direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊1C8, C8⋊1C56, C56⋊5C8, C28.68D8, C28.31Q16, C28.32M4(2), C4⋊C8.5C14, (C2×C8).8C28, (C4×C8).5C14, C4.7(C2×C56), C4.17(C7×D8), C4.9(C7×Q16), C28.47(C2×C8), (C2×C56).14C4, (C4×C56).23C2, C14.12(C4⋊C8), (C2×C28).48Q8, (C2×C28).531D4, C4.5(C7×M4(2)), C42.65(C2×C14), C14.11(C2.D8), C14.8(C8.C4), (C4×C28).349C22, C2.4(C7×C4⋊C8), (C7×C4⋊C8).11C2, (C2×C4).9(C7×Q8), C2.1(C7×C2.D8), (C2×C4).62(C2×C28), (C2×C4).140(C7×D4), C2.2(C7×C8.C4), C22.14(C7×C4⋊C4), (C2×C14).57(C4⋊C4), (C2×C28).323(C2×C4), SmallGroup(448,139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊1C8
G = < a,b,c | a7=b8=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C8⋊1C8
►Regular action on 448 points
Generators in S448
(1 172 49 156 33 140 17)(2 173 50 157 34 141 18)(3 174 51 158 35 142 19)(4 175 52 159 36 143 20)(5 176 53 160 37 144 21)(6 169 54 153 38 137 22)(7 170 55 154 39 138 23)(8 171 56 155 40 139 24)(9 327 440 311 424 295 122)(10 328 433 312 417 296 123)(11 321 434 305 418 289 124)(12 322 435 306 419 290 125)(13 323 436 307 420 291 126)(14 324 437 308 421 292 127)(15 325 438 309 422 293 128)(16 326 439 310 423 294 121)(25 136 184 57 168 41 152)(26 129 177 58 161 42 145)(27 130 178 59 162 43 146)(28 131 179 60 163 44 147)(29 132 180 61 164 45 148)(30 133 181 62 165 46 149)(31 134 182 63 166 47 150)(32 135 183 64 167 48 151)(65 233 220 97 204 81 188)(66 234 221 98 205 82 189)(67 235 222 99 206 83 190)(68 236 223 100 207 84 191)(69 237 224 101 208 85 192)(70 238 217 102 201 86 185)(71 239 218 103 202 87 186)(72 240 219 104 203 88 187)(73 225 338 105 216 89 200)(74 226 339 106 209 90 193)(75 227 340 107 210 91 194)(76 228 341 108 211 92 195)(77 229 342 109 212 93 196)(78 230 343 110 213 94 197)(79 231 344 111 214 95 198)(80 232 337 112 215 96 199)(113 445 330 429 314 413 298)(114 446 331 430 315 414 299)(115 447 332 431 316 415 300)(116 448 333 432 317 416 301)(117 441 334 425 318 409 302)(118 442 335 426 319 410 303)(119 443 336 427 320 411 304)(120 444 329 428 313 412 297)(241 360 346 404 275 388 259)(242 353 347 405 276 389 260)(243 354 348 406 277 390 261)(244 355 349 407 278 391 262)(245 356 350 408 279 392 263)(246 357 351 401 280 385 264)(247 358 352 402 273 386 257)(248 359 345 403 274 387 258)(249 375 396 281 380 265 364)(250 376 397 282 381 266 365)(251 369 398 283 382 267 366)(252 370 399 284 383 268 367)(253 371 400 285 384 269 368)(254 372 393 286 377 270 361)(255 373 394 287 378 271 362)(256 374 395 288 379 272 363)
(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)
(1 300 136 260 66 292 74 256)(2 299 129 259 67 291 75 255)(3 298 130 258 68 290 76 254)(4 297 131 257 69 289 77 253)(5 304 132 264 70 296 78 252)(6 303 133 263 71 295 79 251)(7 302 134 262 72 294 80 250)(8 301 135 261 65 293 73 249)(9 344 398 54 442 62 356 218)(10 343 399 53 443 61 357 217)(11 342 400 52 444 60 358 224)(12 341 393 51 445 59 359 223)(13 340 394 50 446 58 360 222)(14 339 395 49 447 57 353 221)(15 338 396 56 448 64 354 220)(16 337 397 55 441 63 355 219)(17 415 25 389 189 421 193 363)(18 414 26 388 190 420 194 362)(19 413 27 387 191 419 195 361)(20 412 28 386 192 418 196 368)(21 411 29 385 185 417 197 367)(22 410 30 392 186 424 198 366)(23 409 31 391 187 423 199 365)(24 416 32 390 188 422 200 364)(33 431 41 405 205 437 209 379)(34 430 42 404 206 436 210 378)(35 429 43 403 207 435 211 377)(36 428 44 402 208 434 212 384)(37 427 45 401 201 433 213 383)(38 426 46 408 202 440 214 382)(39 425 47 407 203 439 215 381)(40 432 48 406 204 438 216 380)(81 309 89 265 139 317 151 277)(82 308 90 272 140 316 152 276)(83 307 91 271 141 315 145 275)(84 306 92 270 142 314 146 274)(85 305 93 269 143 313 147 273)(86 312 94 268 144 320 148 280)(87 311 95 267 137 319 149 279)(88 310 96 266 138 318 150 278)(97 325 105 281 155 333 167 348)(98 324 106 288 156 332 168 347)(99 323 107 287 157 331 161 346)(100 322 108 286 158 330 162 345)(101 321 109 285 159 329 163 352)(102 328 110 284 160 336 164 351)(103 327 111 283 153 335 165 350)(104 326 112 282 154 334 166 349)(113 178 248 236 125 228 372 174)(114 177 241 235 126 227 373 173)(115 184 242 234 127 226 374 172)(116 183 243 233 128 225 375 171)(117 182 244 240 121 232 376 170)(118 181 245 239 122 231 369 169)(119 180 246 238 123 230 370 176)(120 179 247 237 124 229 371 175)
G:=sub<Sym(448)| (1,172,49,156,33,140,17)(2,173,50,157,34,141,18)(3,174,51,158,35,142,19)(4,175,52,159,36,143,20)(5,176,53,160,37,144,21)(6,169,54,153,38,137,22)(7,170,55,154,39,138,23)(8,171,56,155,40,139,24)(9,327,440,311,424,295,122)(10,328,433,312,417,296,123)(11,321,434,305,418,289,124)(12,322,435,306,419,290,125)(13,323,436,307,420,291,126)(14,324,437,308,421,292,127)(15,325,438,309,422,293,128)(16,326,439,310,423,294,121)(25,136,184,57,168,41,152)(26,129,177,58,161,42,145)(27,130,178,59,162,43,146)(28,131,179,60,163,44,147)(29,132,180,61,164,45,148)(30,133,181,62,165,46,149)(31,134,182,63,166,47,150)(32,135,183,64,167,48,151)(65,233,220,97,204,81,188)(66,234,221,98,205,82,189)(67,235,222,99,206,83,190)(68,236,223,100,207,84,191)(69,237,224,101,208,85,192)(70,238,217,102,201,86,185)(71,239,218,103,202,87,186)(72,240,219,104,203,88,187)(73,225,338,105,216,89,200)(74,226,339,106,209,90,193)(75,227,340,107,210,91,194)(76,228,341,108,211,92,195)(77,229,342,109,212,93,196)(78,230,343,110,213,94,197)(79,231,344,111,214,95,198)(80,232,337,112,215,96,199)(113,445,330,429,314,413,298)(114,446,331,430,315,414,299)(115,447,332,431,316,415,300)(116,448,333,432,317,416,301)(117,441,334,425,318,409,302)(118,442,335,426,319,410,303)(119,443,336,427,320,411,304)(120,444,329,428,313,412,297)(241,360,346,404,275,388,259)(242,353,347,405,276,389,260)(243,354,348,406,277,390,261)(244,355,349,407,278,391,262)(245,356,350,408,279,392,263)(246,357,351,401,280,385,264)(247,358,352,402,273,386,257)(248,359,345,403,274,387,258)(249,375,396,281,380,265,364)(250,376,397,282,381,266,365)(251,369,398,283,382,267,366)(252,370,399,284,383,268,367)(253,371,400,285,384,269,368)(254,372,393,286,377,270,361)(255,373,394,287,378,271,362)(256,374,395,288,379,272,363), (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), (1,300,136,260,66,292,74,256)(2,299,129,259,67,291,75,255)(3,298,130,258,68,290,76,254)(4,297,131,257,69,289,77,253)(5,304,132,264,70,296,78,252)(6,303,133,263,71,295,79,251)(7,302,134,262,72,294,80,250)(8,301,135,261,65,293,73,249)(9,344,398,54,442,62,356,218)(10,343,399,53,443,61,357,217)(11,342,400,52,444,60,358,224)(12,341,393,51,445,59,359,223)(13,340,394,50,446,58,360,222)(14,339,395,49,447,57,353,221)(15,338,396,56,448,64,354,220)(16,337,397,55,441,63,355,219)(17,415,25,389,189,421,193,363)(18,414,26,388,190,420,194,362)(19,413,27,387,191,419,195,361)(20,412,28,386,192,418,196,368)(21,411,29,385,185,417,197,367)(22,410,30,392,186,424,198,366)(23,409,31,391,187,423,199,365)(24,416,32,390,188,422,200,364)(33,431,41,405,205,437,209,379)(34,430,42,404,206,436,210,378)(35,429,43,403,207,435,211,377)(36,428,44,402,208,434,212,384)(37,427,45,401,201,433,213,383)(38,426,46,408,202,440,214,382)(39,425,47,407,203,439,215,381)(40,432,48,406,204,438,216,380)(81,309,89,265,139,317,151,277)(82,308,90,272,140,316,152,276)(83,307,91,271,141,315,145,275)(84,306,92,270,142,314,146,274)(85,305,93,269,143,313,147,273)(86,312,94,268,144,320,148,280)(87,311,95,267,137,319,149,279)(88,310,96,266,138,318,150,278)(97,325,105,281,155,333,167,348)(98,324,106,288,156,332,168,347)(99,323,107,287,157,331,161,346)(100,322,108,286,158,330,162,345)(101,321,109,285,159,329,163,352)(102,328,110,284,160,336,164,351)(103,327,111,283,153,335,165,350)(104,326,112,282,154,334,166,349)(113,178,248,236,125,228,372,174)(114,177,241,235,126,227,373,173)(115,184,242,234,127,226,374,172)(116,183,243,233,128,225,375,171)(117,182,244,240,121,232,376,170)(118,181,245,239,122,231,369,169)(119,180,246,238,123,230,370,176)(120,179,247,237,124,229,371,175)>;
G:=Group( (1,172,49,156,33,140,17)(2,173,50,157,34,141,18)(3,174,51,158,35,142,19)(4,175,52,159,36,143,20)(5,176,53,160,37,144,21)(6,169,54,153,38,137,22)(7,170,55,154,39,138,23)(8,171,56,155,40,139,24)(9,327,440,311,424,295,122)(10,328,433,312,417,296,123)(11,321,434,305,418,289,124)(12,322,435,306,419,290,125)(13,323,436,307,420,291,126)(14,324,437,308,421,292,127)(15,325,438,309,422,293,128)(16,326,439,310,423,294,121)(25,136,184,57,168,41,152)(26,129,177,58,161,42,145)(27,130,178,59,162,43,146)(28,131,179,60,163,44,147)(29,132,180,61,164,45,148)(30,133,181,62,165,46,149)(31,134,182,63,166,47,150)(32,135,183,64,167,48,151)(65,233,220,97,204,81,188)(66,234,221,98,205,82,189)(67,235,222,99,206,83,190)(68,236,223,100,207,84,191)(69,237,224,101,208,85,192)(70,238,217,102,201,86,185)(71,239,218,103,202,87,186)(72,240,219,104,203,88,187)(73,225,338,105,216,89,200)(74,226,339,106,209,90,193)(75,227,340,107,210,91,194)(76,228,341,108,211,92,195)(77,229,342,109,212,93,196)(78,230,343,110,213,94,197)(79,231,344,111,214,95,198)(80,232,337,112,215,96,199)(113,445,330,429,314,413,298)(114,446,331,430,315,414,299)(115,447,332,431,316,415,300)(116,448,333,432,317,416,301)(117,441,334,425,318,409,302)(118,442,335,426,319,410,303)(119,443,336,427,320,411,304)(120,444,329,428,313,412,297)(241,360,346,404,275,388,259)(242,353,347,405,276,389,260)(243,354,348,406,277,390,261)(244,355,349,407,278,391,262)(245,356,350,408,279,392,263)(246,357,351,401,280,385,264)(247,358,352,402,273,386,257)(248,359,345,403,274,387,258)(249,375,396,281,380,265,364)(250,376,397,282,381,266,365)(251,369,398,283,382,267,366)(252,370,399,284,383,268,367)(253,371,400,285,384,269,368)(254,372,393,286,377,270,361)(255,373,394,287,378,271,362)(256,374,395,288,379,272,363), (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), (1,300,136,260,66,292,74,256)(2,299,129,259,67,291,75,255)(3,298,130,258,68,290,76,254)(4,297,131,257,69,289,77,253)(5,304,132,264,70,296,78,252)(6,303,133,263,71,295,79,251)(7,302,134,262,72,294,80,250)(8,301,135,261,65,293,73,249)(9,344,398,54,442,62,356,218)(10,343,399,53,443,61,357,217)(11,342,400,52,444,60,358,224)(12,341,393,51,445,59,359,223)(13,340,394,50,446,58,360,222)(14,339,395,49,447,57,353,221)(15,338,396,56,448,64,354,220)(16,337,397,55,441,63,355,219)(17,415,25,389,189,421,193,363)(18,414,26,388,190,420,194,362)(19,413,27,387,191,419,195,361)(20,412,28,386,192,418,196,368)(21,411,29,385,185,417,197,367)(22,410,30,392,186,424,198,366)(23,409,31,391,187,423,199,365)(24,416,32,390,188,422,200,364)(33,431,41,405,205,437,209,379)(34,430,42,404,206,436,210,378)(35,429,43,403,207,435,211,377)(36,428,44,402,208,434,212,384)(37,427,45,401,201,433,213,383)(38,426,46,408,202,440,214,382)(39,425,47,407,203,439,215,381)(40,432,48,406,204,438,216,380)(81,309,89,265,139,317,151,277)(82,308,90,272,140,316,152,276)(83,307,91,271,141,315,145,275)(84,306,92,270,142,314,146,274)(85,305,93,269,143,313,147,273)(86,312,94,268,144,320,148,280)(87,311,95,267,137,319,149,279)(88,310,96,266,138,318,150,278)(97,325,105,281,155,333,167,348)(98,324,106,288,156,332,168,347)(99,323,107,287,157,331,161,346)(100,322,108,286,158,330,162,345)(101,321,109,285,159,329,163,352)(102,328,110,284,160,336,164,351)(103,327,111,283,153,335,165,350)(104,326,112,282,154,334,166,349)(113,178,248,236,125,228,372,174)(114,177,241,235,126,227,373,173)(115,184,242,234,127,226,374,172)(116,183,243,233,128,225,375,171)(117,182,244,240,121,232,376,170)(118,181,245,239,122,231,369,169)(119,180,246,238,123,230,370,176)(120,179,247,237,124,229,371,175) );
G=PermutationGroup([[(1,172,49,156,33,140,17),(2,173,50,157,34,141,18),(3,174,51,158,35,142,19),(4,175,52,159,36,143,20),(5,176,53,160,37,144,21),(6,169,54,153,38,137,22),(7,170,55,154,39,138,23),(8,171,56,155,40,139,24),(9,327,440,311,424,295,122),(10,328,433,312,417,296,123),(11,321,434,305,418,289,124),(12,322,435,306,419,290,125),(13,323,436,307,420,291,126),(14,324,437,308,421,292,127),(15,325,438,309,422,293,128),(16,326,439,310,423,294,121),(25,136,184,57,168,41,152),(26,129,177,58,161,42,145),(27,130,178,59,162,43,146),(28,131,179,60,163,44,147),(29,132,180,61,164,45,148),(30,133,181,62,165,46,149),(31,134,182,63,166,47,150),(32,135,183,64,167,48,151),(65,233,220,97,204,81,188),(66,234,221,98,205,82,189),(67,235,222,99,206,83,190),(68,236,223,100,207,84,191),(69,237,224,101,208,85,192),(70,238,217,102,201,86,185),(71,239,218,103,202,87,186),(72,240,219,104,203,88,187),(73,225,338,105,216,89,200),(74,226,339,106,209,90,193),(75,227,340,107,210,91,194),(76,228,341,108,211,92,195),(77,229,342,109,212,93,196),(78,230,343,110,213,94,197),(79,231,344,111,214,95,198),(80,232,337,112,215,96,199),(113,445,330,429,314,413,298),(114,446,331,430,315,414,299),(115,447,332,431,316,415,300),(116,448,333,432,317,416,301),(117,441,334,425,318,409,302),(118,442,335,426,319,410,303),(119,443,336,427,320,411,304),(120,444,329,428,313,412,297),(241,360,346,404,275,388,259),(242,353,347,405,276,389,260),(243,354,348,406,277,390,261),(244,355,349,407,278,391,262),(245,356,350,408,279,392,263),(246,357,351,401,280,385,264),(247,358,352,402,273,386,257),(248,359,345,403,274,387,258),(249,375,396,281,380,265,364),(250,376,397,282,381,266,365),(251,369,398,283,382,267,366),(252,370,399,284,383,268,367),(253,371,400,285,384,269,368),(254,372,393,286,377,270,361),(255,373,394,287,378,271,362),(256,374,395,288,379,272,363)], [(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)], [(1,300,136,260,66,292,74,256),(2,299,129,259,67,291,75,255),(3,298,130,258,68,290,76,254),(4,297,131,257,69,289,77,253),(5,304,132,264,70,296,78,252),(6,303,133,263,71,295,79,251),(7,302,134,262,72,294,80,250),(8,301,135,261,65,293,73,249),(9,344,398,54,442,62,356,218),(10,343,399,53,443,61,357,217),(11,342,400,52,444,60,358,224),(12,341,393,51,445,59,359,223),(13,340,394,50,446,58,360,222),(14,339,395,49,447,57,353,221),(15,338,396,56,448,64,354,220),(16,337,397,55,441,63,355,219),(17,415,25,389,189,421,193,363),(18,414,26,388,190,420,194,362),(19,413,27,387,191,419,195,361),(20,412,28,386,192,418,196,368),(21,411,29,385,185,417,197,367),(22,410,30,392,186,424,198,366),(23,409,31,391,187,423,199,365),(24,416,32,390,188,422,200,364),(33,431,41,405,205,437,209,379),(34,430,42,404,206,436,210,378),(35,429,43,403,207,435,211,377),(36,428,44,402,208,434,212,384),(37,427,45,401,201,433,213,383),(38,426,46,408,202,440,214,382),(39,425,47,407,203,439,215,381),(40,432,48,406,204,438,216,380),(81,309,89,265,139,317,151,277),(82,308,90,272,140,316,152,276),(83,307,91,271,141,315,145,275),(84,306,92,270,142,314,146,274),(85,305,93,269,143,313,147,273),(86,312,94,268,144,320,148,280),(87,311,95,267,137,319,149,279),(88,310,96,266,138,318,150,278),(97,325,105,281,155,333,167,348),(98,324,106,288,156,332,168,347),(99,323,107,287,157,331,161,346),(100,322,108,286,158,330,162,345),(101,321,109,285,159,329,163,352),(102,328,110,284,160,336,164,351),(103,327,111,283,153,335,165,350),(104,326,112,282,154,334,166,349),(113,178,248,236,125,228,372,174),(114,177,241,235,126,227,373,173),(115,184,242,234,127,226,374,172),(116,183,243,233,128,225,375,171),(117,182,244,240,121,232,376,170),(118,181,245,239,122,231,369,169),(119,180,246,238,123,230,370,176),(120,179,247,237,124,229,371,175)]])
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 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C8 | C14 | C14 | C28 | C56 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 | C7×D4 | C7×Q8 | C7×M4(2) | C7×D8 | C7×Q16 | C7×C8.C4 |
| kernel | C7×C8⋊1C8 | C4×C56 | C7×C4⋊C8 | C2×C56 | C8⋊1C8 | C56 | C4×C8 | C4⋊C8 | C2×C8 | C8 | C2×C28 | C2×C28 | C28 | C28 | C28 | C14 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 6 | 8 | 6 | 12 | 24 | 48 | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C7×C8⋊1C8 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 106 | 0 |
| 0 | 0 | 106 |
| 1 | 0 | 0 |
| 0 | 82 | 31 |
| 0 | 82 | 82 |
| 44 | 0 | 0 |
| 0 | 104 | 101 |
| 0 | 101 | 9 |
G:=sub<GL(3,GF(113))| [1,0,0,0,106,0,0,0,106],[1,0,0,0,82,82,0,31,82],[44,0,0,0,104,101,0,101,9] >;
C7×C8⋊1C8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_1C_8
% in TeX
G:=Group("C7xC8:1C8"); // GroupNames label
G:=SmallGroup(448,139);
// by ID
G=gap.SmallGroup(448,139);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,988,3923,136,172]);
// 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^-1>;
// generators/relations
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