direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C8⋊4Q8, C56⋊14Q8, C28.21M4(2), C8⋊4(C7×Q8), C4⋊C4.9C28, C4⋊C8.10C14, C2.6(Q8×C28), (C4×C8).15C14, (C4×C56).33C2, (C4×Q8).4C14, (C2×Q8).7C28, C14.34(C4×Q8), C4.25(Q8×C14), C8⋊C4.5C14, (Q8×C28).17C2, (Q8×C14).17C4, C28.131(C2×Q8), C4.3(C7×M4(2)), C14.52(C8○D4), C42.75(C2×C14), C28.357(C4○D4), (C4×C28).253C22, (C2×C56).447C22, (C2×C28).994C23, C2.11(C14×M4(2)), C14.55(C2×M4(2)), C22.49(C22×C28), C2.9(C7×C8○D4), (C7×C4⋊C8).23C2, (C7×C4⋊C4).21C4, C4.55(C7×C4○D4), (C2×C4).30(C2×C28), (C2×C8).55(C2×C14), (C7×C8⋊C4).11C2, (C2×C28).202(C2×C4), (C2×C14).244(C22×C4), (C2×C4).162(C22×C14), SmallGroup(448,854)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊4Q8
G = < a,b,c,d | a7=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c-1 >
Subgroups: 114 in 94 conjugacy classes, 74 normal (38 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C28, C28, C28, C2×C14, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C56, C56, C2×C28, C2×C28, C7×Q8, C8⋊4Q8, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C2×C56, Q8×C14, C4×C56, C7×C8⋊C4, C7×C4⋊C8, C7×C4⋊C8, Q8×C28, C7×C8⋊4Q8
Quotients: C1, C2, C4, C22, C7, C2×C4, Q8, C23, C14, M4(2), C22×C4, C2×Q8, C4○D4, C28, C2×C14, C4×Q8, C2×M4(2), C8○D4, C2×C28, C7×Q8, C22×C14, C8⋊4Q8, C7×M4(2), C22×C28, Q8×C14, C7×C4○D4, Q8×C28, C14×M4(2), C7×C8○D4, C7×C8⋊4Q8
►Regular action on 448 points
Generators in S448
(1 119 111 55 103 47 95)(2 120 112 56 104 48 96)(3 113 105 49 97 41 89)(4 114 106 50 98 42 90)(5 115 107 51 99 43 91)(6 116 108 52 100 44 92)(7 117 109 53 101 45 93)(8 118 110 54 102 46 94)(9 34 88 32 80 24 72)(10 35 81 25 73 17 65)(11 36 82 26 74 18 66)(12 37 83 27 75 19 67)(13 38 84 28 76 20 68)(14 39 85 29 77 21 69)(15 40 86 30 78 22 70)(16 33 87 31 79 23 71)(57 441 392 433 384 425 376)(58 442 385 434 377 426 369)(59 443 386 435 378 427 370)(60 444 387 436 379 428 371)(61 445 388 437 380 429 372)(62 446 389 438 381 430 373)(63 447 390 439 382 431 374)(64 448 391 440 383 432 375)(121 173 194 138 186 130 178)(122 174 195 139 187 131 179)(123 175 196 140 188 132 180)(124 176 197 141 189 133 181)(125 169 198 142 190 134 182)(126 170 199 143 191 135 183)(127 171 200 144 192 136 184)(128 172 193 137 185 129 177)(145 232 217 161 209 153 201)(146 225 218 162 210 154 202)(147 226 219 163 211 155 203)(148 227 220 164 212 156 204)(149 228 221 165 213 157 205)(150 229 222 166 214 158 206)(151 230 223 167 215 159 207)(152 231 224 168 216 160 208)(233 284 311 254 303 246 295)(234 285 312 255 304 247 296)(235 286 305 256 297 248 289)(236 287 306 249 298 241 290)(237 288 307 250 299 242 291)(238 281 308 251 300 243 292)(239 282 309 252 301 244 293)(240 283 310 253 302 245 294)(257 337 330 273 322 265 314)(258 338 331 274 323 266 315)(259 339 332 275 324 267 316)(260 340 333 276 325 268 317)(261 341 334 277 326 269 318)(262 342 335 278 327 270 319)(263 343 336 279 328 271 320)(264 344 329 280 321 272 313)(345 396 423 366 415 358 407)(346 397 424 367 416 359 408)(347 398 417 368 409 360 401)(348 399 418 361 410 353 402)(349 400 419 362 411 354 403)(350 393 420 363 412 355 404)(351 394 421 364 413 356 405)(352 395 422 365 414 357 406)
(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)
(1 183 71 151)(2 184 72 152)(3 177 65 145)(4 178 66 146)(5 179 67 147)(6 180 68 148)(7 181 69 149)(8 182 70 150)(9 231 120 127)(10 232 113 128)(11 225 114 121)(12 226 115 122)(13 227 116 123)(14 228 117 124)(15 229 118 125)(16 230 119 126)(17 201 89 129)(18 202 90 130)(19 203 91 131)(20 204 92 132)(21 205 93 133)(22 206 94 134)(23 207 95 135)(24 208 96 136)(25 209 97 137)(26 210 98 138)(27 211 99 139)(28 212 100 140)(29 213 101 141)(30 214 102 142)(31 215 103 143)(32 216 104 144)(33 223 111 170)(34 224 112 171)(35 217 105 172)(36 218 106 173)(37 219 107 174)(38 220 108 175)(39 221 109 176)(40 222 110 169)(41 185 73 153)(42 186 74 154)(43 187 75 155)(44 188 76 156)(45 189 77 157)(46 190 78 158)(47 191 79 159)(48 192 80 160)(49 193 81 161)(50 194 82 162)(51 195 83 163)(52 196 84 164)(53 197 85 165)(54 198 86 166)(55 199 87 167)(56 200 88 168)(57 237 349 344)(58 238 350 337)(59 239 351 338)(60 240 352 339)(61 233 345 340)(62 234 346 341)(63 235 347 342)(64 236 348 343)(241 353 320 432)(242 354 313 425)(243 355 314 426)(244 356 315 427)(245 357 316 428)(246 358 317 429)(247 359 318 430)(248 360 319 431)(249 361 328 440)(250 362 321 433)(251 363 322 434)(252 364 323 435)(253 365 324 436)(254 366 325 437)(255 367 326 438)(256 368 327 439)(257 369 292 404)(258 370 293 405)(259 371 294 406)(260 372 295 407)(261 373 296 408)(262 374 289 401)(263 375 290 402)(264 376 291 403)(265 377 300 412)(266 378 301 413)(267 379 302 414)(268 380 303 415)(269 381 304 416)(270 382 297 409)(271 383 298 410)(272 384 299 411)(273 385 308 420)(274 386 309 421)(275 387 310 422)(276 388 311 423)(277 389 312 424)(278 390 305 417)(279 391 306 418)(280 392 307 419)(281 393 330 442)(282 394 331 443)(283 395 332 444)(284 396 333 445)(285 397 334 446)(286 398 335 447)(287 399 336 448)(288 400 329 441)
(1 296 71 261)(2 293 72 258)(3 290 65 263)(4 295 66 260)(5 292 67 257)(6 289 68 262)(7 294 69 259)(8 291 70 264)(9 338 120 239)(10 343 113 236)(11 340 114 233)(12 337 115 238)(13 342 116 235)(14 339 117 240)(15 344 118 237)(16 341 119 234)(17 320 89 241)(18 317 90 246)(19 314 91 243)(20 319 92 248)(21 316 93 245)(22 313 94 242)(23 318 95 247)(24 315 96 244)(25 328 97 249)(26 325 98 254)(27 322 99 251)(28 327 100 256)(29 324 101 253)(30 321 102 250)(31 326 103 255)(32 323 104 252)(33 334 111 285)(34 331 112 282)(35 336 105 287)(36 333 106 284)(37 330 107 281)(38 335 108 286)(39 332 109 283)(40 329 110 288)(41 298 73 271)(42 303 74 268)(43 300 75 265)(44 297 76 270)(45 302 77 267)(46 299 78 272)(47 304 79 269)(48 301 80 266)(49 306 81 279)(50 311 82 276)(51 308 83 273)(52 305 84 278)(53 310 85 275)(54 307 86 280)(55 312 87 277)(56 309 88 274)(57 229 349 125)(58 226 350 122)(59 231 351 127)(60 228 352 124)(61 225 345 121)(62 230 346 126)(63 227 347 123)(64 232 348 128)(129 432 201 353)(130 429 202 358)(131 426 203 355)(132 431 204 360)(133 428 205 357)(134 425 206 354)(135 430 207 359)(136 427 208 356)(137 440 209 361)(138 437 210 366)(139 434 211 363)(140 439 212 368)(141 436 213 365)(142 433 214 362)(143 438 215 367)(144 435 216 364)(145 402 177 375)(146 407 178 372)(147 404 179 369)(148 401 180 374)(149 406 181 371)(150 403 182 376)(151 408 183 373)(152 405 184 370)(153 410 185 383)(154 415 186 380)(155 412 187 377)(156 409 188 382)(157 414 189 379)(158 411 190 384)(159 416 191 381)(160 413 192 378)(161 418 193 391)(162 423 194 388)(163 420 195 385)(164 417 196 390)(165 422 197 387)(166 419 198 392)(167 424 199 389)(168 421 200 386)(169 441 222 400)(170 446 223 397)(171 443 224 394)(172 448 217 399)(173 445 218 396)(174 442 219 393)(175 447 220 398)(176 444 221 395)
G:=sub<Sym(448)| (1,119,111,55,103,47,95)(2,120,112,56,104,48,96)(3,113,105,49,97,41,89)(4,114,106,50,98,42,90)(5,115,107,51,99,43,91)(6,116,108,52,100,44,92)(7,117,109,53,101,45,93)(8,118,110,54,102,46,94)(9,34,88,32,80,24,72)(10,35,81,25,73,17,65)(11,36,82,26,74,18,66)(12,37,83,27,75,19,67)(13,38,84,28,76,20,68)(14,39,85,29,77,21,69)(15,40,86,30,78,22,70)(16,33,87,31,79,23,71)(57,441,392,433,384,425,376)(58,442,385,434,377,426,369)(59,443,386,435,378,427,370)(60,444,387,436,379,428,371)(61,445,388,437,380,429,372)(62,446,389,438,381,430,373)(63,447,390,439,382,431,374)(64,448,391,440,383,432,375)(121,173,194,138,186,130,178)(122,174,195,139,187,131,179)(123,175,196,140,188,132,180)(124,176,197,141,189,133,181)(125,169,198,142,190,134,182)(126,170,199,143,191,135,183)(127,171,200,144,192,136,184)(128,172,193,137,185,129,177)(145,232,217,161,209,153,201)(146,225,218,162,210,154,202)(147,226,219,163,211,155,203)(148,227,220,164,212,156,204)(149,228,221,165,213,157,205)(150,229,222,166,214,158,206)(151,230,223,167,215,159,207)(152,231,224,168,216,160,208)(233,284,311,254,303,246,295)(234,285,312,255,304,247,296)(235,286,305,256,297,248,289)(236,287,306,249,298,241,290)(237,288,307,250,299,242,291)(238,281,308,251,300,243,292)(239,282,309,252,301,244,293)(240,283,310,253,302,245,294)(257,337,330,273,322,265,314)(258,338,331,274,323,266,315)(259,339,332,275,324,267,316)(260,340,333,276,325,268,317)(261,341,334,277,326,269,318)(262,342,335,278,327,270,319)(263,343,336,279,328,271,320)(264,344,329,280,321,272,313)(345,396,423,366,415,358,407)(346,397,424,367,416,359,408)(347,398,417,368,409,360,401)(348,399,418,361,410,353,402)(349,400,419,362,411,354,403)(350,393,420,363,412,355,404)(351,394,421,364,413,356,405)(352,395,422,365,414,357,406), (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), (1,183,71,151)(2,184,72,152)(3,177,65,145)(4,178,66,146)(5,179,67,147)(6,180,68,148)(7,181,69,149)(8,182,70,150)(9,231,120,127)(10,232,113,128)(11,225,114,121)(12,226,115,122)(13,227,116,123)(14,228,117,124)(15,229,118,125)(16,230,119,126)(17,201,89,129)(18,202,90,130)(19,203,91,131)(20,204,92,132)(21,205,93,133)(22,206,94,134)(23,207,95,135)(24,208,96,136)(25,209,97,137)(26,210,98,138)(27,211,99,139)(28,212,100,140)(29,213,101,141)(30,214,102,142)(31,215,103,143)(32,216,104,144)(33,223,111,170)(34,224,112,171)(35,217,105,172)(36,218,106,173)(37,219,107,174)(38,220,108,175)(39,221,109,176)(40,222,110,169)(41,185,73,153)(42,186,74,154)(43,187,75,155)(44,188,76,156)(45,189,77,157)(46,190,78,158)(47,191,79,159)(48,192,80,160)(49,193,81,161)(50,194,82,162)(51,195,83,163)(52,196,84,164)(53,197,85,165)(54,198,86,166)(55,199,87,167)(56,200,88,168)(57,237,349,344)(58,238,350,337)(59,239,351,338)(60,240,352,339)(61,233,345,340)(62,234,346,341)(63,235,347,342)(64,236,348,343)(241,353,320,432)(242,354,313,425)(243,355,314,426)(244,356,315,427)(245,357,316,428)(246,358,317,429)(247,359,318,430)(248,360,319,431)(249,361,328,440)(250,362,321,433)(251,363,322,434)(252,364,323,435)(253,365,324,436)(254,366,325,437)(255,367,326,438)(256,368,327,439)(257,369,292,404)(258,370,293,405)(259,371,294,406)(260,372,295,407)(261,373,296,408)(262,374,289,401)(263,375,290,402)(264,376,291,403)(265,377,300,412)(266,378,301,413)(267,379,302,414)(268,380,303,415)(269,381,304,416)(270,382,297,409)(271,383,298,410)(272,384,299,411)(273,385,308,420)(274,386,309,421)(275,387,310,422)(276,388,311,423)(277,389,312,424)(278,390,305,417)(279,391,306,418)(280,392,307,419)(281,393,330,442)(282,394,331,443)(283,395,332,444)(284,396,333,445)(285,397,334,446)(286,398,335,447)(287,399,336,448)(288,400,329,441), (1,296,71,261)(2,293,72,258)(3,290,65,263)(4,295,66,260)(5,292,67,257)(6,289,68,262)(7,294,69,259)(8,291,70,264)(9,338,120,239)(10,343,113,236)(11,340,114,233)(12,337,115,238)(13,342,116,235)(14,339,117,240)(15,344,118,237)(16,341,119,234)(17,320,89,241)(18,317,90,246)(19,314,91,243)(20,319,92,248)(21,316,93,245)(22,313,94,242)(23,318,95,247)(24,315,96,244)(25,328,97,249)(26,325,98,254)(27,322,99,251)(28,327,100,256)(29,324,101,253)(30,321,102,250)(31,326,103,255)(32,323,104,252)(33,334,111,285)(34,331,112,282)(35,336,105,287)(36,333,106,284)(37,330,107,281)(38,335,108,286)(39,332,109,283)(40,329,110,288)(41,298,73,271)(42,303,74,268)(43,300,75,265)(44,297,76,270)(45,302,77,267)(46,299,78,272)(47,304,79,269)(48,301,80,266)(49,306,81,279)(50,311,82,276)(51,308,83,273)(52,305,84,278)(53,310,85,275)(54,307,86,280)(55,312,87,277)(56,309,88,274)(57,229,349,125)(58,226,350,122)(59,231,351,127)(60,228,352,124)(61,225,345,121)(62,230,346,126)(63,227,347,123)(64,232,348,128)(129,432,201,353)(130,429,202,358)(131,426,203,355)(132,431,204,360)(133,428,205,357)(134,425,206,354)(135,430,207,359)(136,427,208,356)(137,440,209,361)(138,437,210,366)(139,434,211,363)(140,439,212,368)(141,436,213,365)(142,433,214,362)(143,438,215,367)(144,435,216,364)(145,402,177,375)(146,407,178,372)(147,404,179,369)(148,401,180,374)(149,406,181,371)(150,403,182,376)(151,408,183,373)(152,405,184,370)(153,410,185,383)(154,415,186,380)(155,412,187,377)(156,409,188,382)(157,414,189,379)(158,411,190,384)(159,416,191,381)(160,413,192,378)(161,418,193,391)(162,423,194,388)(163,420,195,385)(164,417,196,390)(165,422,197,387)(166,419,198,392)(167,424,199,389)(168,421,200,386)(169,441,222,400)(170,446,223,397)(171,443,224,394)(172,448,217,399)(173,445,218,396)(174,442,219,393)(175,447,220,398)(176,444,221,395)>;
G:=Group( (1,119,111,55,103,47,95)(2,120,112,56,104,48,96)(3,113,105,49,97,41,89)(4,114,106,50,98,42,90)(5,115,107,51,99,43,91)(6,116,108,52,100,44,92)(7,117,109,53,101,45,93)(8,118,110,54,102,46,94)(9,34,88,32,80,24,72)(10,35,81,25,73,17,65)(11,36,82,26,74,18,66)(12,37,83,27,75,19,67)(13,38,84,28,76,20,68)(14,39,85,29,77,21,69)(15,40,86,30,78,22,70)(16,33,87,31,79,23,71)(57,441,392,433,384,425,376)(58,442,385,434,377,426,369)(59,443,386,435,378,427,370)(60,444,387,436,379,428,371)(61,445,388,437,380,429,372)(62,446,389,438,381,430,373)(63,447,390,439,382,431,374)(64,448,391,440,383,432,375)(121,173,194,138,186,130,178)(122,174,195,139,187,131,179)(123,175,196,140,188,132,180)(124,176,197,141,189,133,181)(125,169,198,142,190,134,182)(126,170,199,143,191,135,183)(127,171,200,144,192,136,184)(128,172,193,137,185,129,177)(145,232,217,161,209,153,201)(146,225,218,162,210,154,202)(147,226,219,163,211,155,203)(148,227,220,164,212,156,204)(149,228,221,165,213,157,205)(150,229,222,166,214,158,206)(151,230,223,167,215,159,207)(152,231,224,168,216,160,208)(233,284,311,254,303,246,295)(234,285,312,255,304,247,296)(235,286,305,256,297,248,289)(236,287,306,249,298,241,290)(237,288,307,250,299,242,291)(238,281,308,251,300,243,292)(239,282,309,252,301,244,293)(240,283,310,253,302,245,294)(257,337,330,273,322,265,314)(258,338,331,274,323,266,315)(259,339,332,275,324,267,316)(260,340,333,276,325,268,317)(261,341,334,277,326,269,318)(262,342,335,278,327,270,319)(263,343,336,279,328,271,320)(264,344,329,280,321,272,313)(345,396,423,366,415,358,407)(346,397,424,367,416,359,408)(347,398,417,368,409,360,401)(348,399,418,361,410,353,402)(349,400,419,362,411,354,403)(350,393,420,363,412,355,404)(351,394,421,364,413,356,405)(352,395,422,365,414,357,406), (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), (1,183,71,151)(2,184,72,152)(3,177,65,145)(4,178,66,146)(5,179,67,147)(6,180,68,148)(7,181,69,149)(8,182,70,150)(9,231,120,127)(10,232,113,128)(11,225,114,121)(12,226,115,122)(13,227,116,123)(14,228,117,124)(15,229,118,125)(16,230,119,126)(17,201,89,129)(18,202,90,130)(19,203,91,131)(20,204,92,132)(21,205,93,133)(22,206,94,134)(23,207,95,135)(24,208,96,136)(25,209,97,137)(26,210,98,138)(27,211,99,139)(28,212,100,140)(29,213,101,141)(30,214,102,142)(31,215,103,143)(32,216,104,144)(33,223,111,170)(34,224,112,171)(35,217,105,172)(36,218,106,173)(37,219,107,174)(38,220,108,175)(39,221,109,176)(40,222,110,169)(41,185,73,153)(42,186,74,154)(43,187,75,155)(44,188,76,156)(45,189,77,157)(46,190,78,158)(47,191,79,159)(48,192,80,160)(49,193,81,161)(50,194,82,162)(51,195,83,163)(52,196,84,164)(53,197,85,165)(54,198,86,166)(55,199,87,167)(56,200,88,168)(57,237,349,344)(58,238,350,337)(59,239,351,338)(60,240,352,339)(61,233,345,340)(62,234,346,341)(63,235,347,342)(64,236,348,343)(241,353,320,432)(242,354,313,425)(243,355,314,426)(244,356,315,427)(245,357,316,428)(246,358,317,429)(247,359,318,430)(248,360,319,431)(249,361,328,440)(250,362,321,433)(251,363,322,434)(252,364,323,435)(253,365,324,436)(254,366,325,437)(255,367,326,438)(256,368,327,439)(257,369,292,404)(258,370,293,405)(259,371,294,406)(260,372,295,407)(261,373,296,408)(262,374,289,401)(263,375,290,402)(264,376,291,403)(265,377,300,412)(266,378,301,413)(267,379,302,414)(268,380,303,415)(269,381,304,416)(270,382,297,409)(271,383,298,410)(272,384,299,411)(273,385,308,420)(274,386,309,421)(275,387,310,422)(276,388,311,423)(277,389,312,424)(278,390,305,417)(279,391,306,418)(280,392,307,419)(281,393,330,442)(282,394,331,443)(283,395,332,444)(284,396,333,445)(285,397,334,446)(286,398,335,447)(287,399,336,448)(288,400,329,441), (1,296,71,261)(2,293,72,258)(3,290,65,263)(4,295,66,260)(5,292,67,257)(6,289,68,262)(7,294,69,259)(8,291,70,264)(9,338,120,239)(10,343,113,236)(11,340,114,233)(12,337,115,238)(13,342,116,235)(14,339,117,240)(15,344,118,237)(16,341,119,234)(17,320,89,241)(18,317,90,246)(19,314,91,243)(20,319,92,248)(21,316,93,245)(22,313,94,242)(23,318,95,247)(24,315,96,244)(25,328,97,249)(26,325,98,254)(27,322,99,251)(28,327,100,256)(29,324,101,253)(30,321,102,250)(31,326,103,255)(32,323,104,252)(33,334,111,285)(34,331,112,282)(35,336,105,287)(36,333,106,284)(37,330,107,281)(38,335,108,286)(39,332,109,283)(40,329,110,288)(41,298,73,271)(42,303,74,268)(43,300,75,265)(44,297,76,270)(45,302,77,267)(46,299,78,272)(47,304,79,269)(48,301,80,266)(49,306,81,279)(50,311,82,276)(51,308,83,273)(52,305,84,278)(53,310,85,275)(54,307,86,280)(55,312,87,277)(56,309,88,274)(57,229,349,125)(58,226,350,122)(59,231,351,127)(60,228,352,124)(61,225,345,121)(62,230,346,126)(63,227,347,123)(64,232,348,128)(129,432,201,353)(130,429,202,358)(131,426,203,355)(132,431,204,360)(133,428,205,357)(134,425,206,354)(135,430,207,359)(136,427,208,356)(137,440,209,361)(138,437,210,366)(139,434,211,363)(140,439,212,368)(141,436,213,365)(142,433,214,362)(143,438,215,367)(144,435,216,364)(145,402,177,375)(146,407,178,372)(147,404,179,369)(148,401,180,374)(149,406,181,371)(150,403,182,376)(151,408,183,373)(152,405,184,370)(153,410,185,383)(154,415,186,380)(155,412,187,377)(156,409,188,382)(157,414,189,379)(158,411,190,384)(159,416,191,381)(160,413,192,378)(161,418,193,391)(162,423,194,388)(163,420,195,385)(164,417,196,390)(165,422,197,387)(166,419,198,392)(167,424,199,389)(168,421,200,386)(169,441,222,400)(170,446,223,397)(171,443,224,394)(172,448,217,399)(173,445,218,396)(174,442,219,393)(175,447,220,398)(176,444,221,395) );
G=PermutationGroup([[(1,119,111,55,103,47,95),(2,120,112,56,104,48,96),(3,113,105,49,97,41,89),(4,114,106,50,98,42,90),(5,115,107,51,99,43,91),(6,116,108,52,100,44,92),(7,117,109,53,101,45,93),(8,118,110,54,102,46,94),(9,34,88,32,80,24,72),(10,35,81,25,73,17,65),(11,36,82,26,74,18,66),(12,37,83,27,75,19,67),(13,38,84,28,76,20,68),(14,39,85,29,77,21,69),(15,40,86,30,78,22,70),(16,33,87,31,79,23,71),(57,441,392,433,384,425,376),(58,442,385,434,377,426,369),(59,443,386,435,378,427,370),(60,444,387,436,379,428,371),(61,445,388,437,380,429,372),(62,446,389,438,381,430,373),(63,447,390,439,382,431,374),(64,448,391,440,383,432,375),(121,173,194,138,186,130,178),(122,174,195,139,187,131,179),(123,175,196,140,188,132,180),(124,176,197,141,189,133,181),(125,169,198,142,190,134,182),(126,170,199,143,191,135,183),(127,171,200,144,192,136,184),(128,172,193,137,185,129,177),(145,232,217,161,209,153,201),(146,225,218,162,210,154,202),(147,226,219,163,211,155,203),(148,227,220,164,212,156,204),(149,228,221,165,213,157,205),(150,229,222,166,214,158,206),(151,230,223,167,215,159,207),(152,231,224,168,216,160,208),(233,284,311,254,303,246,295),(234,285,312,255,304,247,296),(235,286,305,256,297,248,289),(236,287,306,249,298,241,290),(237,288,307,250,299,242,291),(238,281,308,251,300,243,292),(239,282,309,252,301,244,293),(240,283,310,253,302,245,294),(257,337,330,273,322,265,314),(258,338,331,274,323,266,315),(259,339,332,275,324,267,316),(260,340,333,276,325,268,317),(261,341,334,277,326,269,318),(262,342,335,278,327,270,319),(263,343,336,279,328,271,320),(264,344,329,280,321,272,313),(345,396,423,366,415,358,407),(346,397,424,367,416,359,408),(347,398,417,368,409,360,401),(348,399,418,361,410,353,402),(349,400,419,362,411,354,403),(350,393,420,363,412,355,404),(351,394,421,364,413,356,405),(352,395,422,365,414,357,406)], [(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)], [(1,183,71,151),(2,184,72,152),(3,177,65,145),(4,178,66,146),(5,179,67,147),(6,180,68,148),(7,181,69,149),(8,182,70,150),(9,231,120,127),(10,232,113,128),(11,225,114,121),(12,226,115,122),(13,227,116,123),(14,228,117,124),(15,229,118,125),(16,230,119,126),(17,201,89,129),(18,202,90,130),(19,203,91,131),(20,204,92,132),(21,205,93,133),(22,206,94,134),(23,207,95,135),(24,208,96,136),(25,209,97,137),(26,210,98,138),(27,211,99,139),(28,212,100,140),(29,213,101,141),(30,214,102,142),(31,215,103,143),(32,216,104,144),(33,223,111,170),(34,224,112,171),(35,217,105,172),(36,218,106,173),(37,219,107,174),(38,220,108,175),(39,221,109,176),(40,222,110,169),(41,185,73,153),(42,186,74,154),(43,187,75,155),(44,188,76,156),(45,189,77,157),(46,190,78,158),(47,191,79,159),(48,192,80,160),(49,193,81,161),(50,194,82,162),(51,195,83,163),(52,196,84,164),(53,197,85,165),(54,198,86,166),(55,199,87,167),(56,200,88,168),(57,237,349,344),(58,238,350,337),(59,239,351,338),(60,240,352,339),(61,233,345,340),(62,234,346,341),(63,235,347,342),(64,236,348,343),(241,353,320,432),(242,354,313,425),(243,355,314,426),(244,356,315,427),(245,357,316,428),(246,358,317,429),(247,359,318,430),(248,360,319,431),(249,361,328,440),(250,362,321,433),(251,363,322,434),(252,364,323,435),(253,365,324,436),(254,366,325,437),(255,367,326,438),(256,368,327,439),(257,369,292,404),(258,370,293,405),(259,371,294,406),(260,372,295,407),(261,373,296,408),(262,374,289,401),(263,375,290,402),(264,376,291,403),(265,377,300,412),(266,378,301,413),(267,379,302,414),(268,380,303,415),(269,381,304,416),(270,382,297,409),(271,383,298,410),(272,384,299,411),(273,385,308,420),(274,386,309,421),(275,387,310,422),(276,388,311,423),(277,389,312,424),(278,390,305,417),(279,391,306,418),(280,392,307,419),(281,393,330,442),(282,394,331,443),(283,395,332,444),(284,396,333,445),(285,397,334,446),(286,398,335,447),(287,399,336,448),(288,400,329,441)], [(1,296,71,261),(2,293,72,258),(3,290,65,263),(4,295,66,260),(5,292,67,257),(6,289,68,262),(7,294,69,259),(8,291,70,264),(9,338,120,239),(10,343,113,236),(11,340,114,233),(12,337,115,238),(13,342,116,235),(14,339,117,240),(15,344,118,237),(16,341,119,234),(17,320,89,241),(18,317,90,246),(19,314,91,243),(20,319,92,248),(21,316,93,245),(22,313,94,242),(23,318,95,247),(24,315,96,244),(25,328,97,249),(26,325,98,254),(27,322,99,251),(28,327,100,256),(29,324,101,253),(30,321,102,250),(31,326,103,255),(32,323,104,252),(33,334,111,285),(34,331,112,282),(35,336,105,287),(36,333,106,284),(37,330,107,281),(38,335,108,286),(39,332,109,283),(40,329,110,288),(41,298,73,271),(42,303,74,268),(43,300,75,265),(44,297,76,270),(45,302,77,267),(46,299,78,272),(47,304,79,269),(48,301,80,266),(49,306,81,279),(50,311,82,276),(51,308,83,273),(52,305,84,278),(53,310,85,275),(54,307,86,280),(55,312,87,277),(56,309,88,274),(57,229,349,125),(58,226,350,122),(59,231,351,127),(60,228,352,124),(61,225,345,121),(62,230,346,126),(63,227,347,123),(64,232,348,128),(129,432,201,353),(130,429,202,358),(131,426,203,355),(132,431,204,360),(133,428,205,357),(134,425,206,354),(135,430,207,359),(136,427,208,356),(137,440,209,361),(138,437,210,366),(139,434,211,363),(140,439,212,368),(141,436,213,365),(142,433,214,362),(143,438,215,367),(144,435,216,364),(145,402,177,375),(146,407,178,372),(147,404,179,369),(148,401,180,374),(149,406,181,371),(150,403,182,376),(151,408,183,373),(152,405,184,370),(153,410,185,383),(154,415,186,380),(155,412,187,377),(156,409,188,382),(157,414,189,379),(158,411,190,384),(159,416,191,381),(160,413,192,378),(161,418,193,391),(162,423,194,388),(163,420,195,385),(164,417,196,390),(165,422,197,387),(166,419,198,392),(167,424,199,389),(168,421,200,386),(169,441,222,400),(170,446,223,397),(171,443,224,394),(172,448,217,399),(173,445,218,396),(174,442,219,393),(175,447,220,398),(176,444,221,395)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BT | 56A | ··· | 56AV | 56AW | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C28 | Q8 | M4(2) | C4○D4 | C8○D4 | C7×Q8 | C7×M4(2) | C7×C4○D4 | C7×C8○D4 |
| kernel | C7×C8⋊4Q8 | C4×C56 | C7×C8⋊C4 | C7×C4⋊C8 | Q8×C28 | C7×C4⋊C4 | Q8×C14 | C8⋊4Q8 | C4×C8 | C8⋊C4 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | C56 | C28 | C28 | C14 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 6 | 2 | 6 | 6 | 12 | 18 | 6 | 36 | 12 | 2 | 4 | 2 | 4 | 12 | 24 | 12 | 24 |
Matrix representation of C7×C8⋊4Q8 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 112 | 111 | 0 | 0 |
| 106 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 98 | 0 |
| 44 | 88 | 0 | 0 |
| 82 | 69 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 45 | 1 | 0 | 0 |
| 8 | 68 | 0 | 0 |
| 0 | 0 | 17 | 79 |
| 0 | 0 | 55 | 96 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,30,0,0,0,0,30],[112,106,0,0,111,1,0,0,0,0,0,98,0,0,1,0],[44,82,0,0,88,69,0,0,0,0,112,0,0,0,0,112],[45,8,0,0,1,68,0,0,0,0,17,55,0,0,79,96] >;
C7×C8⋊4Q8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_4Q_8
% in TeX
G:=Group("C7xC8:4Q8"); // GroupNames label
G:=SmallGroup(448,854);
// by ID
G=gap.SmallGroup(448,854);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,4790,898,124]);
// 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^5,d*c*d^-1=c^-1>;
// generators/relations