,374,335,261,213,156,100)(44,428,375,332,262,210,157,97)(45,425,376,329,263,215,158,102)(46,430,369,334,264,212,159,99)(47,427,370,331,257,209,160,104)(48,432,371,336,258,214,153,101)(49,439,380,326,284,221,162,108)(50,436,381,323,285,218,163,105)(51,433,382,328,286,223,164,110)(52,438,383,325,287,220,165,107)(53,435,384,322,288,217,166,112)(54,440,377,327,281,222,167,109)(55,437,378,324,282,219,168,106)(56,434,379,321,283,224,161,111)(57,445,388,349,275,231,170,114)(58,442,389,346,276,228,171,119)(59,447,390,351,277,225,172,116)(60,444,391,348,278,230,173,113)(61,441,392,345,279,227,174,118)(62,446,385,350,280,232,175,115)(63,443,386,347,273,229,176,120)(64,448,387,352,274,226,169,117)>;
G:=Group( (1,397,388,381,372,365,356)(2,398,389,382,373,366,357)(3,399,390,383,374,367,358)(4,400,391,384,375,368,359)(5,393,392,377,376,361,360)(6,394,385,378,369,362,353)(7,395,386,379,370,363,354)(8,396,387,380,371,364,355)(9,445,436,429,420,413,404)(10,446,437,430,421,414,405)(11,447,438,431,422,415,406)(12,448,439,432,423,416,407)(13,441,440,425,424,409,408)(14,442,433,426,417,410,401)(15,443,434,427,418,411,402)(16,444,435,428,419,412,403)(17,65,64,49,48,33,32)(18,66,57,50,41,34,25)(19,67,58,51,42,35,26)(20,68,59,52,43,36,27)(21,69,60,53,44,37,28)(22,70,61,54,45,38,29)(23,71,62,55,46,39,30)(24,72,63,56,47,40,31)(73,121,114,105,98,89,82)(74,122,115,106,99,90,83)(75,123,116,107,100,91,84)(76,124,117,108,101,92,85)(77,125,118,109,102,93,86)(78,126,119,110,103,94,87)(79,127,120,111,104,95,88)(80,128,113,112,97,96,81)(129,177,176,161,160,145,144)(130,178,169,162,153,146,137)(131,179,170,163,154,147,138)(132,180,171,164,155,148,139)(133,181,172,165,156,149,140)(134,182,173,166,157,150,141)(135,183,174,167,158,151,142)(136,184,175,168,159,152,143)(185,237,230,217,210,201,194)(186,238,231,218,211,202,195)(187,239,232,219,212,203,196)(188,240,225,220,213,204,197)(189,233,226,221,214,205,198)(190,234,227,222,215,206,199)(191,235,228,223,216,207,200)(192,236,229,224,209,208,193)(241,251,299,273,283,257,267)(242,252,300,274,284,258,268)(243,253,301,275,285,259,269)(244,254,302,276,286,260,270)(245,255,303,277,287,261,271)(246,256,304,278,288,262,272)(247,249,297,279,281,263,265)(248,250,298,280,282,264,266)(289,337,347,321,331,305,315)(290,338,348,322,332,306,316)(291,339,349,323,333,307,317)(292,340,350,324,334,308,318)(293,341,351,325,335,309,319)(294,342,352,326,336,310,320)(295,343,345,327,329,311,313)(296,344,346,328,330,312,314), (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), (1,291,253,186,131,73,18,404)(2,296,254,191,132,78,19,401)(3,293,255,188,133,75,20,406)(4,290,256,185,134,80,21,403)(5,295,249,190,135,77,22,408)(6,292,250,187,136,74,23,405)(7,289,251,192,129,79,24,402)(8,294,252,189,130,76,17,407)(9,397,339,301,238,179,121,66)(10,394,340,298,239,184,122,71)(11,399,341,303,240,181,123,68)(12,396,342,300,233,178,124,65)(13,393,343,297,234,183,125,70)(14,398,344,302,235,180,126,67)(15,395,337,299,236,177,127,72)(16,400,338,304,237,182,128,69)(25,413,356,317,243,195,138,82)(26,410,357,314,244,200,139,87)(27,415,358,319,245,197,140,84)(28,412,359,316,246,194,141,81)(29,409,360,313,247,199,142,86)(30,414,353,318,248,196,143,83)(31,411,354,315,241,193,144,88)(32,416,355,320,242,198,137,85)(33,423,364,310,268,205,146,92)(34,420,365,307,269,202,147,89)(35,417,366,312,270,207,148,94)(36,422,367,309,271,204,149,91)(37,419,368,306,272,201,150,96)(38,424,361,311,265,206,151,93)(39,421,362,308,266,203,152,90)(40,418,363,305,267,208,145,95)(41,429,372,333,259,211,154,98)(42,426,373,330,260,216,155,103)(43,431,374,335,261,213,156,100)(44,428,375,332,262,210,157,97)(45,425,376,329,263,215,158,102)(46,430,369,334,264,212,159,99)(47,427,370,331,257,209,160,104)(48,432,371,336,258,214,153,101)(49,439,380,326,284,221,162,108)(50,436,381,323,285,218,163,105)(51,433,382,328,286,223,164,110)(52,438,383,325,287,220,165,107)(53,435,384,322,288,217,166,112)(54,440,377,327,281,222,167,109)(55,437,378,324,282,219,168,106)(56,434,379,321,283,224,161,111)(57,445,388,349,275,231,170,114)(58,442,389,346,276,228,171,119)(59,447,390,351,277,225,172,116)(60,444,391,348,278,230,173,113)(61,441,392,345,279,227,174,118)(62,446,385,350,280,232,175,115)(63,443,386,347,273,229,176,120)(64,448,387,352,274,226,169,117) );
G=PermutationGroup([[(1,397,388,381,372,365,356),(2,398,389,382,373,366,357),(3,399,390,383,374,367,358),(4,400,391,384,375,368,359),(5,393,392,377,376,361,360),(6,394,385,378,369,362,353),(7,395,386,379,370,363,354),(8,396,387,380,371,364,355),(9,445,436,429,420,413,404),(10,446,437,430,421,414,405),(11,447,438,431,422,415,406),(12,448,439,432,423,416,407),(13,441,440,425,424,409,408),(14,442,433,426,417,410,401),(15,443,434,427,418,411,402),(16,444,435,428,419,412,403),(17,65,64,49,48,33,32),(18,66,57,50,41,34,25),(19,67,58,51,42,35,26),(20,68,59,52,43,36,27),(21,69,60,53,44,37,28),(22,70,61,54,45,38,29),(23,71,62,55,46,39,30),(24,72,63,56,47,40,31),(73,121,114,105,98,89,82),(74,122,115,106,99,90,83),(75,123,116,107,100,91,84),(76,124,117,108,101,92,85),(77,125,118,109,102,93,86),(78,126,119,110,103,94,87),(79,127,120,111,104,95,88),(80,128,113,112,97,96,81),(129,177,176,161,160,145,144),(130,178,169,162,153,146,137),(131,179,170,163,154,147,138),(132,180,171,164,155,148,139),(133,181,172,165,156,149,140),(134,182,173,166,157,150,141),(135,183,174,167,158,151,142),(136,184,175,168,159,152,143),(185,237,230,217,210,201,194),(186,238,231,218,211,202,195),(187,239,232,219,212,203,196),(188,240,225,220,213,204,197),(189,233,226,221,214,205,198),(190,234,227,222,215,206,199),(191,235,228,223,216,207,200),(192,236,229,224,209,208,193),(241,251,299,273,283,257,267),(242,252,300,274,284,258,268),(243,253,301,275,285,259,269),(244,254,302,276,286,260,270),(245,255,303,277,287,261,271),(246,256,304,278,288,262,272),(247,249,297,279,281,263,265),(248,250,298,280,282,264,266),(289,337,347,321,331,305,315),(290,338,348,322,332,306,316),(291,339,349,323,333,307,317),(292,340,350,324,334,308,318),(293,341,351,325,335,309,319),(294,342,352,326,336,310,320),(295,343,345,327,329,311,313),(296,344,346,328,330,312,314)], [(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)], [(1,291,253,186,131,73,18,404),(2,296,254,191,132,78,19,401),(3,293,255,188,133,75,20,406),(4,290,256,185,134,80,21,403),(5,295,249,190,135,77,22,408),(6,292,250,187,136,74,23,405),(7,289,251,192,129,79,24,402),(8,294,252,189,130,76,17,407),(9,397,339,301,238,179,121,66),(10,394,340,298,239,184,122,71),(11,399,341,303,240,181,123,68),(12,396,342,300,233,178,124,65),(13,393,343,297,234,183,125,70),(14,398,344,302,235,180,126,67),(15,395,337,299,236,177,127,72),(16,400,338,304,237,182,128,69),(25,413,356,317,243,195,138,82),(26,410,357,314,244,200,139,87),(27,415,358,319,245,197,140,84),(28,412,359,316,246,194,141,81),(29,409,360,313,247,199,142,86),(30,414,353,318,248,196,143,83),(31,411,354,315,241,193,144,88),(32,416,355,320,242,198,137,85),(33,423,364,310,268,205,146,92),(34,420,365,307,269,202,147,89),(35,417,366,312,270,207,148,94),(36,422,367,309,271,204,149,91),(37,419,368,306,272,201,150,96),(38,424,361,311,265,206,151,93),(39,421,362,308,266,203,152,90),(40,418,363,305,267,208,145,95),(41,429,372,333,259,211,154,98),(42,426,373,330,260,216,155,103),(43,431,374,335,261,213,156,100),(44,428,375,332,262,210,157,97),(45,425,376,329,263,215,158,102),(46,430,369,334,264,212,159,99),(47,427,370,331,257,209,160,104),(48,432,371,336,258,214,153,101),(49,439,380,326,284,221,162,108),(50,436,381,323,285,218,163,105),(51,433,382,328,286,223,164,110),(52,438,383,325,287,220,165,107),(53,435,384,322,288,217,166,112),(54,440,377,327,281,222,167,109),(55,437,378,324,282,219,168,106),(56,434,379,321,283,224,161,111),(57,445,388,349,275,231,170,114),(58,442,389,346,276,228,171,119),(59,447,390,351,277,225,172,116),(60,444,391,348,278,230,173,113),(61,441,392,345,279,227,174,118),(62,446,385,350,280,232,175,115),(63,443,386,347,273,229,176,120),(64,448,387,352,274,226,169,117)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8X | 14A | ··· | 14R | 28A | ··· | 28BT | 56A | ··· | 56EN |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | ||||||||
| image | C1 | C2 | C4 | C7 | C8 | C14 | C28 | C56 | M4(2) | C7×M4(2) |
| kernel | C7×C8⋊C8 | C4×C56 | C2×C56 | C8⋊C8 | C56 | C4×C8 | C2×C8 | C8 | C28 | C4 |
| # reps | 1 | 3 | 12 | 6 | 16 | 18 | 72 | 96 | 8 | 48 |
Matrix representation of C7×C8⋊C8 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 49 | 10 |
| 0 | 0 | 10 | 64 |
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,49,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,49,10,0,0,10,64],[18,0,0,0,0,1,0,0,0,0,0,112,0,0,1,0] >;
C7×C8⋊C8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes C_8
% in TeX
G:=Group("C7xC8:C8"); // GroupNames label
G:=SmallGroup(448,126);
// by ID
G=gap.SmallGroup(448,126);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,1597,400,136,172]);
// 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^5>;
// generators/relations