direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×C7⋊C8, C7⋊C82, C56⋊3C8, C42.278D14, C14.1(C4×C8), C4.18(C8×D7), (C4×C8).17D7, C28.23(C2×C8), (C4×C56).16C2, (C2×C56).20C4, C2.1(C8×Dic7), (C2×C8).16Dic7, (C2×C14).13C42, (C4×C28).334C22, C22.13(C4×Dic7), C2.1(C4×C7⋊C8), C4.10(C2×C7⋊C8), (C2×C7⋊C8).16C4, (C4×C7⋊C8).18C2, (C2×C4).163(C4×D7), (C2×C28).237(C2×C4), (C2×C4).87(C2×Dic7), SmallGroup(448,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C8×C7⋊C8 |
Generators and relations for C8×C7⋊C8
G = < a,b,c | a8=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 164 in 74 conjugacy classes, 59 normal (13 characteristic)
C1, C2, C2, C4, C22, C7, C8, C8, C2×C4, C2×C4, C14, C14, C42, C2×C8, C2×C8, C28, C2×C14, C4×C8, C4×C8, C7⋊C8, C56, C2×C28, C2×C28, C82, C2×C7⋊C8, C4×C28, C2×C56, C4×C7⋊C8, C4×C56, C8×C7⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D7, C42, C2×C8, Dic7, D14, C4×C8, C7⋊C8, C4×D7, C2×Dic7, C82, C8×D7, C2×C7⋊C8, C4×Dic7, C4×C7⋊C8, C8×Dic7, C8×C7⋊C8
►Regular action on 448 points
Generators in S448
(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)
(1 130 155 87 188 172 13)(2 131 156 88 189 173 14)(3 132 157 81 190 174 15)(4 133 158 82 191 175 16)(5 134 159 83 192 176 9)(6 135 160 84 185 169 10)(7 136 153 85 186 170 11)(8 129 154 86 187 171 12)(17 200 77 126 98 214 208)(18 193 78 127 99 215 201)(19 194 79 128 100 216 202)(20 195 80 121 101 209 203)(21 196 73 122 102 210 204)(22 197 74 123 103 211 205)(23 198 75 124 104 212 206)(24 199 76 125 97 213 207)(25 408 283 334 386 372 36)(26 401 284 335 387 373 37)(27 402 285 336 388 374 38)(28 403 286 329 389 375 39)(29 404 287 330 390 376 40)(30 405 288 331 391 369 33)(31 406 281 332 392 370 34)(32 407 282 333 385 371 35)(41 166 106 63 180 341 352)(42 167 107 64 181 342 345)(43 168 108 57 182 343 346)(44 161 109 58 183 344 347)(45 162 110 59 184 337 348)(46 163 111 60 177 338 349)(47 164 112 61 178 339 350)(48 165 105 62 179 340 351)(49 94 227 258 72 220 236)(50 95 228 259 65 221 237)(51 96 229 260 66 222 238)(52 89 230 261 67 223 239)(53 90 231 262 68 224 240)(54 91 232 263 69 217 233)(55 92 225 264 70 218 234)(56 93 226 257 71 219 235)(113 320 251 143 446 290 152)(114 313 252 144 447 291 145)(115 314 253 137 448 292 146)(116 315 254 138 441 293 147)(117 316 255 139 442 294 148)(118 317 256 140 443 295 149)(119 318 249 141 444 296 150)(120 319 250 142 445 289 151)(241 278 438 322 378 301 424)(242 279 439 323 379 302 417)(243 280 440 324 380 303 418)(244 273 433 325 381 304 419)(245 274 434 326 382 297 420)(246 275 435 327 383 298 421)(247 276 436 328 384 299 422)(248 277 437 321 377 300 423)(265 396 412 311 360 361 429)(266 397 413 312 353 362 430)(267 398 414 305 354 363 431)(268 399 415 306 355 364 432)(269 400 416 307 356 365 425)(270 393 409 308 357 366 426)(271 394 410 309 358 367 427)(272 395 411 310 359 368 428)
(1 294 71 380 121 330 107 428)(2 295 72 381 122 331 108 429)(3 296 65 382 123 332 109 430)(4 289 66 383 124 333 110 431)(5 290 67 384 125 334 111 432)(6 291 68 377 126 335 112 425)(7 292 69 378 127 336 105 426)(8 293 70 379 128 329 106 427)(9 152 261 299 76 386 163 268)(10 145 262 300 77 387 164 269)(11 146 263 301 78 388 165 270)(12 147 264 302 79 389 166 271)(13 148 257 303 80 390 167 272)(14 149 258 304 73 391 168 265)(15 150 259 297 74 392 161 266)(16 151 260 298 75 385 162 267)(17 37 350 416 185 313 90 248)(18 38 351 409 186 314 91 241)(19 39 352 410 187 315 92 242)(20 40 345 411 188 316 93 243)(21 33 346 412 189 317 94 244)(22 34 347 413 190 318 95 245)(23 35 348 414 191 319 96 246)(24 36 349 415 192 320 89 247)(25 338 306 83 251 52 276 207)(26 339 307 84 252 53 277 208)(27 340 308 85 253 54 278 201)(28 341 309 86 254 55 279 202)(29 342 310 87 255 56 280 203)(30 343 311 88 256 49 273 204)(31 344 312 81 249 50 274 205)(32 337 305 82 250 51 275 206)(41 394 171 116 225 417 194 375)(42 395 172 117 226 418 195 376)(43 396 173 118 227 419 196 369)(44 397 174 119 228 420 197 370)(45 398 175 120 229 421 198 371)(46 399 176 113 230 422 199 372)(47 400 169 114 231 423 200 373)(48 393 170 115 232 424 193 374)(57 361 131 443 220 325 102 288)(58 362 132 444 221 326 103 281)(59 363 133 445 222 327 104 282)(60 364 134 446 223 328 97 283)(61 365 135 447 224 321 98 284)(62 366 136 448 217 322 99 285)(63 367 129 441 218 323 100 286)(64 368 130 442 219 324 101 287)(137 233 438 215 402 179 357 153)(138 234 439 216 403 180 358 154)(139 235 440 209 404 181 359 155)(140 236 433 210 405 182 360 156)(141 237 434 211 406 183 353 157)(142 238 435 212 407 184 354 158)(143 239 436 213 408 177 355 159)(144 240 437 214 401 178 356 160)
G:=sub<Sym(448)| (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), (1,130,155,87,188,172,13)(2,131,156,88,189,173,14)(3,132,157,81,190,174,15)(4,133,158,82,191,175,16)(5,134,159,83,192,176,9)(6,135,160,84,185,169,10)(7,136,153,85,186,170,11)(8,129,154,86,187,171,12)(17,200,77,126,98,214,208)(18,193,78,127,99,215,201)(19,194,79,128,100,216,202)(20,195,80,121,101,209,203)(21,196,73,122,102,210,204)(22,197,74,123,103,211,205)(23,198,75,124,104,212,206)(24,199,76,125,97,213,207)(25,408,283,334,386,372,36)(26,401,284,335,387,373,37)(27,402,285,336,388,374,38)(28,403,286,329,389,375,39)(29,404,287,330,390,376,40)(30,405,288,331,391,369,33)(31,406,281,332,392,370,34)(32,407,282,333,385,371,35)(41,166,106,63,180,341,352)(42,167,107,64,181,342,345)(43,168,108,57,182,343,346)(44,161,109,58,183,344,347)(45,162,110,59,184,337,348)(46,163,111,60,177,338,349)(47,164,112,61,178,339,350)(48,165,105,62,179,340,351)(49,94,227,258,72,220,236)(50,95,228,259,65,221,237)(51,96,229,260,66,222,238)(52,89,230,261,67,223,239)(53,90,231,262,68,224,240)(54,91,232,263,69,217,233)(55,92,225,264,70,218,234)(56,93,226,257,71,219,235)(113,320,251,143,446,290,152)(114,313,252,144,447,291,145)(115,314,253,137,448,292,146)(116,315,254,138,441,293,147)(117,316,255,139,442,294,148)(118,317,256,140,443,295,149)(119,318,249,141,444,296,150)(120,319,250,142,445,289,151)(241,278,438,322,378,301,424)(242,279,439,323,379,302,417)(243,280,440,324,380,303,418)(244,273,433,325,381,304,419)(245,274,434,326,382,297,420)(246,275,435,327,383,298,421)(247,276,436,328,384,299,422)(248,277,437,321,377,300,423)(265,396,412,311,360,361,429)(266,397,413,312,353,362,430)(267,398,414,305,354,363,431)(268,399,415,306,355,364,432)(269,400,416,307,356,365,425)(270,393,409,308,357,366,426)(271,394,410,309,358,367,427)(272,395,411,310,359,368,428), (1,294,71,380,121,330,107,428)(2,295,72,381,122,331,108,429)(3,296,65,382,123,332,109,430)(4,289,66,383,124,333,110,431)(5,290,67,384,125,334,111,432)(6,291,68,377,126,335,112,425)(7,292,69,378,127,336,105,426)(8,293,70,379,128,329,106,427)(9,152,261,299,76,386,163,268)(10,145,262,300,77,387,164,269)(11,146,263,301,78,388,165,270)(12,147,264,302,79,389,166,271)(13,148,257,303,80,390,167,272)(14,149,258,304,73,391,168,265)(15,150,259,297,74,392,161,266)(16,151,260,298,75,385,162,267)(17,37,350,416,185,313,90,248)(18,38,351,409,186,314,91,241)(19,39,352,410,187,315,92,242)(20,40,345,411,188,316,93,243)(21,33,346,412,189,317,94,244)(22,34,347,413,190,318,95,245)(23,35,348,414,191,319,96,246)(24,36,349,415,192,320,89,247)(25,338,306,83,251,52,276,207)(26,339,307,84,252,53,277,208)(27,340,308,85,253,54,278,201)(28,341,309,86,254,55,279,202)(29,342,310,87,255,56,280,203)(30,343,311,88,256,49,273,204)(31,344,312,81,249,50,274,205)(32,337,305,82,250,51,275,206)(41,394,171,116,225,417,194,375)(42,395,172,117,226,418,195,376)(43,396,173,118,227,419,196,369)(44,397,174,119,228,420,197,370)(45,398,175,120,229,421,198,371)(46,399,176,113,230,422,199,372)(47,400,169,114,231,423,200,373)(48,393,170,115,232,424,193,374)(57,361,131,443,220,325,102,288)(58,362,132,444,221,326,103,281)(59,363,133,445,222,327,104,282)(60,364,134,446,223,328,97,283)(61,365,135,447,224,321,98,284)(62,366,136,448,217,322,99,285)(63,367,129,441,218,323,100,286)(64,368,130,442,219,324,101,287)(137,233,438,215,402,179,357,153)(138,234,439,216,403,180,358,154)(139,235,440,209,404,181,359,155)(140,236,433,210,405,182,360,156)(141,237,434,211,406,183,353,157)(142,238,435,212,407,184,354,158)(143,239,436,213,408,177,355,159)(144,240,437,214,401,178,356,160)>;
G:=Group( (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), (1,130,155,87,188,172,13)(2,131,156,88,189,173,14)(3,132,157,81,190,174,15)(4,133,158,82,191,175,16)(5,134,159,83,192,176,9)(6,135,160,84,185,169,10)(7,136,153,85,186,170,11)(8,129,154,86,187,171,12)(17,200,77,126,98,214,208)(18,193,78,127,99,215,201)(19,194,79,128,100,216,202)(20,195,80,121,101,209,203)(21,196,73,122,102,210,204)(22,197,74,123,103,211,205)(23,198,75,124,104,212,206)(24,199,76,125,97,213,207)(25,408,283,334,386,372,36)(26,401,284,335,387,373,37)(27,402,285,336,388,374,38)(28,403,286,329,389,375,39)(29,404,287,330,390,376,40)(30,405,288,331,391,369,33)(31,406,281,332,392,370,34)(32,407,282,333,385,371,35)(41,166,106,63,180,341,352)(42,167,107,64,181,342,345)(43,168,108,57,182,343,346)(44,161,109,58,183,344,347)(45,162,110,59,184,337,348)(46,163,111,60,177,338,349)(47,164,112,61,178,339,350)(48,165,105,62,179,340,351)(49,94,227,258,72,220,236)(50,95,228,259,65,221,237)(51,96,229,260,66,222,238)(52,89,230,261,67,223,239)(53,90,231,262,68,224,240)(54,91,232,263,69,217,233)(55,92,225,264,70,218,234)(56,93,226,257,71,219,235)(113,320,251,143,446,290,152)(114,313,252,144,447,291,145)(115,314,253,137,448,292,146)(116,315,254,138,441,293,147)(117,316,255,139,442,294,148)(118,317,256,140,443,295,149)(119,318,249,141,444,296,150)(120,319,250,142,445,289,151)(241,278,438,322,378,301,424)(242,279,439,323,379,302,417)(243,280,440,324,380,303,418)(244,273,433,325,381,304,419)(245,274,434,326,382,297,420)(246,275,435,327,383,298,421)(247,276,436,328,384,299,422)(248,277,437,321,377,300,423)(265,396,412,311,360,361,429)(266,397,413,312,353,362,430)(267,398,414,305,354,363,431)(268,399,415,306,355,364,432)(269,400,416,307,356,365,425)(270,393,409,308,357,366,426)(271,394,410,309,358,367,427)(272,395,411,310,359,368,428), (1,294,71,380,121,330,107,428)(2,295,72,381,122,331,108,429)(3,296,65,382,123,332,109,430)(4,289,66,383,124,333,110,431)(5,290,67,384,125,334,111,432)(6,291,68,377,126,335,112,425)(7,292,69,378,127,336,105,426)(8,293,70,379,128,329,106,427)(9,152,261,299,76,386,163,268)(10,145,262,300,77,387,164,269)(11,146,263,301,78,388,165,270)(12,147,264,302,79,389,166,271)(13,148,257,303,80,390,167,272)(14,149,258,304,73,391,168,265)(15,150,259,297,74,392,161,266)(16,151,260,298,75,385,162,267)(17,37,350,416,185,313,90,248)(18,38,351,409,186,314,91,241)(19,39,352,410,187,315,92,242)(20,40,345,411,188,316,93,243)(21,33,346,412,189,317,94,244)(22,34,347,413,190,318,95,245)(23,35,348,414,191,319,96,246)(24,36,349,415,192,320,89,247)(25,338,306,83,251,52,276,207)(26,339,307,84,252,53,277,208)(27,340,308,85,253,54,278,201)(28,341,309,86,254,55,279,202)(29,342,310,87,255,56,280,203)(30,343,311,88,256,49,273,204)(31,344,312,81,249,50,274,205)(32,337,305,82,250,51,275,206)(41,394,171,116,225,417,194,375)(42,395,172,117,226,418,195,376)(43,396,173,118,227,419,196,369)(44,397,174,119,228,420,197,370)(45,398,175,120,229,421,198,371)(46,399,176,113,230,422,199,372)(47,400,169,114,231,423,200,373)(48,393,170,115,232,424,193,374)(57,361,131,443,220,325,102,288)(58,362,132,444,221,326,103,281)(59,363,133,445,222,327,104,282)(60,364,134,446,223,328,97,283)(61,365,135,447,224,321,98,284)(62,366,136,448,217,322,99,285)(63,367,129,441,218,323,100,286)(64,368,130,442,219,324,101,287)(137,233,438,215,402,179,357,153)(138,234,439,216,403,180,358,154)(139,235,440,209,404,181,359,155)(140,236,433,210,405,182,360,156)(141,237,434,211,406,183,353,157)(142,238,435,212,407,184,354,158)(143,239,436,213,408,177,355,159)(144,240,437,214,401,178,356,160) );
G=PermutationGroup([[(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)], [(1,130,155,87,188,172,13),(2,131,156,88,189,173,14),(3,132,157,81,190,174,15),(4,133,158,82,191,175,16),(5,134,159,83,192,176,9),(6,135,160,84,185,169,10),(7,136,153,85,186,170,11),(8,129,154,86,187,171,12),(17,200,77,126,98,214,208),(18,193,78,127,99,215,201),(19,194,79,128,100,216,202),(20,195,80,121,101,209,203),(21,196,73,122,102,210,204),(22,197,74,123,103,211,205),(23,198,75,124,104,212,206),(24,199,76,125,97,213,207),(25,408,283,334,386,372,36),(26,401,284,335,387,373,37),(27,402,285,336,388,374,38),(28,403,286,329,389,375,39),(29,404,287,330,390,376,40),(30,405,288,331,391,369,33),(31,406,281,332,392,370,34),(32,407,282,333,385,371,35),(41,166,106,63,180,341,352),(42,167,107,64,181,342,345),(43,168,108,57,182,343,346),(44,161,109,58,183,344,347),(45,162,110,59,184,337,348),(46,163,111,60,177,338,349),(47,164,112,61,178,339,350),(48,165,105,62,179,340,351),(49,94,227,258,72,220,236),(50,95,228,259,65,221,237),(51,96,229,260,66,222,238),(52,89,230,261,67,223,239),(53,90,231,262,68,224,240),(54,91,232,263,69,217,233),(55,92,225,264,70,218,234),(56,93,226,257,71,219,235),(113,320,251,143,446,290,152),(114,313,252,144,447,291,145),(115,314,253,137,448,292,146),(116,315,254,138,441,293,147),(117,316,255,139,442,294,148),(118,317,256,140,443,295,149),(119,318,249,141,444,296,150),(120,319,250,142,445,289,151),(241,278,438,322,378,301,424),(242,279,439,323,379,302,417),(243,280,440,324,380,303,418),(244,273,433,325,381,304,419),(245,274,434,326,382,297,420),(246,275,435,327,383,298,421),(247,276,436,328,384,299,422),(248,277,437,321,377,300,423),(265,396,412,311,360,361,429),(266,397,413,312,353,362,430),(267,398,414,305,354,363,431),(268,399,415,306,355,364,432),(269,400,416,307,356,365,425),(270,393,409,308,357,366,426),(271,394,410,309,358,367,427),(272,395,411,310,359,368,428)], [(1,294,71,380,121,330,107,428),(2,295,72,381,122,331,108,429),(3,296,65,382,123,332,109,430),(4,289,66,383,124,333,110,431),(5,290,67,384,125,334,111,432),(6,291,68,377,126,335,112,425),(7,292,69,378,127,336,105,426),(8,293,70,379,128,329,106,427),(9,152,261,299,76,386,163,268),(10,145,262,300,77,387,164,269),(11,146,263,301,78,388,165,270),(12,147,264,302,79,389,166,271),(13,148,257,303,80,390,167,272),(14,149,258,304,73,391,168,265),(15,150,259,297,74,392,161,266),(16,151,260,298,75,385,162,267),(17,37,350,416,185,313,90,248),(18,38,351,409,186,314,91,241),(19,39,352,410,187,315,92,242),(20,40,345,411,188,316,93,243),(21,33,346,412,189,317,94,244),(22,34,347,413,190,318,95,245),(23,35,348,414,191,319,96,246),(24,36,349,415,192,320,89,247),(25,338,306,83,251,52,276,207),(26,339,307,84,252,53,277,208),(27,340,308,85,253,54,278,201),(28,341,309,86,254,55,279,202),(29,342,310,87,255,56,280,203),(30,343,311,88,256,49,273,204),(31,344,312,81,249,50,274,205),(32,337,305,82,250,51,275,206),(41,394,171,116,225,417,194,375),(42,395,172,117,226,418,195,376),(43,396,173,118,227,419,196,369),(44,397,174,119,228,420,197,370),(45,398,175,120,229,421,198,371),(46,399,176,113,230,422,199,372),(47,400,169,114,231,423,200,373),(48,393,170,115,232,424,193,374),(57,361,131,443,220,325,102,288),(58,362,132,444,221,326,103,281),(59,363,133,445,222,327,104,282),(60,364,134,446,223,328,97,283),(61,365,135,447,224,321,98,284),(62,366,136,448,217,322,99,285),(63,367,129,441,218,323,100,286),(64,368,130,442,219,324,101,287),(137,233,438,215,402,179,357,153),(138,234,439,216,403,180,358,154),(139,235,440,209,404,181,359,155),(140,236,433,210,405,182,360,156),(141,237,434,211,406,183,353,157),(142,238,435,212,407,184,354,158),(143,239,436,213,408,177,355,159),(144,240,437,214,401,178,356,160)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8P | 8Q | ··· | 8AV | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D7 | D14 | Dic7 | C7⋊C8 | C4×D7 | C8×D7 |
| kernel | C8×C7⋊C8 | C4×C7⋊C8 | C4×C56 | C2×C7⋊C8 | C2×C56 | C7⋊C8 | C56 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 32 | 16 | 3 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of C8×C7⋊C8 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 69 | 0 |
| 0 | 0 | 69 |
| 1 | 0 | 0 |
| 0 | 24 | 112 |
| 0 | 1 | 0 |
| 18 | 0 | 0 |
| 0 | 67 | 20 |
| 0 | 46 | 46 |
G:=sub<GL(3,GF(113))| [98,0,0,0,69,0,0,0,69],[1,0,0,0,24,1,0,112,0],[18,0,0,0,67,46,0,20,46] >;
C8×C7⋊C8 in GAP, Magma, Sage, TeX
C_8\times C_7\rtimes C_8
% in TeX
G:=Group("C8xC7:C8"); // GroupNames label
G:=SmallGroup(448,10);
// by ID
G=gap.SmallGroup(448,10);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,64,100,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations