direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8.5Q8, C56.21Q8, C8.4(C7×Q8), (C4×C8).9C14, C4.6(Q8×C14), (C4×C56).27C2, C28.95(C2×Q8), C2.D8.6C14, C4.Q8.7C14, (C2×C28).368D4, C14.41(C4⋊Q8), C42.84(C2×C14), C42.C2.4C14, C14.132(C4○D8), (C2×C56).441C22, (C4×C28).368C22, (C2×C28).955C23, C22.120(D4×C14), C2.7(C7×C4⋊Q8), C2.19(C7×C4○D8), (C2×C4).58(C7×D4), C4⋊C4.24(C2×C14), (C2×C8).82(C2×C14), (C7×C4.Q8).14C2, (C7×C2.D8).15C2, (C2×C14).676(C2×D4), (C7×C4⋊C4).244C22, (C7×C42.C2).11C2, (C2×C4).130(C22×C14), SmallGroup(448,907)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.5Q8
G = < a,b,c,d | a7=b8=c4=1, d2=b4c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >
Subgroups: 130 in 86 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C28, C28, C2×C14, C4×C8, C4.Q8, C2.D8, C42.C2, C56, C2×C28, C2×C28, C2×C28, C8.5Q8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C4×C56, C7×C4.Q8, C7×C2.D8, C7×C42.C2, C7×C8.5Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C2×C14, C4⋊Q8, C4○D8, C7×D4, C7×Q8, C22×C14, C8.5Q8, D4×C14, Q8×C14, C7×C4⋊Q8, C7×C4○D8, C7×C8.5Q8
►Regular action on 448 points
Generators in S448
(1 71 12 87 31 79 23)(2 72 13 88 32 80 24)(3 65 14 81 25 73 17)(4 66 15 82 26 74 18)(5 67 16 83 27 75 19)(6 68 9 84 28 76 20)(7 69 10 85 29 77 21)(8 70 11 86 30 78 22)(33 113 105 49 97 41 89)(34 114 106 50 98 42 90)(35 115 107 51 99 43 91)(36 116 108 52 100 44 92)(37 117 109 53 101 45 93)(38 118 110 54 102 46 94)(39 119 111 55 103 47 95)(40 120 112 56 104 48 96)(57 445 387 437 379 429 371)(58 446 388 438 380 430 372)(59 447 389 439 381 431 373)(60 448 390 440 382 432 374)(61 441 391 433 383 425 375)(62 442 392 434 384 426 376)(63 443 385 435 377 427 369)(64 444 386 436 378 428 370)(121 174 194 138 186 130 178)(122 175 195 139 187 131 179)(123 176 196 140 188 132 180)(124 169 197 141 189 133 181)(125 170 198 142 190 134 182)(126 171 199 143 191 135 183)(127 172 200 144 192 136 184)(128 173 193 137 185 129 177)(145 231 217 161 209 153 201)(146 232 218 162 210 154 202)(147 225 219 163 211 155 203)(148 226 220 164 212 156 204)(149 227 221 165 213 157 205)(150 228 222 166 214 158 206)(151 229 223 167 215 159 207)(152 230 224 168 216 160 208)(233 309 251 301 243 290 287)(234 310 252 302 244 291 288)(235 311 253 303 245 292 281)(236 312 254 304 246 293 282)(237 305 255 297 247 294 283)(238 306 256 298 248 295 284)(239 307 249 299 241 296 285)(240 308 250 300 242 289 286)(257 337 331 273 323 265 315)(258 338 332 274 324 266 316)(259 339 333 275 325 267 317)(260 340 334 276 326 268 318)(261 341 335 277 327 269 319)(262 342 336 278 328 270 320)(263 343 329 279 321 271 313)(264 344 330 280 322 272 314)(345 421 363 413 355 402 399)(346 422 364 414 356 403 400)(347 423 365 415 357 404 393)(348 424 366 416 358 405 394)(349 417 367 409 359 406 395)(350 418 368 410 360 407 396)(351 419 361 411 353 408 397)(352 420 362 412 354 401 398)
(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)
(1 147 35 183)(2 148 36 184)(3 149 37 177)(4 150 38 178)(5 151 39 179)(6 152 40 180)(7 145 33 181)(8 146 34 182)(9 224 112 176)(10 217 105 169)(11 218 106 170)(12 219 107 171)(13 220 108 172)(14 221 109 173)(15 222 110 174)(16 223 111 175)(17 205 93 129)(18 206 94 130)(19 207 95 131)(20 208 96 132)(21 201 89 133)(22 202 90 134)(23 203 91 135)(24 204 92 136)(25 213 101 137)(26 214 102 138)(27 215 103 139)(28 216 104 140)(29 209 97 141)(30 210 98 142)(31 211 99 143)(32 212 100 144)(41 189 77 153)(42 190 78 154)(43 191 79 155)(44 192 80 156)(45 185 73 157)(46 186 74 158)(47 187 75 159)(48 188 76 160)(49 197 85 161)(50 198 86 162)(51 199 87 163)(52 200 88 164)(53 193 81 165)(54 194 82 166)(55 195 83 167)(56 196 84 168)(57 292 408 343)(58 293 401 344)(59 294 402 337)(60 295 403 338)(61 296 404 339)(62 289 405 340)(63 290 406 341)(64 291 407 342)(65 227 117 128)(66 228 118 121)(67 229 119 122)(68 230 120 123)(69 231 113 124)(70 232 114 125)(71 225 115 126)(72 226 116 127)(233 349 277 385)(234 350 278 386)(235 351 279 387)(236 352 280 388)(237 345 273 389)(238 346 274 390)(239 347 275 391)(240 348 276 392)(241 357 259 375)(242 358 260 376)(243 359 261 369)(244 360 262 370)(245 353 263 371)(246 354 264 372)(247 355 257 373)(248 356 258 374)(249 365 267 383)(250 366 268 384)(251 367 269 377)(252 368 270 378)(253 361 271 379)(254 362 272 380)(255 363 265 381)(256 364 266 382)(281 397 329 445)(282 398 330 446)(283 399 331 447)(284 400 332 448)(285 393 333 441)(286 394 334 442)(287 395 335 443)(288 396 336 444)(297 413 315 431)(298 414 316 432)(299 415 317 425)(300 416 318 426)(301 409 319 427)(302 410 320 428)(303 411 313 429)(304 412 314 430)(305 421 323 439)(306 422 324 440)(307 423 325 433)(308 424 326 434)(309 417 327 435)(310 418 328 436)(311 419 321 437)(312 420 322 438)
(1 259 39 245)(2 262 40 248)(3 257 33 243)(4 260 34 246)(5 263 35 241)(6 258 36 244)(7 261 37 247)(8 264 38 242)(9 332 108 288)(10 335 109 283)(11 330 110 286)(12 333 111 281)(13 336 112 284)(14 331 105 287)(15 334 106 282)(16 329 107 285)(17 315 89 301)(18 318 90 304)(19 313 91 299)(20 316 92 302)(21 319 93 297)(22 314 94 300)(23 317 95 303)(24 320 96 298)(25 323 97 309)(26 326 98 312)(27 321 99 307)(28 324 100 310)(29 327 101 305)(30 322 102 308)(31 325 103 311)(32 328 104 306)(41 251 73 265)(42 254 74 268)(43 249 75 271)(44 252 76 266)(45 255 77 269)(46 250 78 272)(47 253 79 267)(48 256 80 270)(49 233 81 273)(50 236 82 276)(51 239 83 279)(52 234 84 274)(53 237 85 277)(54 240 86 280)(55 235 87 275)(56 238 88 278)(57 229 404 126)(58 232 405 121)(59 227 406 124)(60 230 407 127)(61 225 408 122)(62 228 401 125)(63 231 402 128)(64 226 403 123)(65 337 113 290)(66 340 114 293)(67 343 115 296)(68 338 116 291)(69 341 117 294)(70 344 118 289)(71 339 119 292)(72 342 120 295)(129 427 201 413)(130 430 202 416)(131 425 203 411)(132 428 204 414)(133 431 205 409)(134 426 206 412)(135 429 207 415)(136 432 208 410)(137 435 209 421)(138 438 210 424)(139 433 211 419)(140 436 212 422)(141 439 213 417)(142 434 214 420)(143 437 215 423)(144 440 216 418)(145 355 177 369)(146 358 178 372)(147 353 179 375)(148 356 180 370)(149 359 181 373)(150 354 182 376)(151 357 183 371)(152 360 184 374)(153 363 185 377)(154 366 186 380)(155 361 187 383)(156 364 188 378)(157 367 189 381)(158 362 190 384)(159 365 191 379)(160 368 192 382)(161 345 193 385)(162 348 194 388)(163 351 195 391)(164 346 196 386)(165 349 197 389)(166 352 198 392)(167 347 199 387)(168 350 200 390)(169 447 221 395)(170 442 222 398)(171 445 223 393)(172 448 224 396)(173 443 217 399)(174 446 218 394)(175 441 219 397)(176 444 220 400)
G:=sub<Sym(448)| (1,71,12,87,31,79,23)(2,72,13,88,32,80,24)(3,65,14,81,25,73,17)(4,66,15,82,26,74,18)(5,67,16,83,27,75,19)(6,68,9,84,28,76,20)(7,69,10,85,29,77,21)(8,70,11,86,30,78,22)(33,113,105,49,97,41,89)(34,114,106,50,98,42,90)(35,115,107,51,99,43,91)(36,116,108,52,100,44,92)(37,117,109,53,101,45,93)(38,118,110,54,102,46,94)(39,119,111,55,103,47,95)(40,120,112,56,104,48,96)(57,445,387,437,379,429,371)(58,446,388,438,380,430,372)(59,447,389,439,381,431,373)(60,448,390,440,382,432,374)(61,441,391,433,383,425,375)(62,442,392,434,384,426,376)(63,443,385,435,377,427,369)(64,444,386,436,378,428,370)(121,174,194,138,186,130,178)(122,175,195,139,187,131,179)(123,176,196,140,188,132,180)(124,169,197,141,189,133,181)(125,170,198,142,190,134,182)(126,171,199,143,191,135,183)(127,172,200,144,192,136,184)(128,173,193,137,185,129,177)(145,231,217,161,209,153,201)(146,232,218,162,210,154,202)(147,225,219,163,211,155,203)(148,226,220,164,212,156,204)(149,227,221,165,213,157,205)(150,228,222,166,214,158,206)(151,229,223,167,215,159,207)(152,230,224,168,216,160,208)(233,309,251,301,243,290,287)(234,310,252,302,244,291,288)(235,311,253,303,245,292,281)(236,312,254,304,246,293,282)(237,305,255,297,247,294,283)(238,306,256,298,248,295,284)(239,307,249,299,241,296,285)(240,308,250,300,242,289,286)(257,337,331,273,323,265,315)(258,338,332,274,324,266,316)(259,339,333,275,325,267,317)(260,340,334,276,326,268,318)(261,341,335,277,327,269,319)(262,342,336,278,328,270,320)(263,343,329,279,321,271,313)(264,344,330,280,322,272,314)(345,421,363,413,355,402,399)(346,422,364,414,356,403,400)(347,423,365,415,357,404,393)(348,424,366,416,358,405,394)(349,417,367,409,359,406,395)(350,418,368,410,360,407,396)(351,419,361,411,353,408,397)(352,420,362,412,354,401,398), (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), (1,147,35,183)(2,148,36,184)(3,149,37,177)(4,150,38,178)(5,151,39,179)(6,152,40,180)(7,145,33,181)(8,146,34,182)(9,224,112,176)(10,217,105,169)(11,218,106,170)(12,219,107,171)(13,220,108,172)(14,221,109,173)(15,222,110,174)(16,223,111,175)(17,205,93,129)(18,206,94,130)(19,207,95,131)(20,208,96,132)(21,201,89,133)(22,202,90,134)(23,203,91,135)(24,204,92,136)(25,213,101,137)(26,214,102,138)(27,215,103,139)(28,216,104,140)(29,209,97,141)(30,210,98,142)(31,211,99,143)(32,212,100,144)(41,189,77,153)(42,190,78,154)(43,191,79,155)(44,192,80,156)(45,185,73,157)(46,186,74,158)(47,187,75,159)(48,188,76,160)(49,197,85,161)(50,198,86,162)(51,199,87,163)(52,200,88,164)(53,193,81,165)(54,194,82,166)(55,195,83,167)(56,196,84,168)(57,292,408,343)(58,293,401,344)(59,294,402,337)(60,295,403,338)(61,296,404,339)(62,289,405,340)(63,290,406,341)(64,291,407,342)(65,227,117,128)(66,228,118,121)(67,229,119,122)(68,230,120,123)(69,231,113,124)(70,232,114,125)(71,225,115,126)(72,226,116,127)(233,349,277,385)(234,350,278,386)(235,351,279,387)(236,352,280,388)(237,345,273,389)(238,346,274,390)(239,347,275,391)(240,348,276,392)(241,357,259,375)(242,358,260,376)(243,359,261,369)(244,360,262,370)(245,353,263,371)(246,354,264,372)(247,355,257,373)(248,356,258,374)(249,365,267,383)(250,366,268,384)(251,367,269,377)(252,368,270,378)(253,361,271,379)(254,362,272,380)(255,363,265,381)(256,364,266,382)(281,397,329,445)(282,398,330,446)(283,399,331,447)(284,400,332,448)(285,393,333,441)(286,394,334,442)(287,395,335,443)(288,396,336,444)(297,413,315,431)(298,414,316,432)(299,415,317,425)(300,416,318,426)(301,409,319,427)(302,410,320,428)(303,411,313,429)(304,412,314,430)(305,421,323,439)(306,422,324,440)(307,423,325,433)(308,424,326,434)(309,417,327,435)(310,418,328,436)(311,419,321,437)(312,420,322,438), (1,259,39,245)(2,262,40,248)(3,257,33,243)(4,260,34,246)(5,263,35,241)(6,258,36,244)(7,261,37,247)(8,264,38,242)(9,332,108,288)(10,335,109,283)(11,330,110,286)(12,333,111,281)(13,336,112,284)(14,331,105,287)(15,334,106,282)(16,329,107,285)(17,315,89,301)(18,318,90,304)(19,313,91,299)(20,316,92,302)(21,319,93,297)(22,314,94,300)(23,317,95,303)(24,320,96,298)(25,323,97,309)(26,326,98,312)(27,321,99,307)(28,324,100,310)(29,327,101,305)(30,322,102,308)(31,325,103,311)(32,328,104,306)(41,251,73,265)(42,254,74,268)(43,249,75,271)(44,252,76,266)(45,255,77,269)(46,250,78,272)(47,253,79,267)(48,256,80,270)(49,233,81,273)(50,236,82,276)(51,239,83,279)(52,234,84,274)(53,237,85,277)(54,240,86,280)(55,235,87,275)(56,238,88,278)(57,229,404,126)(58,232,405,121)(59,227,406,124)(60,230,407,127)(61,225,408,122)(62,228,401,125)(63,231,402,128)(64,226,403,123)(65,337,113,290)(66,340,114,293)(67,343,115,296)(68,338,116,291)(69,341,117,294)(70,344,118,289)(71,339,119,292)(72,342,120,295)(129,427,201,413)(130,430,202,416)(131,425,203,411)(132,428,204,414)(133,431,205,409)(134,426,206,412)(135,429,207,415)(136,432,208,410)(137,435,209,421)(138,438,210,424)(139,433,211,419)(140,436,212,422)(141,439,213,417)(142,434,214,420)(143,437,215,423)(144,440,216,418)(145,355,177,369)(146,358,178,372)(147,353,179,375)(148,356,180,370)(149,359,181,373)(150,354,182,376)(151,357,183,371)(152,360,184,374)(153,363,185,377)(154,366,186,380)(155,361,187,383)(156,364,188,378)(157,367,189,381)(158,362,190,384)(159,365,191,379)(160,368,192,382)(161,345,193,385)(162,348,194,388)(163,351,195,391)(164,346,196,386)(165,349,197,389)(166,352,198,392)(167,347,199,387)(168,350,200,390)(169,447,221,395)(170,442,222,398)(171,445,223,393)(172,448,224,396)(173,443,217,399)(174,446,218,394)(175,441,219,397)(176,444,220,400)>;
G:=Group( (1,71,12,87,31,79,23)(2,72,13,88,32,80,24)(3,65,14,81,25,73,17)(4,66,15,82,26,74,18)(5,67,16,83,27,75,19)(6,68,9,84,28,76,20)(7,69,10,85,29,77,21)(8,70,11,86,30,78,22)(33,113,105,49,97,41,89)(34,114,106,50,98,42,90)(35,115,107,51,99,43,91)(36,116,108,52,100,44,92)(37,117,109,53,101,45,93)(38,118,110,54,102,46,94)(39,119,111,55,103,47,95)(40,120,112,56,104,48,96)(57,445,387,437,379,429,371)(58,446,388,438,380,430,372)(59,447,389,439,381,431,373)(60,448,390,440,382,432,374)(61,441,391,433,383,425,375)(62,442,392,434,384,426,376)(63,443,385,435,377,427,369)(64,444,386,436,378,428,370)(121,174,194,138,186,130,178)(122,175,195,139,187,131,179)(123,176,196,140,188,132,180)(124,169,197,141,189,133,181)(125,170,198,142,190,134,182)(126,171,199,143,191,135,183)(127,172,200,144,192,136,184)(128,173,193,137,185,129,177)(145,231,217,161,209,153,201)(146,232,218,162,210,154,202)(147,225,219,163,211,155,203)(148,226,220,164,212,156,204)(149,227,221,165,213,157,205)(150,228,222,166,214,158,206)(151,229,223,167,215,159,207)(152,230,224,168,216,160,208)(233,309,251,301,243,290,287)(234,310,252,302,244,291,288)(235,311,253,303,245,292,281)(236,312,254,304,246,293,282)(237,305,255,297,247,294,283)(238,306,256,298,248,295,284)(239,307,249,299,241,296,285)(240,308,250,300,242,289,286)(257,337,331,273,323,265,315)(258,338,332,274,324,266,316)(259,339,333,275,325,267,317)(260,340,334,276,326,268,318)(261,341,335,277,327,269,319)(262,342,336,278,328,270,320)(263,343,329,279,321,271,313)(264,344,330,280,322,272,314)(345,421,363,413,355,402,399)(346,422,364,414,356,403,400)(347,423,365,415,357,404,393)(348,424,366,416,358,405,394)(349,417,367,409,359,406,395)(350,418,368,410,360,407,396)(351,419,361,411,353,408,397)(352,420,362,412,354,401,398), (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), (1,147,35,183)(2,148,36,184)(3,149,37,177)(4,150,38,178)(5,151,39,179)(6,152,40,180)(7,145,33,181)(8,146,34,182)(9,224,112,176)(10,217,105,169)(11,218,106,170)(12,219,107,171)(13,220,108,172)(14,221,109,173)(15,222,110,174)(16,223,111,175)(17,205,93,129)(18,206,94,130)(19,207,95,131)(20,208,96,132)(21,201,89,133)(22,202,90,134)(23,203,91,135)(24,204,92,136)(25,213,101,137)(26,214,102,138)(27,215,103,139)(28,216,104,140)(29,209,97,141)(30,210,98,142)(31,211,99,143)(32,212,100,144)(41,189,77,153)(42,190,78,154)(43,191,79,155)(44,192,80,156)(45,185,73,157)(46,186,74,158)(47,187,75,159)(48,188,76,160)(49,197,85,161)(50,198,86,162)(51,199,87,163)(52,200,88,164)(53,193,81,165)(54,194,82,166)(55,195,83,167)(56,196,84,168)(57,292,408,343)(58,293,401,344)(59,294,402,337)(60,295,403,338)(61,296,404,339)(62,289,405,340)(63,290,406,341)(64,291,407,342)(65,227,117,128)(66,228,118,121)(67,229,119,122)(68,230,120,123)(69,231,113,124)(70,232,114,125)(71,225,115,126)(72,226,116,127)(233,349,277,385)(234,350,278,386)(235,351,279,387)(236,352,280,388)(237,345,273,389)(238,346,274,390)(239,347,275,391)(240,348,276,392)(241,357,259,375)(242,358,260,376)(243,359,261,369)(244,360,262,370)(245,353,263,371)(246,354,264,372)(247,355,257,373)(248,356,258,374)(249,365,267,383)(250,366,268,384)(251,367,269,377)(252,368,270,378)(253,361,271,379)(254,362,272,380)(255,363,265,381)(256,364,266,382)(281,397,329,445)(282,398,330,446)(283,399,331,447)(284,400,332,448)(285,393,333,441)(286,394,334,442)(287,395,335,443)(288,396,336,444)(297,413,315,431)(298,414,316,432)(299,415,317,425)(300,416,318,426)(301,409,319,427)(302,410,320,428)(303,411,313,429)(304,412,314,430)(305,421,323,439)(306,422,324,440)(307,423,325,433)(308,424,326,434)(309,417,327,435)(310,418,328,436)(311,419,321,437)(312,420,322,438), (1,259,39,245)(2,262,40,248)(3,257,33,243)(4,260,34,246)(5,263,35,241)(6,258,36,244)(7,261,37,247)(8,264,38,242)(9,332,108,288)(10,335,109,283)(11,330,110,286)(12,333,111,281)(13,336,112,284)(14,331,105,287)(15,334,106,282)(16,329,107,285)(17,315,89,301)(18,318,90,304)(19,313,91,299)(20,316,92,302)(21,319,93,297)(22,314,94,300)(23,317,95,303)(24,320,96,298)(25,323,97,309)(26,326,98,312)(27,321,99,307)(28,324,100,310)(29,327,101,305)(30,322,102,308)(31,325,103,311)(32,328,104,306)(41,251,73,265)(42,254,74,268)(43,249,75,271)(44,252,76,266)(45,255,77,269)(46,250,78,272)(47,253,79,267)(48,256,80,270)(49,233,81,273)(50,236,82,276)(51,239,83,279)(52,234,84,274)(53,237,85,277)(54,240,86,280)(55,235,87,275)(56,238,88,278)(57,229,404,126)(58,232,405,121)(59,227,406,124)(60,230,407,127)(61,225,408,122)(62,228,401,125)(63,231,402,128)(64,226,403,123)(65,337,113,290)(66,340,114,293)(67,343,115,296)(68,338,116,291)(69,341,117,294)(70,344,118,289)(71,339,119,292)(72,342,120,295)(129,427,201,413)(130,430,202,416)(131,425,203,411)(132,428,204,414)(133,431,205,409)(134,426,206,412)(135,429,207,415)(136,432,208,410)(137,435,209,421)(138,438,210,424)(139,433,211,419)(140,436,212,422)(141,439,213,417)(142,434,214,420)(143,437,215,423)(144,440,216,418)(145,355,177,369)(146,358,178,372)(147,353,179,375)(148,356,180,370)(149,359,181,373)(150,354,182,376)(151,357,183,371)(152,360,184,374)(153,363,185,377)(154,366,186,380)(155,361,187,383)(156,364,188,378)(157,367,189,381)(158,362,190,384)(159,365,191,379)(160,368,192,382)(161,345,193,385)(162,348,194,388)(163,351,195,391)(164,346,196,386)(165,349,197,389)(166,352,198,392)(167,347,199,387)(168,350,200,390)(169,447,221,395)(170,442,222,398)(171,445,223,393)(172,448,224,396)(173,443,217,399)(174,446,218,394)(175,441,219,397)(176,444,220,400) );
G=PermutationGroup([[(1,71,12,87,31,79,23),(2,72,13,88,32,80,24),(3,65,14,81,25,73,17),(4,66,15,82,26,74,18),(5,67,16,83,27,75,19),(6,68,9,84,28,76,20),(7,69,10,85,29,77,21),(8,70,11,86,30,78,22),(33,113,105,49,97,41,89),(34,114,106,50,98,42,90),(35,115,107,51,99,43,91),(36,116,108,52,100,44,92),(37,117,109,53,101,45,93),(38,118,110,54,102,46,94),(39,119,111,55,103,47,95),(40,120,112,56,104,48,96),(57,445,387,437,379,429,371),(58,446,388,438,380,430,372),(59,447,389,439,381,431,373),(60,448,390,440,382,432,374),(61,441,391,433,383,425,375),(62,442,392,434,384,426,376),(63,443,385,435,377,427,369),(64,444,386,436,378,428,370),(121,174,194,138,186,130,178),(122,175,195,139,187,131,179),(123,176,196,140,188,132,180),(124,169,197,141,189,133,181),(125,170,198,142,190,134,182),(126,171,199,143,191,135,183),(127,172,200,144,192,136,184),(128,173,193,137,185,129,177),(145,231,217,161,209,153,201),(146,232,218,162,210,154,202),(147,225,219,163,211,155,203),(148,226,220,164,212,156,204),(149,227,221,165,213,157,205),(150,228,222,166,214,158,206),(151,229,223,167,215,159,207),(152,230,224,168,216,160,208),(233,309,251,301,243,290,287),(234,310,252,302,244,291,288),(235,311,253,303,245,292,281),(236,312,254,304,246,293,282),(237,305,255,297,247,294,283),(238,306,256,298,248,295,284),(239,307,249,299,241,296,285),(240,308,250,300,242,289,286),(257,337,331,273,323,265,315),(258,338,332,274,324,266,316),(259,339,333,275,325,267,317),(260,340,334,276,326,268,318),(261,341,335,277,327,269,319),(262,342,336,278,328,270,320),(263,343,329,279,321,271,313),(264,344,330,280,322,272,314),(345,421,363,413,355,402,399),(346,422,364,414,356,403,400),(347,423,365,415,357,404,393),(348,424,366,416,358,405,394),(349,417,367,409,359,406,395),(350,418,368,410,360,407,396),(351,419,361,411,353,408,397),(352,420,362,412,354,401,398)], [(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)], [(1,147,35,183),(2,148,36,184),(3,149,37,177),(4,150,38,178),(5,151,39,179),(6,152,40,180),(7,145,33,181),(8,146,34,182),(9,224,112,176),(10,217,105,169),(11,218,106,170),(12,219,107,171),(13,220,108,172),(14,221,109,173),(15,222,110,174),(16,223,111,175),(17,205,93,129),(18,206,94,130),(19,207,95,131),(20,208,96,132),(21,201,89,133),(22,202,90,134),(23,203,91,135),(24,204,92,136),(25,213,101,137),(26,214,102,138),(27,215,103,139),(28,216,104,140),(29,209,97,141),(30,210,98,142),(31,211,99,143),(32,212,100,144),(41,189,77,153),(42,190,78,154),(43,191,79,155),(44,192,80,156),(45,185,73,157),(46,186,74,158),(47,187,75,159),(48,188,76,160),(49,197,85,161),(50,198,86,162),(51,199,87,163),(52,200,88,164),(53,193,81,165),(54,194,82,166),(55,195,83,167),(56,196,84,168),(57,292,408,343),(58,293,401,344),(59,294,402,337),(60,295,403,338),(61,296,404,339),(62,289,405,340),(63,290,406,341),(64,291,407,342),(65,227,117,128),(66,228,118,121),(67,229,119,122),(68,230,120,123),(69,231,113,124),(70,232,114,125),(71,225,115,126),(72,226,116,127),(233,349,277,385),(234,350,278,386),(235,351,279,387),(236,352,280,388),(237,345,273,389),(238,346,274,390),(239,347,275,391),(240,348,276,392),(241,357,259,375),(242,358,260,376),(243,359,261,369),(244,360,262,370),(245,353,263,371),(246,354,264,372),(247,355,257,373),(248,356,258,374),(249,365,267,383),(250,366,268,384),(251,367,269,377),(252,368,270,378),(253,361,271,379),(254,362,272,380),(255,363,265,381),(256,364,266,382),(281,397,329,445),(282,398,330,446),(283,399,331,447),(284,400,332,448),(285,393,333,441),(286,394,334,442),(287,395,335,443),(288,396,336,444),(297,413,315,431),(298,414,316,432),(299,415,317,425),(300,416,318,426),(301,409,319,427),(302,410,320,428),(303,411,313,429),(304,412,314,430),(305,421,323,439),(306,422,324,440),(307,423,325,433),(308,424,326,434),(309,417,327,435),(310,418,328,436),(311,419,321,437),(312,420,322,438)], [(1,259,39,245),(2,262,40,248),(3,257,33,243),(4,260,34,246),(5,263,35,241),(6,258,36,244),(7,261,37,247),(8,264,38,242),(9,332,108,288),(10,335,109,283),(11,330,110,286),(12,333,111,281),(13,336,112,284),(14,331,105,287),(15,334,106,282),(16,329,107,285),(17,315,89,301),(18,318,90,304),(19,313,91,299),(20,316,92,302),(21,319,93,297),(22,314,94,300),(23,317,95,303),(24,320,96,298),(25,323,97,309),(26,326,98,312),(27,321,99,307),(28,324,100,310),(29,327,101,305),(30,322,102,308),(31,325,103,311),(32,328,104,306),(41,251,73,265),(42,254,74,268),(43,249,75,271),(44,252,76,266),(45,255,77,269),(46,250,78,272),(47,253,79,267),(48,256,80,270),(49,233,81,273),(50,236,82,276),(51,239,83,279),(52,234,84,274),(53,237,85,277),(54,240,86,280),(55,235,87,275),(56,238,88,278),(57,229,404,126),(58,232,405,121),(59,227,406,124),(60,230,407,127),(61,225,408,122),(62,228,401,125),(63,231,402,128),(64,226,403,123),(65,337,113,290),(66,340,114,293),(67,343,115,296),(68,338,116,291),(69,341,117,294),(70,344,118,289),(71,339,119,292),(72,342,120,295),(129,427,201,413),(130,430,202,416),(131,425,203,411),(132,428,204,414),(133,431,205,409),(134,426,206,412),(135,429,207,415),(136,432,208,410),(137,435,209,421),(138,438,210,424),(139,433,211,419),(140,436,212,422),(141,439,213,417),(142,434,214,420),(143,437,215,423),(144,440,216,418),(145,355,177,369),(146,358,178,372),(147,353,179,375),(148,356,180,370),(149,359,181,373),(150,354,182,376),(151,357,183,371),(152,360,184,374),(153,363,185,377),(154,366,186,380),(155,361,187,383),(156,364,188,378),(157,367,189,381),(158,362,190,384),(159,365,191,379),(160,368,192,382),(161,345,193,385),(162,348,194,388),(163,351,195,391),(164,346,196,386),(165,349,197,389),(166,352,198,392),(167,347,199,387),(168,350,200,390),(169,447,221,395),(170,442,222,398),(171,445,223,393),(172,448,224,396),(173,443,217,399),(174,446,218,394),(175,441,219,397),(176,444,220,400)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28AJ | 28AK | ··· | 28BH | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | Q8 | D4 | C4○D8 | C7×Q8 | C7×D4 | C7×C4○D8 |
| kernel | C7×C8.5Q8 | C4×C56 | C7×C4.Q8 | C7×C2.D8 | C7×C42.C2 | C8.5Q8 | C4×C8 | C4.Q8 | C2.D8 | C42.C2 | C56 | C2×C28 | C14 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 4 | 2 | 8 | 24 | 12 | 48 |
Matrix representation of C7×C8.5Q8 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 100 | 13 |
| 0 | 0 | 100 | 100 |
| 1 | 2 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 60 | 10 | 0 | 0 |
| 58 | 53 | 0 | 0 |
| 0 | 0 | 86 | 96 |
| 0 | 0 | 96 | 27 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,16,0,0,0,0,16],[112,0,0,0,0,112,0,0,0,0,100,100,0,0,13,100],[1,112,0,0,2,112,0,0,0,0,15,0,0,0,0,15],[60,58,0,0,10,53,0,0,0,0,86,96,0,0,96,27] >;
C7×C8.5Q8 in GAP, Magma, Sage, TeX
C_7\times C_8._5Q_8
% in TeX
G:=Group("C7xC8.5Q8"); // GroupNames label
G:=SmallGroup(448,907);
// by ID
G=gap.SmallGroup(448,907);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,1968,2438,604,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=1,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations