direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×C7⋊C8, C42.209D14, C7⋊3(C8×Q8), (C7×Q8)⋊2C8, C4.57(Q8×D7), C28.14(C2×C8), (Q8×C28).5C2, (C4×Q8).13D7, (Q8×C14).8C4, C14.26(C4×Q8), C2.2(Q8×Dic7), C4⋊C4.12Dic7, C28.115(C2×Q8), (C2×Q8).9Dic7, C28⋊C8.17C2, C14.41(C8○D4), C14.25(C22×C8), (C4×C28).94C22, C28.338(C4○D4), (C2×C28).851C23, C4.58(Q8⋊2D7), C2.3(Q8.Dic7), C22.23(C22×Dic7), C4.4(C2×C7⋊C8), (C4×C7⋊C8).7C2, C2.6(C22×C7⋊C8), (C7×C4⋊C4).13C4, (C2×C28).165(C2×C4), (C2×C7⋊C8).316C22, (C2×C4).44(C2×Dic7), (C2×C4).793(C22×D7), (C2×C14).188(C22×C4), SmallGroup(448,557)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C7⋊C8
G = < a,b,c,d | a4=c7=d8=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 228 in 102 conjugacy classes, 77 normal (22 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, C28, C28, C28, C2×C14, C4×C8, C4⋊C8, C4×Q8, C7⋊C8, C7⋊C8, C2×C28, C2×C28, C7×Q8, C8×Q8, C2×C7⋊C8, C2×C7⋊C8, C4×C28, C7×C4⋊C4, Q8×C14, C4×C7⋊C8, C28⋊C8, Q8×C28, Q8×C7⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, D7, C2×C8, C22×C4, C2×Q8, C4○D4, Dic7, D14, C4×Q8, C22×C8, C8○D4, C7⋊C8, C2×Dic7, C22×D7, C8×Q8, C2×C7⋊C8, Q8×D7, Q8⋊2D7, C22×Dic7, C22×C7⋊C8, Q8×Dic7, Q8.Dic7, Q8×C7⋊C8
►Regular action on 448 points
Generators in S448
(1 69 237 165)(2 70 238 166)(3 71 239 167)(4 72 240 168)(5 65 233 161)(6 66 234 162)(7 67 235 163)(8 68 236 164)(9 203 136 277)(10 204 129 278)(11 205 130 279)(12 206 131 280)(13 207 132 273)(14 208 133 274)(15 201 134 275)(16 202 135 276)(17 404 357 92)(18 405 358 93)(19 406 359 94)(20 407 360 95)(21 408 353 96)(22 401 354 89)(23 402 355 90)(24 403 356 91)(25 110 45 200)(26 111 46 193)(27 112 47 194)(28 105 48 195)(29 106 41 196)(30 107 42 197)(31 108 43 198)(32 109 44 199)(33 78 145 410)(34 79 146 411)(35 80 147 412)(36 73 148 413)(37 74 149 414)(38 75 150 415)(39 76 151 416)(40 77 152 409)(49 396 139 313)(50 397 140 314)(51 398 141 315)(52 399 142 316)(53 400 143 317)(54 393 144 318)(55 394 137 319)(56 395 138 320)(57 241 392 160)(58 242 385 153)(59 243 386 154)(60 244 387 155)(61 245 388 156)(62 246 389 157)(63 247 390 158)(64 248 391 159)(81 282 174 365)(82 283 175 366)(83 284 176 367)(84 285 169 368)(85 286 170 361)(86 287 171 362)(87 288 172 363)(88 281 173 364)(97 329 188 259)(98 330 189 260)(99 331 190 261)(100 332 191 262)(101 333 192 263)(102 334 185 264)(103 335 186 257)(104 336 187 258)(113 216 448 312)(114 209 441 305)(115 210 442 306)(116 211 443 307)(117 212 444 308)(118 213 445 309)(119 214 446 310)(120 215 447 311)(121 425 217 325)(122 426 218 326)(123 427 219 327)(124 428 220 328)(125 429 221 321)(126 430 222 322)(127 431 223 323)(128 432 224 324)(177 381 295 267)(178 382 296 268)(179 383 289 269)(180 384 290 270)(181 377 291 271)(182 378 292 272)(183 379 293 265)(184 380 294 266)(225 337 370 420)(226 338 371 421)(227 339 372 422)(228 340 373 423)(229 341 374 424)(230 342 375 417)(231 343 376 418)(232 344 369 419)(249 303 350 436)(250 304 351 437)(251 297 352 438)(252 298 345 439)(253 299 346 440)(254 300 347 433)(255 301 348 434)(256 302 349 435)
(1 325 237 425)(2 326 238 426)(3 327 239 427)(4 328 240 428)(5 321 233 429)(6 322 234 430)(7 323 235 431)(8 324 236 432)(9 188 136 97)(10 189 129 98)(11 190 130 99)(12 191 131 100)(13 192 132 101)(14 185 133 102)(15 186 134 103)(16 187 135 104)(17 409 357 77)(18 410 358 78)(19 411 359 79)(20 412 360 80)(21 413 353 73)(22 414 354 74)(23 415 355 75)(24 416 356 76)(25 397 45 314)(26 398 46 315)(27 399 47 316)(28 400 48 317)(29 393 41 318)(30 394 42 319)(31 395 43 320)(32 396 44 313)(33 405 145 93)(34 406 146 94)(35 407 147 95)(36 408 148 96)(37 401 149 89)(38 402 150 90)(39 403 151 91)(40 404 152 92)(49 199 139 109)(50 200 140 110)(51 193 141 111)(52 194 142 112)(53 195 143 105)(54 196 144 106)(55 197 137 107)(56 198 138 108)(57 444 392 117)(58 445 385 118)(59 446 386 119)(60 447 387 120)(61 448 388 113)(62 441 389 114)(63 442 390 115)(64 443 391 116)(65 221 161 125)(66 222 162 126)(67 223 163 127)(68 224 164 128)(69 217 165 121)(70 218 166 122)(71 219 167 123)(72 220 168 124)(81 230 174 375)(82 231 175 376)(83 232 176 369)(84 225 169 370)(85 226 170 371)(86 227 171 372)(87 228 172 373)(88 229 173 374)(153 309 242 213)(154 310 243 214)(155 311 244 215)(156 312 245 216)(157 305 246 209)(158 306 247 210)(159 307 248 211)(160 308 241 212)(177 347 295 254)(178 348 296 255)(179 349 289 256)(180 350 290 249)(181 351 291 250)(182 352 292 251)(183 345 293 252)(184 346 294 253)(201 335 275 257)(202 336 276 258)(203 329 277 259)(204 330 278 260)(205 331 279 261)(206 332 280 262)(207 333 273 263)(208 334 274 264)(265 439 379 298)(266 440 380 299)(267 433 381 300)(268 434 382 301)(269 435 383 302)(270 436 384 303)(271 437 377 304)(272 438 378 297)(281 424 364 341)(282 417 365 342)(283 418 366 343)(284 419 367 344)(285 420 368 337)(286 421 361 338)(287 422 362 339)(288 423 363 340)
(1 242 267 143 356 284 13)(2 14 285 357 144 268 243)(3 244 269 137 358 286 15)(4 16 287 359 138 270 245)(5 246 271 139 360 288 9)(6 10 281 353 140 272 247)(7 248 265 141 354 282 11)(8 12 283 355 142 266 241)(17 54 382 154 238 133 368)(18 361 134 239 155 383 55)(19 56 384 156 240 135 362)(20 363 136 233 157 377 49)(21 50 378 158 234 129 364)(22 365 130 235 159 379 51)(23 52 380 160 236 131 366)(24 367 132 237 153 381 53)(25 352 115 222 330 374 36)(26 37 375 331 223 116 345)(27 346 117 224 332 376 38)(28 39 369 333 217 118 347)(29 348 119 218 334 370 40)(30 33 371 335 219 120 349)(31 350 113 220 336 372 34)(32 35 373 329 221 114 351)(41 255 446 122 264 225 152)(42 145 226 257 123 447 256)(43 249 448 124 258 227 146)(44 147 228 259 125 441 250)(45 251 442 126 260 229 148)(46 149 230 261 127 443 252)(47 253 444 128 262 231 150)(48 151 232 263 121 445 254)(57 164 280 82 402 399 294)(58 295 400 403 83 273 165)(59 166 274 84 404 393 296)(60 289 394 405 85 275 167)(61 168 276 86 406 395 290)(62 291 396 407 87 277 161)(63 162 278 88 408 397 292)(64 293 398 401 81 279 163)(65 389 181 313 95 172 203)(66 204 173 96 314 182 390)(67 391 183 315 89 174 205)(68 206 175 90 316 184 392)(69 385 177 317 91 176 207)(70 208 169 92 318 178 386)(71 387 179 319 93 170 201)(72 202 171 94 320 180 388)(73 110 438 210 322 189 424)(74 417 190 323 211 439 111)(75 112 440 212 324 191 418)(76 419 192 325 213 433 105)(77 106 434 214 326 185 420)(78 421 186 327 215 435 107)(79 108 436 216 328 187 422)(80 423 188 321 209 437 109)(97 429 305 304 199 412 340)(98 341 413 200 297 306 430)(99 431 307 298 193 414 342)(100 343 415 194 299 308 432)(101 425 309 300 195 416 344)(102 337 409 196 301 310 426)(103 427 311 302 197 410 338)(104 339 411 198 303 312 428)
(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)
G:=sub<Sym(448)| (1,69,237,165)(2,70,238,166)(3,71,239,167)(4,72,240,168)(5,65,233,161)(6,66,234,162)(7,67,235,163)(8,68,236,164)(9,203,136,277)(10,204,129,278)(11,205,130,279)(12,206,131,280)(13,207,132,273)(14,208,133,274)(15,201,134,275)(16,202,135,276)(17,404,357,92)(18,405,358,93)(19,406,359,94)(20,407,360,95)(21,408,353,96)(22,401,354,89)(23,402,355,90)(24,403,356,91)(25,110,45,200)(26,111,46,193)(27,112,47,194)(28,105,48,195)(29,106,41,196)(30,107,42,197)(31,108,43,198)(32,109,44,199)(33,78,145,410)(34,79,146,411)(35,80,147,412)(36,73,148,413)(37,74,149,414)(38,75,150,415)(39,76,151,416)(40,77,152,409)(49,396,139,313)(50,397,140,314)(51,398,141,315)(52,399,142,316)(53,400,143,317)(54,393,144,318)(55,394,137,319)(56,395,138,320)(57,241,392,160)(58,242,385,153)(59,243,386,154)(60,244,387,155)(61,245,388,156)(62,246,389,157)(63,247,390,158)(64,248,391,159)(81,282,174,365)(82,283,175,366)(83,284,176,367)(84,285,169,368)(85,286,170,361)(86,287,171,362)(87,288,172,363)(88,281,173,364)(97,329,188,259)(98,330,189,260)(99,331,190,261)(100,332,191,262)(101,333,192,263)(102,334,185,264)(103,335,186,257)(104,336,187,258)(113,216,448,312)(114,209,441,305)(115,210,442,306)(116,211,443,307)(117,212,444,308)(118,213,445,309)(119,214,446,310)(120,215,447,311)(121,425,217,325)(122,426,218,326)(123,427,219,327)(124,428,220,328)(125,429,221,321)(126,430,222,322)(127,431,223,323)(128,432,224,324)(177,381,295,267)(178,382,296,268)(179,383,289,269)(180,384,290,270)(181,377,291,271)(182,378,292,272)(183,379,293,265)(184,380,294,266)(225,337,370,420)(226,338,371,421)(227,339,372,422)(228,340,373,423)(229,341,374,424)(230,342,375,417)(231,343,376,418)(232,344,369,419)(249,303,350,436)(250,304,351,437)(251,297,352,438)(252,298,345,439)(253,299,346,440)(254,300,347,433)(255,301,348,434)(256,302,349,435), (1,325,237,425)(2,326,238,426)(3,327,239,427)(4,328,240,428)(5,321,233,429)(6,322,234,430)(7,323,235,431)(8,324,236,432)(9,188,136,97)(10,189,129,98)(11,190,130,99)(12,191,131,100)(13,192,132,101)(14,185,133,102)(15,186,134,103)(16,187,135,104)(17,409,357,77)(18,410,358,78)(19,411,359,79)(20,412,360,80)(21,413,353,73)(22,414,354,74)(23,415,355,75)(24,416,356,76)(25,397,45,314)(26,398,46,315)(27,399,47,316)(28,400,48,317)(29,393,41,318)(30,394,42,319)(31,395,43,320)(32,396,44,313)(33,405,145,93)(34,406,146,94)(35,407,147,95)(36,408,148,96)(37,401,149,89)(38,402,150,90)(39,403,151,91)(40,404,152,92)(49,199,139,109)(50,200,140,110)(51,193,141,111)(52,194,142,112)(53,195,143,105)(54,196,144,106)(55,197,137,107)(56,198,138,108)(57,444,392,117)(58,445,385,118)(59,446,386,119)(60,447,387,120)(61,448,388,113)(62,441,389,114)(63,442,390,115)(64,443,391,116)(65,221,161,125)(66,222,162,126)(67,223,163,127)(68,224,164,128)(69,217,165,121)(70,218,166,122)(71,219,167,123)(72,220,168,124)(81,230,174,375)(82,231,175,376)(83,232,176,369)(84,225,169,370)(85,226,170,371)(86,227,171,372)(87,228,172,373)(88,229,173,374)(153,309,242,213)(154,310,243,214)(155,311,244,215)(156,312,245,216)(157,305,246,209)(158,306,247,210)(159,307,248,211)(160,308,241,212)(177,347,295,254)(178,348,296,255)(179,349,289,256)(180,350,290,249)(181,351,291,250)(182,352,292,251)(183,345,293,252)(184,346,294,253)(201,335,275,257)(202,336,276,258)(203,329,277,259)(204,330,278,260)(205,331,279,261)(206,332,280,262)(207,333,273,263)(208,334,274,264)(265,439,379,298)(266,440,380,299)(267,433,381,300)(268,434,382,301)(269,435,383,302)(270,436,384,303)(271,437,377,304)(272,438,378,297)(281,424,364,341)(282,417,365,342)(283,418,366,343)(284,419,367,344)(285,420,368,337)(286,421,361,338)(287,422,362,339)(288,423,363,340), (1,242,267,143,356,284,13)(2,14,285,357,144,268,243)(3,244,269,137,358,286,15)(4,16,287,359,138,270,245)(5,246,271,139,360,288,9)(6,10,281,353,140,272,247)(7,248,265,141,354,282,11)(8,12,283,355,142,266,241)(17,54,382,154,238,133,368)(18,361,134,239,155,383,55)(19,56,384,156,240,135,362)(20,363,136,233,157,377,49)(21,50,378,158,234,129,364)(22,365,130,235,159,379,51)(23,52,380,160,236,131,366)(24,367,132,237,153,381,53)(25,352,115,222,330,374,36)(26,37,375,331,223,116,345)(27,346,117,224,332,376,38)(28,39,369,333,217,118,347)(29,348,119,218,334,370,40)(30,33,371,335,219,120,349)(31,350,113,220,336,372,34)(32,35,373,329,221,114,351)(41,255,446,122,264,225,152)(42,145,226,257,123,447,256)(43,249,448,124,258,227,146)(44,147,228,259,125,441,250)(45,251,442,126,260,229,148)(46,149,230,261,127,443,252)(47,253,444,128,262,231,150)(48,151,232,263,121,445,254)(57,164,280,82,402,399,294)(58,295,400,403,83,273,165)(59,166,274,84,404,393,296)(60,289,394,405,85,275,167)(61,168,276,86,406,395,290)(62,291,396,407,87,277,161)(63,162,278,88,408,397,292)(64,293,398,401,81,279,163)(65,389,181,313,95,172,203)(66,204,173,96,314,182,390)(67,391,183,315,89,174,205)(68,206,175,90,316,184,392)(69,385,177,317,91,176,207)(70,208,169,92,318,178,386)(71,387,179,319,93,170,201)(72,202,171,94,320,180,388)(73,110,438,210,322,189,424)(74,417,190,323,211,439,111)(75,112,440,212,324,191,418)(76,419,192,325,213,433,105)(77,106,434,214,326,185,420)(78,421,186,327,215,435,107)(79,108,436,216,328,187,422)(80,423,188,321,209,437,109)(97,429,305,304,199,412,340)(98,341,413,200,297,306,430)(99,431,307,298,193,414,342)(100,343,415,194,299,308,432)(101,425,309,300,195,416,344)(102,337,409,196,301,310,426)(103,427,311,302,197,410,338)(104,339,411,198,303,312,428), (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)>;
G:=Group( (1,69,237,165)(2,70,238,166)(3,71,239,167)(4,72,240,168)(5,65,233,161)(6,66,234,162)(7,67,235,163)(8,68,236,164)(9,203,136,277)(10,204,129,278)(11,205,130,279)(12,206,131,280)(13,207,132,273)(14,208,133,274)(15,201,134,275)(16,202,135,276)(17,404,357,92)(18,405,358,93)(19,406,359,94)(20,407,360,95)(21,408,353,96)(22,401,354,89)(23,402,355,90)(24,403,356,91)(25,110,45,200)(26,111,46,193)(27,112,47,194)(28,105,48,195)(29,106,41,196)(30,107,42,197)(31,108,43,198)(32,109,44,199)(33,78,145,410)(34,79,146,411)(35,80,147,412)(36,73,148,413)(37,74,149,414)(38,75,150,415)(39,76,151,416)(40,77,152,409)(49,396,139,313)(50,397,140,314)(51,398,141,315)(52,399,142,316)(53,400,143,317)(54,393,144,318)(55,394,137,319)(56,395,138,320)(57,241,392,160)(58,242,385,153)(59,243,386,154)(60,244,387,155)(61,245,388,156)(62,246,389,157)(63,247,390,158)(64,248,391,159)(81,282,174,365)(82,283,175,366)(83,284,176,367)(84,285,169,368)(85,286,170,361)(86,287,171,362)(87,288,172,363)(88,281,173,364)(97,329,188,259)(98,330,189,260)(99,331,190,261)(100,332,191,262)(101,333,192,263)(102,334,185,264)(103,335,186,257)(104,336,187,258)(113,216,448,312)(114,209,441,305)(115,210,442,306)(116,211,443,307)(117,212,444,308)(118,213,445,309)(119,214,446,310)(120,215,447,311)(121,425,217,325)(122,426,218,326)(123,427,219,327)(124,428,220,328)(125,429,221,321)(126,430,222,322)(127,431,223,323)(128,432,224,324)(177,381,295,267)(178,382,296,268)(179,383,289,269)(180,384,290,270)(181,377,291,271)(182,378,292,272)(183,379,293,265)(184,380,294,266)(225,337,370,420)(226,338,371,421)(227,339,372,422)(228,340,373,423)(229,341,374,424)(230,342,375,417)(231,343,376,418)(232,344,369,419)(249,303,350,436)(250,304,351,437)(251,297,352,438)(252,298,345,439)(253,299,346,440)(254,300,347,433)(255,301,348,434)(256,302,349,435), (1,325,237,425)(2,326,238,426)(3,327,239,427)(4,328,240,428)(5,321,233,429)(6,322,234,430)(7,323,235,431)(8,324,236,432)(9,188,136,97)(10,189,129,98)(11,190,130,99)(12,191,131,100)(13,192,132,101)(14,185,133,102)(15,186,134,103)(16,187,135,104)(17,409,357,77)(18,410,358,78)(19,411,359,79)(20,412,360,80)(21,413,353,73)(22,414,354,74)(23,415,355,75)(24,416,356,76)(25,397,45,314)(26,398,46,315)(27,399,47,316)(28,400,48,317)(29,393,41,318)(30,394,42,319)(31,395,43,320)(32,396,44,313)(33,405,145,93)(34,406,146,94)(35,407,147,95)(36,408,148,96)(37,401,149,89)(38,402,150,90)(39,403,151,91)(40,404,152,92)(49,199,139,109)(50,200,140,110)(51,193,141,111)(52,194,142,112)(53,195,143,105)(54,196,144,106)(55,197,137,107)(56,198,138,108)(57,444,392,117)(58,445,385,118)(59,446,386,119)(60,447,387,120)(61,448,388,113)(62,441,389,114)(63,442,390,115)(64,443,391,116)(65,221,161,125)(66,222,162,126)(67,223,163,127)(68,224,164,128)(69,217,165,121)(70,218,166,122)(71,219,167,123)(72,220,168,124)(81,230,174,375)(82,231,175,376)(83,232,176,369)(84,225,169,370)(85,226,170,371)(86,227,171,372)(87,228,172,373)(88,229,173,374)(153,309,242,213)(154,310,243,214)(155,311,244,215)(156,312,245,216)(157,305,246,209)(158,306,247,210)(159,307,248,211)(160,308,241,212)(177,347,295,254)(178,348,296,255)(179,349,289,256)(180,350,290,249)(181,351,291,250)(182,352,292,251)(183,345,293,252)(184,346,294,253)(201,335,275,257)(202,336,276,258)(203,329,277,259)(204,330,278,260)(205,331,279,261)(206,332,280,262)(207,333,273,263)(208,334,274,264)(265,439,379,298)(266,440,380,299)(267,433,381,300)(268,434,382,301)(269,435,383,302)(270,436,384,303)(271,437,377,304)(272,438,378,297)(281,424,364,341)(282,417,365,342)(283,418,366,343)(284,419,367,344)(285,420,368,337)(286,421,361,338)(287,422,362,339)(288,423,363,340), (1,242,267,143,356,284,13)(2,14,285,357,144,268,243)(3,244,269,137,358,286,15)(4,16,287,359,138,270,245)(5,246,271,139,360,288,9)(6,10,281,353,140,272,247)(7,248,265,141,354,282,11)(8,12,283,355,142,266,241)(17,54,382,154,238,133,368)(18,361,134,239,155,383,55)(19,56,384,156,240,135,362)(20,363,136,233,157,377,49)(21,50,378,158,234,129,364)(22,365,130,235,159,379,51)(23,52,380,160,236,131,366)(24,367,132,237,153,381,53)(25,352,115,222,330,374,36)(26,37,375,331,223,116,345)(27,346,117,224,332,376,38)(28,39,369,333,217,118,347)(29,348,119,218,334,370,40)(30,33,371,335,219,120,349)(31,350,113,220,336,372,34)(32,35,373,329,221,114,351)(41,255,446,122,264,225,152)(42,145,226,257,123,447,256)(43,249,448,124,258,227,146)(44,147,228,259,125,441,250)(45,251,442,126,260,229,148)(46,149,230,261,127,443,252)(47,253,444,128,262,231,150)(48,151,232,263,121,445,254)(57,164,280,82,402,399,294)(58,295,400,403,83,273,165)(59,166,274,84,404,393,296)(60,289,394,405,85,275,167)(61,168,276,86,406,395,290)(62,291,396,407,87,277,161)(63,162,278,88,408,397,292)(64,293,398,401,81,279,163)(65,389,181,313,95,172,203)(66,204,173,96,314,182,390)(67,391,183,315,89,174,205)(68,206,175,90,316,184,392)(69,385,177,317,91,176,207)(70,208,169,92,318,178,386)(71,387,179,319,93,170,201)(72,202,171,94,320,180,388)(73,110,438,210,322,189,424)(74,417,190,323,211,439,111)(75,112,440,212,324,191,418)(76,419,192,325,213,433,105)(77,106,434,214,326,185,420)(78,421,186,327,215,435,107)(79,108,436,216,328,187,422)(80,423,188,321,209,437,109)(97,429,305,304,199,412,340)(98,341,413,200,297,306,430)(99,431,307,298,193,414,342)(100,343,415,194,299,308,432)(101,425,309,300,195,416,344)(102,337,409,196,301,310,426)(103,427,311,302,197,410,338)(104,339,411,198,303,312,428), (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) );
G=PermutationGroup([[(1,69,237,165),(2,70,238,166),(3,71,239,167),(4,72,240,168),(5,65,233,161),(6,66,234,162),(7,67,235,163),(8,68,236,164),(9,203,136,277),(10,204,129,278),(11,205,130,279),(12,206,131,280),(13,207,132,273),(14,208,133,274),(15,201,134,275),(16,202,135,276),(17,404,357,92),(18,405,358,93),(19,406,359,94),(20,407,360,95),(21,408,353,96),(22,401,354,89),(23,402,355,90),(24,403,356,91),(25,110,45,200),(26,111,46,193),(27,112,47,194),(28,105,48,195),(29,106,41,196),(30,107,42,197),(31,108,43,198),(32,109,44,199),(33,78,145,410),(34,79,146,411),(35,80,147,412),(36,73,148,413),(37,74,149,414),(38,75,150,415),(39,76,151,416),(40,77,152,409),(49,396,139,313),(50,397,140,314),(51,398,141,315),(52,399,142,316),(53,400,143,317),(54,393,144,318),(55,394,137,319),(56,395,138,320),(57,241,392,160),(58,242,385,153),(59,243,386,154),(60,244,387,155),(61,245,388,156),(62,246,389,157),(63,247,390,158),(64,248,391,159),(81,282,174,365),(82,283,175,366),(83,284,176,367),(84,285,169,368),(85,286,170,361),(86,287,171,362),(87,288,172,363),(88,281,173,364),(97,329,188,259),(98,330,189,260),(99,331,190,261),(100,332,191,262),(101,333,192,263),(102,334,185,264),(103,335,186,257),(104,336,187,258),(113,216,448,312),(114,209,441,305),(115,210,442,306),(116,211,443,307),(117,212,444,308),(118,213,445,309),(119,214,446,310),(120,215,447,311),(121,425,217,325),(122,426,218,326),(123,427,219,327),(124,428,220,328),(125,429,221,321),(126,430,222,322),(127,431,223,323),(128,432,224,324),(177,381,295,267),(178,382,296,268),(179,383,289,269),(180,384,290,270),(181,377,291,271),(182,378,292,272),(183,379,293,265),(184,380,294,266),(225,337,370,420),(226,338,371,421),(227,339,372,422),(228,340,373,423),(229,341,374,424),(230,342,375,417),(231,343,376,418),(232,344,369,419),(249,303,350,436),(250,304,351,437),(251,297,352,438),(252,298,345,439),(253,299,346,440),(254,300,347,433),(255,301,348,434),(256,302,349,435)], [(1,325,237,425),(2,326,238,426),(3,327,239,427),(4,328,240,428),(5,321,233,429),(6,322,234,430),(7,323,235,431),(8,324,236,432),(9,188,136,97),(10,189,129,98),(11,190,130,99),(12,191,131,100),(13,192,132,101),(14,185,133,102),(15,186,134,103),(16,187,135,104),(17,409,357,77),(18,410,358,78),(19,411,359,79),(20,412,360,80),(21,413,353,73),(22,414,354,74),(23,415,355,75),(24,416,356,76),(25,397,45,314),(26,398,46,315),(27,399,47,316),(28,400,48,317),(29,393,41,318),(30,394,42,319),(31,395,43,320),(32,396,44,313),(33,405,145,93),(34,406,146,94),(35,407,147,95),(36,408,148,96),(37,401,149,89),(38,402,150,90),(39,403,151,91),(40,404,152,92),(49,199,139,109),(50,200,140,110),(51,193,141,111),(52,194,142,112),(53,195,143,105),(54,196,144,106),(55,197,137,107),(56,198,138,108),(57,444,392,117),(58,445,385,118),(59,446,386,119),(60,447,387,120),(61,448,388,113),(62,441,389,114),(63,442,390,115),(64,443,391,116),(65,221,161,125),(66,222,162,126),(67,223,163,127),(68,224,164,128),(69,217,165,121),(70,218,166,122),(71,219,167,123),(72,220,168,124),(81,230,174,375),(82,231,175,376),(83,232,176,369),(84,225,169,370),(85,226,170,371),(86,227,171,372),(87,228,172,373),(88,229,173,374),(153,309,242,213),(154,310,243,214),(155,311,244,215),(156,312,245,216),(157,305,246,209),(158,306,247,210),(159,307,248,211),(160,308,241,212),(177,347,295,254),(178,348,296,255),(179,349,289,256),(180,350,290,249),(181,351,291,250),(182,352,292,251),(183,345,293,252),(184,346,294,253),(201,335,275,257),(202,336,276,258),(203,329,277,259),(204,330,278,260),(205,331,279,261),(206,332,280,262),(207,333,273,263),(208,334,274,264),(265,439,379,298),(266,440,380,299),(267,433,381,300),(268,434,382,301),(269,435,383,302),(270,436,384,303),(271,437,377,304),(272,438,378,297),(281,424,364,341),(282,417,365,342),(283,418,366,343),(284,419,367,344),(285,420,368,337),(286,421,361,338),(287,422,362,339),(288,423,363,340)], [(1,242,267,143,356,284,13),(2,14,285,357,144,268,243),(3,244,269,137,358,286,15),(4,16,287,359,138,270,245),(5,246,271,139,360,288,9),(6,10,281,353,140,272,247),(7,248,265,141,354,282,11),(8,12,283,355,142,266,241),(17,54,382,154,238,133,368),(18,361,134,239,155,383,55),(19,56,384,156,240,135,362),(20,363,136,233,157,377,49),(21,50,378,158,234,129,364),(22,365,130,235,159,379,51),(23,52,380,160,236,131,366),(24,367,132,237,153,381,53),(25,352,115,222,330,374,36),(26,37,375,331,223,116,345),(27,346,117,224,332,376,38),(28,39,369,333,217,118,347),(29,348,119,218,334,370,40),(30,33,371,335,219,120,349),(31,350,113,220,336,372,34),(32,35,373,329,221,114,351),(41,255,446,122,264,225,152),(42,145,226,257,123,447,256),(43,249,448,124,258,227,146),(44,147,228,259,125,441,250),(45,251,442,126,260,229,148),(46,149,230,261,127,443,252),(47,253,444,128,262,231,150),(48,151,232,263,121,445,254),(57,164,280,82,402,399,294),(58,295,400,403,83,273,165),(59,166,274,84,404,393,296),(60,289,394,405,85,275,167),(61,168,276,86,406,395,290),(62,291,396,407,87,277,161),(63,162,278,88,408,397,292),(64,293,398,401,81,279,163),(65,389,181,313,95,172,203),(66,204,173,96,314,182,390),(67,391,183,315,89,174,205),(68,206,175,90,316,184,392),(69,385,177,317,91,176,207),(70,208,169,92,318,178,386),(71,387,179,319,93,170,201),(72,202,171,94,320,180,388),(73,110,438,210,322,189,424),(74,417,190,323,211,439,111),(75,112,440,212,324,191,418),(76,419,192,325,213,433,105),(77,106,434,214,326,185,420),(78,421,186,327,215,435,107),(79,108,436,216,328,187,422),(80,423,188,321,209,437,109),(97,429,305,304,199,412,340),(98,341,413,200,297,306,430),(99,431,307,298,193,414,342),(100,343,415,194,299,308,432),(101,425,309,300,195,416,344),(102,337,409,196,301,310,426),(103,427,311,302,197,410,338),(104,339,411,198,303,312,428)], [(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)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | Q8 | D7 | C4○D4 | D14 | Dic7 | Dic7 | C8○D4 | C7⋊C8 | Q8×D7 | Q8⋊2D7 | Q8.Dic7 |
| kernel | Q8×C7⋊C8 | C4×C7⋊C8 | C28⋊C8 | Q8×C28 | C7×C4⋊C4 | Q8×C14 | C7×Q8 | C7⋊C8 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 16 | 2 | 3 | 2 | 9 | 9 | 3 | 4 | 24 | 3 | 3 | 6 |
Matrix representation of Q8×C7⋊C8 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 32 |
| 0 | 0 | 7 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 76 | 35 |
| 0 | 0 | 90 | 37 |
| 24 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 80 | 0 | 0 |
| 46 | 105 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,7,0,0,32,112],[1,0,0,0,0,1,0,0,0,0,76,90,0,0,35,37],[24,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[8,46,0,0,80,105,0,0,0,0,112,0,0,0,0,112] >;
Q8×C7⋊C8 in GAP, Magma, Sage, TeX
Q_8\times C_7\rtimes C_8
% in TeX
G:=Group("Q8xC7:C8"); // GroupNames label
G:=SmallGroup(448,557);
// by ID
G=gap.SmallGroup(448,557);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,219,100,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^7=d^8=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations