direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊2Q8, C56⋊12Q8, C28.43D8, C28.21Q16, C8⋊2(C7×Q8), C4.5(C7×D8), (C4×C8).8C14, C4.4(C7×Q16), C4.7(Q8×C14), (C4×C56).26C2, C2.11(C14×D8), C14.83(C2×D8), C4⋊Q8.11C14, C28.96(C2×Q8), C2.D8.7C14, (C2×C28).425D4, C14.42(C4⋊Q8), C14.58(C2×Q16), C2.11(C14×Q16), C42.85(C2×C14), (C4×C28).369C22, (C2×C28).956C23, (C2×C56).425C22, C22.121(D4×C14), C2.8(C7×C4⋊Q8), (C2×C4).81(C7×D4), (C7×C4⋊Q8).26C2, C4⋊C4.25(C2×C14), (C2×C8).83(C2×C14), (C7×C2.D8).16C2, (C2×C14).677(C2×D4), (C7×C4⋊C4).245C22, (C2×C4).131(C22×C14), SmallGroup(448,908)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊2Q8
G = < a,b,c,d | a7=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 162 in 98 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4×C8, C2.D8, C4⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C8⋊2Q8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, C4×C56, C7×C2.D8, C7×C4⋊Q8, C7×C8⋊2Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, D8, Q16, C2×D4, C2×Q8, C2×C14, C4⋊Q8, C2×D8, C2×Q16, C7×D4, C7×Q8, C22×C14, C8⋊2Q8, C7×D8, C7×Q16, D4×C14, Q8×C14, C7×C4⋊Q8, C14×D8, C14×Q16, C7×C8⋊2Q8
►Regular action on 448 points
Generators in S448
(1 17 9 70 85 29 77)(2 18 10 71 86 30 78)(3 19 11 72 87 31 79)(4 20 12 65 88 32 80)(5 21 13 66 81 25 73)(6 22 14 67 82 26 74)(7 23 15 68 83 27 75)(8 24 16 69 84 28 76)(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)(57 446 388 438 380 430 372)(58 447 389 439 381 431 373)(59 448 390 440 382 432 374)(60 441 391 433 383 425 375)(61 442 392 434 384 426 376)(62 443 385 435 377 427 369)(63 444 386 436 378 428 370)(64 445 387 437 379 429 371)(121 172 197 141 189 133 180)(122 173 198 142 190 134 181)(123 174 199 143 191 135 182)(124 175 200 144 192 136 183)(125 176 193 137 185 129 184)(126 169 194 138 186 130 177)(127 170 195 139 187 131 178)(128 171 196 140 188 132 179)(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)(233 284 307 249 299 241 291)(234 285 308 250 300 242 292)(235 286 309 251 301 243 293)(236 287 310 252 302 244 294)(237 288 311 253 303 245 295)(238 281 312 254 304 246 296)(239 282 305 255 297 247 289)(240 283 306 256 298 248 290)(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)(345 396 419 361 411 353 403)(346 397 420 362 412 354 404)(347 398 421 363 413 355 405)(348 399 422 364 414 356 406)(349 400 423 365 415 357 407)(350 393 424 366 416 358 408)(351 394 417 367 409 359 401)(352 395 418 368 410 360 402)
(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 35 133)(2 148 36 134)(3 149 37 135)(4 150 38 136)(5 151 39 129)(6 152 40 130)(7 145 33 131)(8 146 34 132)(9 219 107 121)(10 220 108 122)(11 221 109 123)(12 222 110 124)(13 223 111 125)(14 224 112 126)(15 217 105 127)(16 218 106 128)(17 225 115 180)(18 226 116 181)(19 227 117 182)(20 228 118 183)(21 229 119 184)(22 230 120 177)(23 231 113 178)(24 232 114 179)(25 159 47 137)(26 160 48 138)(27 153 41 139)(28 154 42 140)(29 155 43 141)(30 156 44 142)(31 157 45 143)(32 158 46 144)(49 170 68 161)(50 171 69 162)(51 172 70 163)(52 173 71 164)(53 174 72 165)(54 175 65 166)(55 176 66 167)(56 169 67 168)(57 281 393 340)(58 282 394 341)(59 283 395 342)(60 284 396 343)(61 285 397 344)(62 286 398 337)(63 287 399 338)(64 288 400 339)(73 207 95 185)(74 208 96 186)(75 201 89 187)(76 202 90 188)(77 203 91 189)(78 204 92 190)(79 205 93 191)(80 206 94 192)(81 215 103 193)(82 216 104 194)(83 209 97 195)(84 210 98 196)(85 211 99 197)(86 212 100 198)(87 213 101 199)(88 214 102 200)(233 345 263 375)(234 346 264 376)(235 347 257 369)(236 348 258 370)(237 349 259 371)(238 350 260 372)(239 351 261 373)(240 352 262 374)(241 353 271 383)(242 354 272 384)(243 355 265 377)(244 356 266 378)(245 357 267 379)(246 358 268 380)(247 359 269 381)(248 360 270 382)(249 361 279 391)(250 362 280 392)(251 363 273 385)(252 364 274 386)(253 365 275 387)(254 366 276 388)(255 367 277 389)(256 368 278 390)(289 401 319 431)(290 402 320 432)(291 403 313 425)(292 404 314 426)(293 405 315 427)(294 406 316 428)(295 407 317 429)(296 408 318 430)(297 409 327 439)(298 410 328 440)(299 411 321 433)(300 412 322 434)(301 413 323 435)(302 414 324 436)(303 415 325 437)(304 416 326 438)(305 417 335 447)(306 418 336 448)(307 419 329 441)(308 420 330 442)(309 421 331 443)(310 422 332 444)(311 423 333 445)(312 424 334 446)
(1 259 35 237)(2 258 36 236)(3 257 37 235)(4 264 38 234)(5 263 39 233)(6 262 40 240)(7 261 33 239)(8 260 34 238)(9 333 107 311)(10 332 108 310)(11 331 109 309)(12 330 110 308)(13 329 111 307)(14 336 112 306)(15 335 105 305)(16 334 106 312)(17 339 115 288)(18 338 116 287)(19 337 117 286)(20 344 118 285)(21 343 119 284)(22 342 120 283)(23 341 113 282)(24 340 114 281)(25 271 47 241)(26 270 48 248)(27 269 41 247)(28 268 42 246)(29 267 43 245)(30 266 44 244)(31 265 45 243)(32 272 46 242)(49 255 68 277)(50 254 69 276)(51 253 70 275)(52 252 71 274)(53 251 72 273)(54 250 65 280)(55 249 66 279)(56 256 67 278)(57 232 393 179)(58 231 394 178)(59 230 395 177)(60 229 396 184)(61 228 397 183)(62 227 398 182)(63 226 399 181)(64 225 400 180)(73 313 95 291)(74 320 96 290)(75 319 89 289)(76 318 90 296)(77 317 91 295)(78 316 92 294)(79 315 93 293)(80 314 94 292)(81 321 103 299)(82 328 104 298)(83 327 97 297)(84 326 98 304)(85 325 99 303)(86 324 100 302)(87 323 101 301)(88 322 102 300)(121 445 219 423)(122 444 220 422)(123 443 221 421)(124 442 222 420)(125 441 223 419)(126 448 224 418)(127 447 217 417)(128 446 218 424)(129 375 151 345)(130 374 152 352)(131 373 145 351)(132 372 146 350)(133 371 147 349)(134 370 148 348)(135 369 149 347)(136 376 150 346)(137 383 159 353)(138 382 160 360)(139 381 153 359)(140 380 154 358)(141 379 155 357)(142 378 156 356)(143 377 157 355)(144 384 158 354)(161 367 170 389)(162 366 171 388)(163 365 172 387)(164 364 173 386)(165 363 174 385)(166 362 175 392)(167 361 176 391)(168 368 169 390)(185 425 207 403)(186 432 208 402)(187 431 201 401)(188 430 202 408)(189 429 203 407)(190 428 204 406)(191 427 205 405)(192 426 206 404)(193 433 215 411)(194 440 216 410)(195 439 209 409)(196 438 210 416)(197 437 211 415)(198 436 212 414)(199 435 213 413)(200 434 214 412)
G:=sub<Sym(448)| (1,17,9,70,85,29,77)(2,18,10,71,86,30,78)(3,19,11,72,87,31,79)(4,20,12,65,88,32,80)(5,21,13,66,81,25,73)(6,22,14,67,82,26,74)(7,23,15,68,83,27,75)(8,24,16,69,84,28,76)(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)(57,446,388,438,380,430,372)(58,447,389,439,381,431,373)(59,448,390,440,382,432,374)(60,441,391,433,383,425,375)(61,442,392,434,384,426,376)(62,443,385,435,377,427,369)(63,444,386,436,378,428,370)(64,445,387,437,379,429,371)(121,172,197,141,189,133,180)(122,173,198,142,190,134,181)(123,174,199,143,191,135,182)(124,175,200,144,192,136,183)(125,176,193,137,185,129,184)(126,169,194,138,186,130,177)(127,170,195,139,187,131,178)(128,171,196,140,188,132,179)(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)(233,284,307,249,299,241,291)(234,285,308,250,300,242,292)(235,286,309,251,301,243,293)(236,287,310,252,302,244,294)(237,288,311,253,303,245,295)(238,281,312,254,304,246,296)(239,282,305,255,297,247,289)(240,283,306,256,298,248,290)(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)(345,396,419,361,411,353,403)(346,397,420,362,412,354,404)(347,398,421,363,413,355,405)(348,399,422,364,414,356,406)(349,400,423,365,415,357,407)(350,393,424,366,416,358,408)(351,394,417,367,409,359,401)(352,395,418,368,410,360,402), (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), (1,147,35,133)(2,148,36,134)(3,149,37,135)(4,150,38,136)(5,151,39,129)(6,152,40,130)(7,145,33,131)(8,146,34,132)(9,219,107,121)(10,220,108,122)(11,221,109,123)(12,222,110,124)(13,223,111,125)(14,224,112,126)(15,217,105,127)(16,218,106,128)(17,225,115,180)(18,226,116,181)(19,227,117,182)(20,228,118,183)(21,229,119,184)(22,230,120,177)(23,231,113,178)(24,232,114,179)(25,159,47,137)(26,160,48,138)(27,153,41,139)(28,154,42,140)(29,155,43,141)(30,156,44,142)(31,157,45,143)(32,158,46,144)(49,170,68,161)(50,171,69,162)(51,172,70,163)(52,173,71,164)(53,174,72,165)(54,175,65,166)(55,176,66,167)(56,169,67,168)(57,281,393,340)(58,282,394,341)(59,283,395,342)(60,284,396,343)(61,285,397,344)(62,286,398,337)(63,287,399,338)(64,288,400,339)(73,207,95,185)(74,208,96,186)(75,201,89,187)(76,202,90,188)(77,203,91,189)(78,204,92,190)(79,205,93,191)(80,206,94,192)(81,215,103,193)(82,216,104,194)(83,209,97,195)(84,210,98,196)(85,211,99,197)(86,212,100,198)(87,213,101,199)(88,214,102,200)(233,345,263,375)(234,346,264,376)(235,347,257,369)(236,348,258,370)(237,349,259,371)(238,350,260,372)(239,351,261,373)(240,352,262,374)(241,353,271,383)(242,354,272,384)(243,355,265,377)(244,356,266,378)(245,357,267,379)(246,358,268,380)(247,359,269,381)(248,360,270,382)(249,361,279,391)(250,362,280,392)(251,363,273,385)(252,364,274,386)(253,365,275,387)(254,366,276,388)(255,367,277,389)(256,368,278,390)(289,401,319,431)(290,402,320,432)(291,403,313,425)(292,404,314,426)(293,405,315,427)(294,406,316,428)(295,407,317,429)(296,408,318,430)(297,409,327,439)(298,410,328,440)(299,411,321,433)(300,412,322,434)(301,413,323,435)(302,414,324,436)(303,415,325,437)(304,416,326,438)(305,417,335,447)(306,418,336,448)(307,419,329,441)(308,420,330,442)(309,421,331,443)(310,422,332,444)(311,423,333,445)(312,424,334,446), (1,259,35,237)(2,258,36,236)(3,257,37,235)(4,264,38,234)(5,263,39,233)(6,262,40,240)(7,261,33,239)(8,260,34,238)(9,333,107,311)(10,332,108,310)(11,331,109,309)(12,330,110,308)(13,329,111,307)(14,336,112,306)(15,335,105,305)(16,334,106,312)(17,339,115,288)(18,338,116,287)(19,337,117,286)(20,344,118,285)(21,343,119,284)(22,342,120,283)(23,341,113,282)(24,340,114,281)(25,271,47,241)(26,270,48,248)(27,269,41,247)(28,268,42,246)(29,267,43,245)(30,266,44,244)(31,265,45,243)(32,272,46,242)(49,255,68,277)(50,254,69,276)(51,253,70,275)(52,252,71,274)(53,251,72,273)(54,250,65,280)(55,249,66,279)(56,256,67,278)(57,232,393,179)(58,231,394,178)(59,230,395,177)(60,229,396,184)(61,228,397,183)(62,227,398,182)(63,226,399,181)(64,225,400,180)(73,313,95,291)(74,320,96,290)(75,319,89,289)(76,318,90,296)(77,317,91,295)(78,316,92,294)(79,315,93,293)(80,314,94,292)(81,321,103,299)(82,328,104,298)(83,327,97,297)(84,326,98,304)(85,325,99,303)(86,324,100,302)(87,323,101,301)(88,322,102,300)(121,445,219,423)(122,444,220,422)(123,443,221,421)(124,442,222,420)(125,441,223,419)(126,448,224,418)(127,447,217,417)(128,446,218,424)(129,375,151,345)(130,374,152,352)(131,373,145,351)(132,372,146,350)(133,371,147,349)(134,370,148,348)(135,369,149,347)(136,376,150,346)(137,383,159,353)(138,382,160,360)(139,381,153,359)(140,380,154,358)(141,379,155,357)(142,378,156,356)(143,377,157,355)(144,384,158,354)(161,367,170,389)(162,366,171,388)(163,365,172,387)(164,364,173,386)(165,363,174,385)(166,362,175,392)(167,361,176,391)(168,368,169,390)(185,425,207,403)(186,432,208,402)(187,431,201,401)(188,430,202,408)(189,429,203,407)(190,428,204,406)(191,427,205,405)(192,426,206,404)(193,433,215,411)(194,440,216,410)(195,439,209,409)(196,438,210,416)(197,437,211,415)(198,436,212,414)(199,435,213,413)(200,434,214,412)>;
G:=Group( (1,17,9,70,85,29,77)(2,18,10,71,86,30,78)(3,19,11,72,87,31,79)(4,20,12,65,88,32,80)(5,21,13,66,81,25,73)(6,22,14,67,82,26,74)(7,23,15,68,83,27,75)(8,24,16,69,84,28,76)(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)(57,446,388,438,380,430,372)(58,447,389,439,381,431,373)(59,448,390,440,382,432,374)(60,441,391,433,383,425,375)(61,442,392,434,384,426,376)(62,443,385,435,377,427,369)(63,444,386,436,378,428,370)(64,445,387,437,379,429,371)(121,172,197,141,189,133,180)(122,173,198,142,190,134,181)(123,174,199,143,191,135,182)(124,175,200,144,192,136,183)(125,176,193,137,185,129,184)(126,169,194,138,186,130,177)(127,170,195,139,187,131,178)(128,171,196,140,188,132,179)(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)(233,284,307,249,299,241,291)(234,285,308,250,300,242,292)(235,286,309,251,301,243,293)(236,287,310,252,302,244,294)(237,288,311,253,303,245,295)(238,281,312,254,304,246,296)(239,282,305,255,297,247,289)(240,283,306,256,298,248,290)(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)(345,396,419,361,411,353,403)(346,397,420,362,412,354,404)(347,398,421,363,413,355,405)(348,399,422,364,414,356,406)(349,400,423,365,415,357,407)(350,393,424,366,416,358,408)(351,394,417,367,409,359,401)(352,395,418,368,410,360,402), (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), (1,147,35,133)(2,148,36,134)(3,149,37,135)(4,150,38,136)(5,151,39,129)(6,152,40,130)(7,145,33,131)(8,146,34,132)(9,219,107,121)(10,220,108,122)(11,221,109,123)(12,222,110,124)(13,223,111,125)(14,224,112,126)(15,217,105,127)(16,218,106,128)(17,225,115,180)(18,226,116,181)(19,227,117,182)(20,228,118,183)(21,229,119,184)(22,230,120,177)(23,231,113,178)(24,232,114,179)(25,159,47,137)(26,160,48,138)(27,153,41,139)(28,154,42,140)(29,155,43,141)(30,156,44,142)(31,157,45,143)(32,158,46,144)(49,170,68,161)(50,171,69,162)(51,172,70,163)(52,173,71,164)(53,174,72,165)(54,175,65,166)(55,176,66,167)(56,169,67,168)(57,281,393,340)(58,282,394,341)(59,283,395,342)(60,284,396,343)(61,285,397,344)(62,286,398,337)(63,287,399,338)(64,288,400,339)(73,207,95,185)(74,208,96,186)(75,201,89,187)(76,202,90,188)(77,203,91,189)(78,204,92,190)(79,205,93,191)(80,206,94,192)(81,215,103,193)(82,216,104,194)(83,209,97,195)(84,210,98,196)(85,211,99,197)(86,212,100,198)(87,213,101,199)(88,214,102,200)(233,345,263,375)(234,346,264,376)(235,347,257,369)(236,348,258,370)(237,349,259,371)(238,350,260,372)(239,351,261,373)(240,352,262,374)(241,353,271,383)(242,354,272,384)(243,355,265,377)(244,356,266,378)(245,357,267,379)(246,358,268,380)(247,359,269,381)(248,360,270,382)(249,361,279,391)(250,362,280,392)(251,363,273,385)(252,364,274,386)(253,365,275,387)(254,366,276,388)(255,367,277,389)(256,368,278,390)(289,401,319,431)(290,402,320,432)(291,403,313,425)(292,404,314,426)(293,405,315,427)(294,406,316,428)(295,407,317,429)(296,408,318,430)(297,409,327,439)(298,410,328,440)(299,411,321,433)(300,412,322,434)(301,413,323,435)(302,414,324,436)(303,415,325,437)(304,416,326,438)(305,417,335,447)(306,418,336,448)(307,419,329,441)(308,420,330,442)(309,421,331,443)(310,422,332,444)(311,423,333,445)(312,424,334,446), (1,259,35,237)(2,258,36,236)(3,257,37,235)(4,264,38,234)(5,263,39,233)(6,262,40,240)(7,261,33,239)(8,260,34,238)(9,333,107,311)(10,332,108,310)(11,331,109,309)(12,330,110,308)(13,329,111,307)(14,336,112,306)(15,335,105,305)(16,334,106,312)(17,339,115,288)(18,338,116,287)(19,337,117,286)(20,344,118,285)(21,343,119,284)(22,342,120,283)(23,341,113,282)(24,340,114,281)(25,271,47,241)(26,270,48,248)(27,269,41,247)(28,268,42,246)(29,267,43,245)(30,266,44,244)(31,265,45,243)(32,272,46,242)(49,255,68,277)(50,254,69,276)(51,253,70,275)(52,252,71,274)(53,251,72,273)(54,250,65,280)(55,249,66,279)(56,256,67,278)(57,232,393,179)(58,231,394,178)(59,230,395,177)(60,229,396,184)(61,228,397,183)(62,227,398,182)(63,226,399,181)(64,225,400,180)(73,313,95,291)(74,320,96,290)(75,319,89,289)(76,318,90,296)(77,317,91,295)(78,316,92,294)(79,315,93,293)(80,314,94,292)(81,321,103,299)(82,328,104,298)(83,327,97,297)(84,326,98,304)(85,325,99,303)(86,324,100,302)(87,323,101,301)(88,322,102,300)(121,445,219,423)(122,444,220,422)(123,443,221,421)(124,442,222,420)(125,441,223,419)(126,448,224,418)(127,447,217,417)(128,446,218,424)(129,375,151,345)(130,374,152,352)(131,373,145,351)(132,372,146,350)(133,371,147,349)(134,370,148,348)(135,369,149,347)(136,376,150,346)(137,383,159,353)(138,382,160,360)(139,381,153,359)(140,380,154,358)(141,379,155,357)(142,378,156,356)(143,377,157,355)(144,384,158,354)(161,367,170,389)(162,366,171,388)(163,365,172,387)(164,364,173,386)(165,363,174,385)(166,362,175,392)(167,361,176,391)(168,368,169,390)(185,425,207,403)(186,432,208,402)(187,431,201,401)(188,430,202,408)(189,429,203,407)(190,428,204,406)(191,427,205,405)(192,426,206,404)(193,433,215,411)(194,440,216,410)(195,439,209,409)(196,438,210,416)(197,437,211,415)(198,436,212,414)(199,435,213,413)(200,434,214,412) );
G=PermutationGroup([[(1,17,9,70,85,29,77),(2,18,10,71,86,30,78),(3,19,11,72,87,31,79),(4,20,12,65,88,32,80),(5,21,13,66,81,25,73),(6,22,14,67,82,26,74),(7,23,15,68,83,27,75),(8,24,16,69,84,28,76),(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),(57,446,388,438,380,430,372),(58,447,389,439,381,431,373),(59,448,390,440,382,432,374),(60,441,391,433,383,425,375),(61,442,392,434,384,426,376),(62,443,385,435,377,427,369),(63,444,386,436,378,428,370),(64,445,387,437,379,429,371),(121,172,197,141,189,133,180),(122,173,198,142,190,134,181),(123,174,199,143,191,135,182),(124,175,200,144,192,136,183),(125,176,193,137,185,129,184),(126,169,194,138,186,130,177),(127,170,195,139,187,131,178),(128,171,196,140,188,132,179),(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),(233,284,307,249,299,241,291),(234,285,308,250,300,242,292),(235,286,309,251,301,243,293),(236,287,310,252,302,244,294),(237,288,311,253,303,245,295),(238,281,312,254,304,246,296),(239,282,305,255,297,247,289),(240,283,306,256,298,248,290),(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),(345,396,419,361,411,353,403),(346,397,420,362,412,354,404),(347,398,421,363,413,355,405),(348,399,422,364,414,356,406),(349,400,423,365,415,357,407),(350,393,424,366,416,358,408),(351,394,417,367,409,359,401),(352,395,418,368,410,360,402)], [(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)], [(1,147,35,133),(2,148,36,134),(3,149,37,135),(4,150,38,136),(5,151,39,129),(6,152,40,130),(7,145,33,131),(8,146,34,132),(9,219,107,121),(10,220,108,122),(11,221,109,123),(12,222,110,124),(13,223,111,125),(14,224,112,126),(15,217,105,127),(16,218,106,128),(17,225,115,180),(18,226,116,181),(19,227,117,182),(20,228,118,183),(21,229,119,184),(22,230,120,177),(23,231,113,178),(24,232,114,179),(25,159,47,137),(26,160,48,138),(27,153,41,139),(28,154,42,140),(29,155,43,141),(30,156,44,142),(31,157,45,143),(32,158,46,144),(49,170,68,161),(50,171,69,162),(51,172,70,163),(52,173,71,164),(53,174,72,165),(54,175,65,166),(55,176,66,167),(56,169,67,168),(57,281,393,340),(58,282,394,341),(59,283,395,342),(60,284,396,343),(61,285,397,344),(62,286,398,337),(63,287,399,338),(64,288,400,339),(73,207,95,185),(74,208,96,186),(75,201,89,187),(76,202,90,188),(77,203,91,189),(78,204,92,190),(79,205,93,191),(80,206,94,192),(81,215,103,193),(82,216,104,194),(83,209,97,195),(84,210,98,196),(85,211,99,197),(86,212,100,198),(87,213,101,199),(88,214,102,200),(233,345,263,375),(234,346,264,376),(235,347,257,369),(236,348,258,370),(237,349,259,371),(238,350,260,372),(239,351,261,373),(240,352,262,374),(241,353,271,383),(242,354,272,384),(243,355,265,377),(244,356,266,378),(245,357,267,379),(246,358,268,380),(247,359,269,381),(248,360,270,382),(249,361,279,391),(250,362,280,392),(251,363,273,385),(252,364,274,386),(253,365,275,387),(254,366,276,388),(255,367,277,389),(256,368,278,390),(289,401,319,431),(290,402,320,432),(291,403,313,425),(292,404,314,426),(293,405,315,427),(294,406,316,428),(295,407,317,429),(296,408,318,430),(297,409,327,439),(298,410,328,440),(299,411,321,433),(300,412,322,434),(301,413,323,435),(302,414,324,436),(303,415,325,437),(304,416,326,438),(305,417,335,447),(306,418,336,448),(307,419,329,441),(308,420,330,442),(309,421,331,443),(310,422,332,444),(311,423,333,445),(312,424,334,446)], [(1,259,35,237),(2,258,36,236),(3,257,37,235),(4,264,38,234),(5,263,39,233),(6,262,40,240),(7,261,33,239),(8,260,34,238),(9,333,107,311),(10,332,108,310),(11,331,109,309),(12,330,110,308),(13,329,111,307),(14,336,112,306),(15,335,105,305),(16,334,106,312),(17,339,115,288),(18,338,116,287),(19,337,117,286),(20,344,118,285),(21,343,119,284),(22,342,120,283),(23,341,113,282),(24,340,114,281),(25,271,47,241),(26,270,48,248),(27,269,41,247),(28,268,42,246),(29,267,43,245),(30,266,44,244),(31,265,45,243),(32,272,46,242),(49,255,68,277),(50,254,69,276),(51,253,70,275),(52,252,71,274),(53,251,72,273),(54,250,65,280),(55,249,66,279),(56,256,67,278),(57,232,393,179),(58,231,394,178),(59,230,395,177),(60,229,396,184),(61,228,397,183),(62,227,398,182),(63,226,399,181),(64,225,400,180),(73,313,95,291),(74,320,96,290),(75,319,89,289),(76,318,90,296),(77,317,91,295),(78,316,92,294),(79,315,93,293),(80,314,94,292),(81,321,103,299),(82,328,104,298),(83,327,97,297),(84,326,98,304),(85,325,99,303),(86,324,100,302),(87,323,101,301),(88,322,102,300),(121,445,219,423),(122,444,220,422),(123,443,221,421),(124,442,222,420),(125,441,223,419),(126,448,224,418),(127,447,217,417),(128,446,218,424),(129,375,151,345),(130,374,152,352),(131,373,145,351),(132,372,146,350),(133,371,147,349),(134,370,148,348),(135,369,149,347),(136,376,150,346),(137,383,159,353),(138,382,160,360),(139,381,153,359),(140,380,154,358),(141,379,155,357),(142,378,156,356),(143,377,157,355),(144,384,158,354),(161,367,170,389),(162,366,171,388),(163,365,172,387),(164,364,173,386),(165,363,174,385),(166,362,175,392),(167,361,176,391),(168,368,169,390),(185,425,207,403),(186,432,208,402),(187,431,201,401),(188,430,202,408),(189,429,203,407),(190,428,204,406),(191,427,205,405),(192,426,206,404),(193,433,215,411),(194,440,216,410),(195,439,209,409),(196,438,210,416),(197,437,211,415),(198,436,212,414),(199,435,213,413),(200,434,214,412)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | Q8 | D4 | D8 | Q16 | C7×Q8 | C7×D4 | C7×D8 | C7×Q16 |
| kernel | C7×C8⋊2Q8 | C4×C56 | C7×C2.D8 | C7×C4⋊Q8 | C8⋊2Q8 | C4×C8 | C2.D8 | C4⋊Q8 | C56 | C2×C28 | C28 | C28 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 4 | 2 | 4 | 4 | 24 | 12 | 24 | 24 |
Matrix representation of C7×C8⋊2Q8 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 31 | 82 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 31 | 82 |
| 0 | 0 | 31 | 31 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 18 | 50 | 0 | 0 |
| 50 | 95 | 0 | 0 |
| 0 | 0 | 22 | 67 |
| 0 | 0 | 67 | 91 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,106,0,0,0,0,106],[31,31,0,0,82,31,0,0,0,0,31,31,0,0,82,31],[0,1,0,0,112,0,0,0,0,0,112,0,0,0,0,112],[18,50,0,0,50,95,0,0,0,0,22,67,0,0,67,91] >;
C7×C8⋊2Q8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_2Q_8
% in TeX
G:=Group("C7xC8:2Q8"); // GroupNames label
G:=SmallGroup(448,908);
// by ID
G=gap.SmallGroup(448,908);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,400,2438,1780,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations