direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊Q8, C56⋊7Q8, C8⋊(C7×Q8), C4.8(Q8×C14), C4⋊Q8.12C14, C28.97(C2×Q8), C8⋊C4.2C14, C4.Q8.3C14, C2.D8.8C14, (C2×C28).345D4, C14.43(C4⋊Q8), C42.31(C2×C14), C42.C2.5C14, (C2×C28).957C23, (C2×C56).277C22, (C4×C28).273C22, C22.122(D4×C14), C14.148(C8⋊C22), C14.148(C8.C22), C2.9(C7×C4⋊Q8), (C2×C4).46(C7×D4), (C7×C4⋊Q8).27C2, (C7×C4.Q8).8C2, (C7×C8⋊C4).6C2, C4⋊C4.26(C2×C14), (C2×C8).29(C2×C14), C2.23(C7×C8⋊C22), (C7×C2.D8).17C2, (C2×C14).678(C2×D4), C2.23(C7×C8.C22), (C7×C4⋊C4).246C22, (C7×C42.C2).12C2, (C2×C4).132(C22×C14), SmallGroup(448,909)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊Q8
G = < a,b,c,d | a7=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=c-1 >
Subgroups: 146 in 90 conjugacy classes, 58 normal (30 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C8⋊Q8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, C7×C8⋊C4, C7×C4.Q8, C7×C2.D8, C7×C42.C2, C7×C4⋊Q8, C7×C8⋊Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C2×C14, C4⋊Q8, C8⋊C22, C8.C22, C7×D4, C7×Q8, C22×C14, C8⋊Q8, D4×C14, Q8×C14, C7×C4⋊Q8, C7×C8⋊C22, C7×C8.C22, C7×C8⋊Q8
►Regular action on 448 points
Generators in S448
(1 71 14 87 31 79 23)(2 72 15 88 32 80 24)(3 65 16 81 25 73 17)(4 66 9 82 26 74 18)(5 67 10 83 27 75 19)(6 68 11 84 28 76 20)(7 69 12 85 29 77 21)(8 70 13 86 30 78 22)(33 120 105 49 97 41 89)(34 113 106 50 98 42 90)(35 114 107 51 99 43 91)(36 115 108 52 100 44 92)(37 116 109 53 101 45 93)(38 117 110 54 102 46 94)(39 118 111 55 103 47 95)(40 119 112 56 104 48 96)(57 445 386 437 378 429 370)(58 446 387 438 379 430 371)(59 447 388 439 380 431 372)(60 448 389 440 381 432 373)(61 441 390 433 382 425 374)(62 442 391 434 383 426 375)(63 443 392 435 384 427 376)(64 444 385 436 377 428 369)(121 173 194 137 186 129 178)(122 174 195 138 187 130 179)(123 175 196 139 188 131 180)(124 176 197 140 189 132 181)(125 169 198 141 190 133 182)(126 170 199 142 191 134 183)(127 171 200 143 192 135 184)(128 172 193 144 185 136 177)(145 232 218 161 210 153 202)(146 225 219 162 211 154 203)(147 226 220 163 212 155 204)(148 227 221 164 213 156 205)(149 228 222 165 214 157 206)(150 229 223 166 215 158 207)(151 230 224 167 216 159 208)(152 231 217 168 209 160 201)(233 287 310 252 302 244 294)(234 288 311 253 303 245 295)(235 281 312 254 304 246 296)(236 282 305 255 297 247 289)(237 283 306 256 298 248 290)(238 284 307 249 299 241 291)(239 285 308 250 300 242 292)(240 286 309 251 301 243 293)(257 344 331 273 323 265 315)(258 337 332 274 324 266 316)(259 338 333 275 325 267 317)(260 339 334 276 326 268 318)(261 340 335 277 327 269 319)(262 341 336 278 328 270 320)(263 342 329 279 321 271 313)(264 343 330 280 322 272 314)(345 397 420 361 412 353 404)(346 398 421 362 413 354 405)(347 399 422 363 414 355 406)(348 400 423 364 415 356 407)(349 393 424 365 416 357 408)(350 394 417 366 409 358 401)(351 395 418 367 410 359 402)(352 396 419 368 411 360 403)
(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)
(1 146 39 128)(2 151 40 125)(3 148 33 122)(4 145 34 127)(5 150 35 124)(6 147 36 121)(7 152 37 126)(8 149 38 123)(9 218 106 200)(10 223 107 197)(11 220 108 194)(12 217 109 199)(13 222 110 196)(14 219 111 193)(15 224 112 198)(16 221 105 195)(17 205 89 179)(18 202 90 184)(19 207 91 181)(20 204 92 178)(21 201 93 183)(22 206 94 180)(23 203 95 177)(24 208 96 182)(25 213 97 187)(26 210 98 192)(27 215 99 189)(28 212 100 186)(29 209 101 191)(30 214 102 188)(31 211 103 185)(32 216 104 190)(41 130 73 156)(42 135 74 153)(43 132 75 158)(44 129 76 155)(45 134 77 160)(46 131 78 157)(47 136 79 154)(48 133 80 159)(49 138 81 164)(50 143 82 161)(51 140 83 166)(52 137 84 163)(53 142 85 168)(54 139 86 165)(55 144 87 162)(56 141 88 167)(57 240 396 342)(58 237 397 339)(59 234 398 344)(60 239 399 341)(61 236 400 338)(62 233 393 343)(63 238 394 340)(64 235 395 337)(65 227 120 174)(66 232 113 171)(67 229 114 176)(68 226 115 173)(69 231 116 170)(70 228 117 175)(71 225 118 172)(72 230 119 169)(241 401 319 427)(242 406 320 432)(243 403 313 429)(244 408 314 426)(245 405 315 431)(246 402 316 428)(247 407 317 425)(248 404 318 430)(249 409 327 435)(250 414 328 440)(251 411 321 437)(252 416 322 434)(253 413 323 439)(254 410 324 436)(255 415 325 433)(256 412 326 438)(257 372 295 346)(258 369 296 351)(259 374 289 348)(260 371 290 345)(261 376 291 350)(262 373 292 347)(263 370 293 352)(264 375 294 349)(265 380 303 354)(266 377 304 359)(267 382 297 356)(268 379 298 353)(269 384 299 358)(270 381 300 355)(271 378 301 360)(272 383 302 357)(273 388 311 362)(274 385 312 367)(275 390 305 364)(276 387 306 361)(277 392 307 366)(278 389 308 363)(279 386 309 368)(280 391 310 365)(281 418 332 444)(282 423 333 441)(283 420 334 446)(284 417 335 443)(285 422 336 448)(286 419 329 445)(287 424 330 442)(288 421 331 447)
(1 259 39 289)(2 262 40 292)(3 257 33 295)(4 260 34 290)(5 263 35 293)(6 258 36 296)(7 261 37 291)(8 264 38 294)(9 334 106 283)(10 329 107 286)(11 332 108 281)(12 335 109 284)(13 330 110 287)(14 333 111 282)(15 336 112 285)(16 331 105 288)(17 315 89 245)(18 318 90 248)(19 313 91 243)(20 316 92 246)(21 319 93 241)(22 314 94 244)(23 317 95 247)(24 320 96 242)(25 323 97 253)(26 326 98 256)(27 321 99 251)(28 324 100 254)(29 327 101 249)(30 322 102 252)(31 325 103 255)(32 328 104 250)(41 303 73 265)(42 298 74 268)(43 301 75 271)(44 304 76 266)(45 299 77 269)(46 302 78 272)(47 297 79 267)(48 300 80 270)(49 311 81 273)(50 306 82 276)(51 309 83 279)(52 312 84 274)(53 307 85 277)(54 310 86 280)(55 305 87 275)(56 308 88 278)(57 229 396 176)(58 232 397 171)(59 227 398 174)(60 230 399 169)(61 225 400 172)(62 228 393 175)(63 231 394 170)(64 226 395 173)(65 344 120 234)(66 339 113 237)(67 342 114 240)(68 337 115 235)(69 340 116 238)(70 343 117 233)(71 338 118 236)(72 341 119 239)(121 369 147 351)(122 372 148 346)(123 375 149 349)(124 370 150 352)(125 373 151 347)(126 376 152 350)(127 371 145 345)(128 374 146 348)(129 377 155 359)(130 380 156 354)(131 383 157 357)(132 378 158 360)(133 381 159 355)(134 384 160 358)(135 379 153 353)(136 382 154 356)(137 385 163 367)(138 388 164 362)(139 391 165 365)(140 386 166 368)(141 389 167 363)(142 392 168 366)(143 387 161 361)(144 390 162 364)(177 425 203 407)(178 428 204 402)(179 431 205 405)(180 426 206 408)(181 429 207 403)(182 432 208 406)(183 427 201 401)(184 430 202 404)(185 433 211 415)(186 436 212 410)(187 439 213 413)(188 434 214 416)(189 437 215 411)(190 440 216 414)(191 435 209 409)(192 438 210 412)(193 441 219 423)(194 444 220 418)(195 447 221 421)(196 442 222 424)(197 445 223 419)(198 448 224 422)(199 443 217 417)(200 446 218 420)
G:=sub<Sym(448)| (1,71,14,87,31,79,23)(2,72,15,88,32,80,24)(3,65,16,81,25,73,17)(4,66,9,82,26,74,18)(5,67,10,83,27,75,19)(6,68,11,84,28,76,20)(7,69,12,85,29,77,21)(8,70,13,86,30,78,22)(33,120,105,49,97,41,89)(34,113,106,50,98,42,90)(35,114,107,51,99,43,91)(36,115,108,52,100,44,92)(37,116,109,53,101,45,93)(38,117,110,54,102,46,94)(39,118,111,55,103,47,95)(40,119,112,56,104,48,96)(57,445,386,437,378,429,370)(58,446,387,438,379,430,371)(59,447,388,439,380,431,372)(60,448,389,440,381,432,373)(61,441,390,433,382,425,374)(62,442,391,434,383,426,375)(63,443,392,435,384,427,376)(64,444,385,436,377,428,369)(121,173,194,137,186,129,178)(122,174,195,138,187,130,179)(123,175,196,139,188,131,180)(124,176,197,140,189,132,181)(125,169,198,141,190,133,182)(126,170,199,142,191,134,183)(127,171,200,143,192,135,184)(128,172,193,144,185,136,177)(145,232,218,161,210,153,202)(146,225,219,162,211,154,203)(147,226,220,163,212,155,204)(148,227,221,164,213,156,205)(149,228,222,165,214,157,206)(150,229,223,166,215,158,207)(151,230,224,167,216,159,208)(152,231,217,168,209,160,201)(233,287,310,252,302,244,294)(234,288,311,253,303,245,295)(235,281,312,254,304,246,296)(236,282,305,255,297,247,289)(237,283,306,256,298,248,290)(238,284,307,249,299,241,291)(239,285,308,250,300,242,292)(240,286,309,251,301,243,293)(257,344,331,273,323,265,315)(258,337,332,274,324,266,316)(259,338,333,275,325,267,317)(260,339,334,276,326,268,318)(261,340,335,277,327,269,319)(262,341,336,278,328,270,320)(263,342,329,279,321,271,313)(264,343,330,280,322,272,314)(345,397,420,361,412,353,404)(346,398,421,362,413,354,405)(347,399,422,363,414,355,406)(348,400,423,364,415,356,407)(349,393,424,365,416,357,408)(350,394,417,366,409,358,401)(351,395,418,367,410,359,402)(352,396,419,368,411,360,403), (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), (1,146,39,128)(2,151,40,125)(3,148,33,122)(4,145,34,127)(5,150,35,124)(6,147,36,121)(7,152,37,126)(8,149,38,123)(9,218,106,200)(10,223,107,197)(11,220,108,194)(12,217,109,199)(13,222,110,196)(14,219,111,193)(15,224,112,198)(16,221,105,195)(17,205,89,179)(18,202,90,184)(19,207,91,181)(20,204,92,178)(21,201,93,183)(22,206,94,180)(23,203,95,177)(24,208,96,182)(25,213,97,187)(26,210,98,192)(27,215,99,189)(28,212,100,186)(29,209,101,191)(30,214,102,188)(31,211,103,185)(32,216,104,190)(41,130,73,156)(42,135,74,153)(43,132,75,158)(44,129,76,155)(45,134,77,160)(46,131,78,157)(47,136,79,154)(48,133,80,159)(49,138,81,164)(50,143,82,161)(51,140,83,166)(52,137,84,163)(53,142,85,168)(54,139,86,165)(55,144,87,162)(56,141,88,167)(57,240,396,342)(58,237,397,339)(59,234,398,344)(60,239,399,341)(61,236,400,338)(62,233,393,343)(63,238,394,340)(64,235,395,337)(65,227,120,174)(66,232,113,171)(67,229,114,176)(68,226,115,173)(69,231,116,170)(70,228,117,175)(71,225,118,172)(72,230,119,169)(241,401,319,427)(242,406,320,432)(243,403,313,429)(244,408,314,426)(245,405,315,431)(246,402,316,428)(247,407,317,425)(248,404,318,430)(249,409,327,435)(250,414,328,440)(251,411,321,437)(252,416,322,434)(253,413,323,439)(254,410,324,436)(255,415,325,433)(256,412,326,438)(257,372,295,346)(258,369,296,351)(259,374,289,348)(260,371,290,345)(261,376,291,350)(262,373,292,347)(263,370,293,352)(264,375,294,349)(265,380,303,354)(266,377,304,359)(267,382,297,356)(268,379,298,353)(269,384,299,358)(270,381,300,355)(271,378,301,360)(272,383,302,357)(273,388,311,362)(274,385,312,367)(275,390,305,364)(276,387,306,361)(277,392,307,366)(278,389,308,363)(279,386,309,368)(280,391,310,365)(281,418,332,444)(282,423,333,441)(283,420,334,446)(284,417,335,443)(285,422,336,448)(286,419,329,445)(287,424,330,442)(288,421,331,447), (1,259,39,289)(2,262,40,292)(3,257,33,295)(4,260,34,290)(5,263,35,293)(6,258,36,296)(7,261,37,291)(8,264,38,294)(9,334,106,283)(10,329,107,286)(11,332,108,281)(12,335,109,284)(13,330,110,287)(14,333,111,282)(15,336,112,285)(16,331,105,288)(17,315,89,245)(18,318,90,248)(19,313,91,243)(20,316,92,246)(21,319,93,241)(22,314,94,244)(23,317,95,247)(24,320,96,242)(25,323,97,253)(26,326,98,256)(27,321,99,251)(28,324,100,254)(29,327,101,249)(30,322,102,252)(31,325,103,255)(32,328,104,250)(41,303,73,265)(42,298,74,268)(43,301,75,271)(44,304,76,266)(45,299,77,269)(46,302,78,272)(47,297,79,267)(48,300,80,270)(49,311,81,273)(50,306,82,276)(51,309,83,279)(52,312,84,274)(53,307,85,277)(54,310,86,280)(55,305,87,275)(56,308,88,278)(57,229,396,176)(58,232,397,171)(59,227,398,174)(60,230,399,169)(61,225,400,172)(62,228,393,175)(63,231,394,170)(64,226,395,173)(65,344,120,234)(66,339,113,237)(67,342,114,240)(68,337,115,235)(69,340,116,238)(70,343,117,233)(71,338,118,236)(72,341,119,239)(121,369,147,351)(122,372,148,346)(123,375,149,349)(124,370,150,352)(125,373,151,347)(126,376,152,350)(127,371,145,345)(128,374,146,348)(129,377,155,359)(130,380,156,354)(131,383,157,357)(132,378,158,360)(133,381,159,355)(134,384,160,358)(135,379,153,353)(136,382,154,356)(137,385,163,367)(138,388,164,362)(139,391,165,365)(140,386,166,368)(141,389,167,363)(142,392,168,366)(143,387,161,361)(144,390,162,364)(177,425,203,407)(178,428,204,402)(179,431,205,405)(180,426,206,408)(181,429,207,403)(182,432,208,406)(183,427,201,401)(184,430,202,404)(185,433,211,415)(186,436,212,410)(187,439,213,413)(188,434,214,416)(189,437,215,411)(190,440,216,414)(191,435,209,409)(192,438,210,412)(193,441,219,423)(194,444,220,418)(195,447,221,421)(196,442,222,424)(197,445,223,419)(198,448,224,422)(199,443,217,417)(200,446,218,420)>;
G:=Group( (1,71,14,87,31,79,23)(2,72,15,88,32,80,24)(3,65,16,81,25,73,17)(4,66,9,82,26,74,18)(5,67,10,83,27,75,19)(6,68,11,84,28,76,20)(7,69,12,85,29,77,21)(8,70,13,86,30,78,22)(33,120,105,49,97,41,89)(34,113,106,50,98,42,90)(35,114,107,51,99,43,91)(36,115,108,52,100,44,92)(37,116,109,53,101,45,93)(38,117,110,54,102,46,94)(39,118,111,55,103,47,95)(40,119,112,56,104,48,96)(57,445,386,437,378,429,370)(58,446,387,438,379,430,371)(59,447,388,439,380,431,372)(60,448,389,440,381,432,373)(61,441,390,433,382,425,374)(62,442,391,434,383,426,375)(63,443,392,435,384,427,376)(64,444,385,436,377,428,369)(121,173,194,137,186,129,178)(122,174,195,138,187,130,179)(123,175,196,139,188,131,180)(124,176,197,140,189,132,181)(125,169,198,141,190,133,182)(126,170,199,142,191,134,183)(127,171,200,143,192,135,184)(128,172,193,144,185,136,177)(145,232,218,161,210,153,202)(146,225,219,162,211,154,203)(147,226,220,163,212,155,204)(148,227,221,164,213,156,205)(149,228,222,165,214,157,206)(150,229,223,166,215,158,207)(151,230,224,167,216,159,208)(152,231,217,168,209,160,201)(233,287,310,252,302,244,294)(234,288,311,253,303,245,295)(235,281,312,254,304,246,296)(236,282,305,255,297,247,289)(237,283,306,256,298,248,290)(238,284,307,249,299,241,291)(239,285,308,250,300,242,292)(240,286,309,251,301,243,293)(257,344,331,273,323,265,315)(258,337,332,274,324,266,316)(259,338,333,275,325,267,317)(260,339,334,276,326,268,318)(261,340,335,277,327,269,319)(262,341,336,278,328,270,320)(263,342,329,279,321,271,313)(264,343,330,280,322,272,314)(345,397,420,361,412,353,404)(346,398,421,362,413,354,405)(347,399,422,363,414,355,406)(348,400,423,364,415,356,407)(349,393,424,365,416,357,408)(350,394,417,366,409,358,401)(351,395,418,367,410,359,402)(352,396,419,368,411,360,403), (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), (1,146,39,128)(2,151,40,125)(3,148,33,122)(4,145,34,127)(5,150,35,124)(6,147,36,121)(7,152,37,126)(8,149,38,123)(9,218,106,200)(10,223,107,197)(11,220,108,194)(12,217,109,199)(13,222,110,196)(14,219,111,193)(15,224,112,198)(16,221,105,195)(17,205,89,179)(18,202,90,184)(19,207,91,181)(20,204,92,178)(21,201,93,183)(22,206,94,180)(23,203,95,177)(24,208,96,182)(25,213,97,187)(26,210,98,192)(27,215,99,189)(28,212,100,186)(29,209,101,191)(30,214,102,188)(31,211,103,185)(32,216,104,190)(41,130,73,156)(42,135,74,153)(43,132,75,158)(44,129,76,155)(45,134,77,160)(46,131,78,157)(47,136,79,154)(48,133,80,159)(49,138,81,164)(50,143,82,161)(51,140,83,166)(52,137,84,163)(53,142,85,168)(54,139,86,165)(55,144,87,162)(56,141,88,167)(57,240,396,342)(58,237,397,339)(59,234,398,344)(60,239,399,341)(61,236,400,338)(62,233,393,343)(63,238,394,340)(64,235,395,337)(65,227,120,174)(66,232,113,171)(67,229,114,176)(68,226,115,173)(69,231,116,170)(70,228,117,175)(71,225,118,172)(72,230,119,169)(241,401,319,427)(242,406,320,432)(243,403,313,429)(244,408,314,426)(245,405,315,431)(246,402,316,428)(247,407,317,425)(248,404,318,430)(249,409,327,435)(250,414,328,440)(251,411,321,437)(252,416,322,434)(253,413,323,439)(254,410,324,436)(255,415,325,433)(256,412,326,438)(257,372,295,346)(258,369,296,351)(259,374,289,348)(260,371,290,345)(261,376,291,350)(262,373,292,347)(263,370,293,352)(264,375,294,349)(265,380,303,354)(266,377,304,359)(267,382,297,356)(268,379,298,353)(269,384,299,358)(270,381,300,355)(271,378,301,360)(272,383,302,357)(273,388,311,362)(274,385,312,367)(275,390,305,364)(276,387,306,361)(277,392,307,366)(278,389,308,363)(279,386,309,368)(280,391,310,365)(281,418,332,444)(282,423,333,441)(283,420,334,446)(284,417,335,443)(285,422,336,448)(286,419,329,445)(287,424,330,442)(288,421,331,447), (1,259,39,289)(2,262,40,292)(3,257,33,295)(4,260,34,290)(5,263,35,293)(6,258,36,296)(7,261,37,291)(8,264,38,294)(9,334,106,283)(10,329,107,286)(11,332,108,281)(12,335,109,284)(13,330,110,287)(14,333,111,282)(15,336,112,285)(16,331,105,288)(17,315,89,245)(18,318,90,248)(19,313,91,243)(20,316,92,246)(21,319,93,241)(22,314,94,244)(23,317,95,247)(24,320,96,242)(25,323,97,253)(26,326,98,256)(27,321,99,251)(28,324,100,254)(29,327,101,249)(30,322,102,252)(31,325,103,255)(32,328,104,250)(41,303,73,265)(42,298,74,268)(43,301,75,271)(44,304,76,266)(45,299,77,269)(46,302,78,272)(47,297,79,267)(48,300,80,270)(49,311,81,273)(50,306,82,276)(51,309,83,279)(52,312,84,274)(53,307,85,277)(54,310,86,280)(55,305,87,275)(56,308,88,278)(57,229,396,176)(58,232,397,171)(59,227,398,174)(60,230,399,169)(61,225,400,172)(62,228,393,175)(63,231,394,170)(64,226,395,173)(65,344,120,234)(66,339,113,237)(67,342,114,240)(68,337,115,235)(69,340,116,238)(70,343,117,233)(71,338,118,236)(72,341,119,239)(121,369,147,351)(122,372,148,346)(123,375,149,349)(124,370,150,352)(125,373,151,347)(126,376,152,350)(127,371,145,345)(128,374,146,348)(129,377,155,359)(130,380,156,354)(131,383,157,357)(132,378,158,360)(133,381,159,355)(134,384,160,358)(135,379,153,353)(136,382,154,356)(137,385,163,367)(138,388,164,362)(139,391,165,365)(140,386,166,368)(141,389,167,363)(142,392,168,366)(143,387,161,361)(144,390,162,364)(177,425,203,407)(178,428,204,402)(179,431,205,405)(180,426,206,408)(181,429,207,403)(182,432,208,406)(183,427,201,401)(184,430,202,404)(185,433,211,415)(186,436,212,410)(187,439,213,413)(188,434,214,416)(189,437,215,411)(190,440,216,414)(191,435,209,409)(192,438,210,412)(193,441,219,423)(194,444,220,418)(195,447,221,421)(196,442,222,424)(197,445,223,419)(198,448,224,422)(199,443,217,417)(200,446,218,420) );
G=PermutationGroup([[(1,71,14,87,31,79,23),(2,72,15,88,32,80,24),(3,65,16,81,25,73,17),(4,66,9,82,26,74,18),(5,67,10,83,27,75,19),(6,68,11,84,28,76,20),(7,69,12,85,29,77,21),(8,70,13,86,30,78,22),(33,120,105,49,97,41,89),(34,113,106,50,98,42,90),(35,114,107,51,99,43,91),(36,115,108,52,100,44,92),(37,116,109,53,101,45,93),(38,117,110,54,102,46,94),(39,118,111,55,103,47,95),(40,119,112,56,104,48,96),(57,445,386,437,378,429,370),(58,446,387,438,379,430,371),(59,447,388,439,380,431,372),(60,448,389,440,381,432,373),(61,441,390,433,382,425,374),(62,442,391,434,383,426,375),(63,443,392,435,384,427,376),(64,444,385,436,377,428,369),(121,173,194,137,186,129,178),(122,174,195,138,187,130,179),(123,175,196,139,188,131,180),(124,176,197,140,189,132,181),(125,169,198,141,190,133,182),(126,170,199,142,191,134,183),(127,171,200,143,192,135,184),(128,172,193,144,185,136,177),(145,232,218,161,210,153,202),(146,225,219,162,211,154,203),(147,226,220,163,212,155,204),(148,227,221,164,213,156,205),(149,228,222,165,214,157,206),(150,229,223,166,215,158,207),(151,230,224,167,216,159,208),(152,231,217,168,209,160,201),(233,287,310,252,302,244,294),(234,288,311,253,303,245,295),(235,281,312,254,304,246,296),(236,282,305,255,297,247,289),(237,283,306,256,298,248,290),(238,284,307,249,299,241,291),(239,285,308,250,300,242,292),(240,286,309,251,301,243,293),(257,344,331,273,323,265,315),(258,337,332,274,324,266,316),(259,338,333,275,325,267,317),(260,339,334,276,326,268,318),(261,340,335,277,327,269,319),(262,341,336,278,328,270,320),(263,342,329,279,321,271,313),(264,343,330,280,322,272,314),(345,397,420,361,412,353,404),(346,398,421,362,413,354,405),(347,399,422,363,414,355,406),(348,400,423,364,415,356,407),(349,393,424,365,416,357,408),(350,394,417,366,409,358,401),(351,395,418,367,410,359,402),(352,396,419,368,411,360,403)], [(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)], [(1,146,39,128),(2,151,40,125),(3,148,33,122),(4,145,34,127),(5,150,35,124),(6,147,36,121),(7,152,37,126),(8,149,38,123),(9,218,106,200),(10,223,107,197),(11,220,108,194),(12,217,109,199),(13,222,110,196),(14,219,111,193),(15,224,112,198),(16,221,105,195),(17,205,89,179),(18,202,90,184),(19,207,91,181),(20,204,92,178),(21,201,93,183),(22,206,94,180),(23,203,95,177),(24,208,96,182),(25,213,97,187),(26,210,98,192),(27,215,99,189),(28,212,100,186),(29,209,101,191),(30,214,102,188),(31,211,103,185),(32,216,104,190),(41,130,73,156),(42,135,74,153),(43,132,75,158),(44,129,76,155),(45,134,77,160),(46,131,78,157),(47,136,79,154),(48,133,80,159),(49,138,81,164),(50,143,82,161),(51,140,83,166),(52,137,84,163),(53,142,85,168),(54,139,86,165),(55,144,87,162),(56,141,88,167),(57,240,396,342),(58,237,397,339),(59,234,398,344),(60,239,399,341),(61,236,400,338),(62,233,393,343),(63,238,394,340),(64,235,395,337),(65,227,120,174),(66,232,113,171),(67,229,114,176),(68,226,115,173),(69,231,116,170),(70,228,117,175),(71,225,118,172),(72,230,119,169),(241,401,319,427),(242,406,320,432),(243,403,313,429),(244,408,314,426),(245,405,315,431),(246,402,316,428),(247,407,317,425),(248,404,318,430),(249,409,327,435),(250,414,328,440),(251,411,321,437),(252,416,322,434),(253,413,323,439),(254,410,324,436),(255,415,325,433),(256,412,326,438),(257,372,295,346),(258,369,296,351),(259,374,289,348),(260,371,290,345),(261,376,291,350),(262,373,292,347),(263,370,293,352),(264,375,294,349),(265,380,303,354),(266,377,304,359),(267,382,297,356),(268,379,298,353),(269,384,299,358),(270,381,300,355),(271,378,301,360),(272,383,302,357),(273,388,311,362),(274,385,312,367),(275,390,305,364),(276,387,306,361),(277,392,307,366),(278,389,308,363),(279,386,309,368),(280,391,310,365),(281,418,332,444),(282,423,333,441),(283,420,334,446),(284,417,335,443),(285,422,336,448),(286,419,329,445),(287,424,330,442),(288,421,331,447)], [(1,259,39,289),(2,262,40,292),(3,257,33,295),(4,260,34,290),(5,263,35,293),(6,258,36,296),(7,261,37,291),(8,264,38,294),(9,334,106,283),(10,329,107,286),(11,332,108,281),(12,335,109,284),(13,330,110,287),(14,333,111,282),(15,336,112,285),(16,331,105,288),(17,315,89,245),(18,318,90,248),(19,313,91,243),(20,316,92,246),(21,319,93,241),(22,314,94,244),(23,317,95,247),(24,320,96,242),(25,323,97,253),(26,326,98,256),(27,321,99,251),(28,324,100,254),(29,327,101,249),(30,322,102,252),(31,325,103,255),(32,328,104,250),(41,303,73,265),(42,298,74,268),(43,301,75,271),(44,304,76,266),(45,299,77,269),(46,302,78,272),(47,297,79,267),(48,300,80,270),(49,311,81,273),(50,306,82,276),(51,309,83,279),(52,312,84,274),(53,307,85,277),(54,310,86,280),(55,305,87,275),(56,308,88,278),(57,229,396,176),(58,232,397,171),(59,227,398,174),(60,230,399,169),(61,225,400,172),(62,228,393,175),(63,231,394,170),(64,226,395,173),(65,344,120,234),(66,339,113,237),(67,342,114,240),(68,337,115,235),(69,340,116,238),(70,343,117,233),(71,338,118,236),(72,341,119,239),(121,369,147,351),(122,372,148,346),(123,375,149,349),(124,370,150,352),(125,373,151,347),(126,376,152,350),(127,371,145,345),(128,374,146,348),(129,377,155,359),(130,380,156,354),(131,383,157,357),(132,378,158,360),(133,381,159,355),(134,384,160,358),(135,379,153,353),(136,382,154,356),(137,385,163,367),(138,388,164,362),(139,391,165,365),(140,386,166,368),(141,389,167,363),(142,392,168,366),(143,387,161,361),(144,390,162,364),(177,425,203,407),(178,428,204,402),(179,431,205,405),(180,426,206,408),(181,429,207,403),(182,432,208,406),(183,427,201,401),(184,430,202,404),(185,433,211,415),(186,436,212,410),(187,439,213,413),(188,434,214,416),(189,437,215,411),(190,440,216,414),(191,435,209,409),(192,438,210,412),(193,441,219,423),(194,444,220,418),(195,447,221,421),(196,442,222,424),(197,445,223,419),(198,448,224,422),(199,443,217,417),(200,446,218,420)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 28A | ··· | 28L | 28M | ··· | 28X | 28Y | ··· | 28AV | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | Q8 | D4 | C7×Q8 | C7×D4 | C8⋊C22 | C8.C22 | C7×C8⋊C22 | C7×C8.C22 |
| kernel | C7×C8⋊Q8 | C7×C8⋊C4 | C7×C4.Q8 | C7×C2.D8 | C7×C42.C2 | C7×C4⋊Q8 | C8⋊Q8 | C8⋊C4 | C4.Q8 | C2.D8 | C42.C2 | C4⋊Q8 | C56 | C2×C28 | C8 | C2×C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 4 | 2 | 24 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C7×C8⋊Q8 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 0 | 0 | 0 |
| 0 | 0 | 0 | 106 | 0 | 0 |
| 0 | 0 | 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 0 | 0 | 106 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 6 | 80 | 55 |
| 0 | 0 | 107 | 27 | 58 | 80 |
| 0 | 0 | 55 | 33 | 86 | 107 |
| 0 | 0 | 80 | 55 | 6 | 86 |
| 58 | 108 | 0 | 0 | 0 | 0 |
| 108 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 74 | 44 | 63 | 74 |
| 0 | 0 | 44 | 39 | 74 | 50 |
| 0 | 0 | 50 | 39 | 69 | 74 |
| 0 | 0 | 39 | 63 | 74 | 44 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,27,107,55,80,0,0,6,27,33,55,0,0,80,58,86,6,0,0,55,80,107,86],[58,108,0,0,0,0,108,55,0,0,0,0,0,0,74,44,50,39,0,0,44,39,39,63,0,0,63,74,69,74,0,0,74,50,74,44] >;
C7×C8⋊Q8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes Q_8
% in TeX
G:=Group("C7xC8:Q8"); // GroupNames label
G:=SmallGroup(448,909);
// by ID
G=gap.SmallGroup(448,909);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,400,2438,2403,604,14117,124]);
// 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,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations