direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊3Q8, C56⋊13Q8, C28.34SD16, C8⋊3(C7×Q8), C4.5(Q8×C14), (C4×C56).30C2, (C4×C8).12C14, C4⋊Q8.10C14, C28.94(C2×Q8), C4.Q8.6C14, C4.5(C7×SD16), (C2×C28).424D4, C14.40(C4⋊Q8), C42.83(C2×C14), C2.17(C14×SD16), C14.97(C2×SD16), (C4×C28).367C22, (C2×C28).954C23, (C2×C56).440C22, C22.119(D4×C14), C2.6(C7×C4⋊Q8), (C2×C4).80(C7×D4), (C7×C4⋊Q8).25C2, C4⋊C4.23(C2×C14), (C2×C8).95(C2×C14), (C7×C4.Q8).13C2, (C2×C14).675(C2×D4), (C7×C4⋊C4).243C22, (C2×C4).129(C22×C14), SmallGroup(448,906)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊3Q8
G = < a,b,c,d | a7=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >
Subgroups: 162 in 98 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4×C8, C4.Q8, C4⋊Q8, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C8⋊3Q8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, C4×C56, C7×C4.Q8, C7×C4⋊Q8, C7×C8⋊3Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, SD16, C2×D4, C2×Q8, C2×C14, C4⋊Q8, C2×SD16, C7×D4, C7×Q8, C22×C14, C8⋊3Q8, C7×SD16, D4×C14, Q8×C14, C7×C4⋊Q8, C14×SD16, C7×C8⋊3Q8
►Regular action on 448 points
Generators in S448
(1 20 65 16 77 85 29)(2 21 66 9 78 86 30)(3 22 67 10 79 87 31)(4 23 68 11 80 88 32)(5 24 69 12 73 81 25)(6 17 70 13 74 82 26)(7 18 71 14 75 83 27)(8 19 72 15 76 84 28)(33 113 105 49 97 41 89)(34 114 106 50 98 42 90)(35 115 107 51 99 43 91)(36 116 108 52 100 44 92)(37 117 109 53 101 45 93)(38 118 110 54 102 46 94)(39 119 111 55 103 47 95)(40 120 112 56 104 48 96)(57 445 387 437 379 429 371)(58 446 388 438 380 430 372)(59 447 389 439 381 431 373)(60 448 390 440 382 432 374)(61 441 391 433 383 425 375)(62 442 392 434 384 426 376)(63 443 385 435 377 427 369)(64 444 386 436 378 428 370)(121 199 143 191 134 179 171)(122 200 144 192 135 180 172)(123 193 137 185 136 181 173)(124 194 138 186 129 182 174)(125 195 139 187 130 183 175)(126 196 140 188 131 184 176)(127 197 141 189 132 177 169)(128 198 142 190 133 178 170)(145 231 217 161 209 153 201)(146 232 218 162 210 154 202)(147 225 219 163 211 155 203)(148 226 220 164 212 156 204)(149 227 221 165 213 157 205)(150 228 222 166 214 158 206)(151 229 223 167 215 159 207)(152 230 224 168 216 160 208)(233 288 307 249 299 241 291)(234 281 308 250 300 242 292)(235 282 309 251 301 243 293)(236 283 310 252 302 244 294)(237 284 311 253 303 245 295)(238 285 312 254 304 246 296)(239 286 305 255 297 247 289)(240 287 306 256 298 248 290)(257 337 331 273 323 265 315)(258 338 332 274 324 266 316)(259 339 333 275 325 267 317)(260 340 334 276 326 268 318)(261 341 335 277 327 269 319)(262 342 336 278 328 270 320)(263 343 329 279 321 271 313)(264 344 330 280 322 272 314)(345 400 419 361 411 353 403)(346 393 420 362 412 354 404)(347 394 421 363 413 355 405)(348 395 422 364 414 356 406)(349 396 423 365 415 357 407)(350 397 424 366 416 358 408)(351 398 417 367 409 359 401)(352 399 418 368 410 360 402)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248)(249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296)(297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320)(321 322 323 324 325 326 327 328)(329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344)(345 346 347 348 349 350 351 352)(353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368)(369 370 371 372 373 374 375 376)(377 378 379 380 381 382 383 384)(385 386 387 388 389 390 391 392)(393 394 395 396 397 398 399 400)(401 402 403 404 405 406 407 408)(409 410 411 412 413 414 415 416)(417 418 419 420 421 422 423 424)(425 426 427 428 429 430 431 432)(433 434 435 436 437 438 439 440)(441 442 443 444 445 446 447 448)
(1 151 39 189)(2 152 40 190)(3 145 33 191)(4 146 34 192)(5 147 35 185)(6 148 36 186)(7 149 37 187)(8 150 38 188)(9 168 56 170)(10 161 49 171)(11 162 50 172)(12 163 51 173)(13 164 52 174)(14 165 53 175)(15 166 54 176)(16 167 55 169)(17 226 116 129)(18 227 117 130)(19 228 118 131)(20 229 119 132)(21 230 120 133)(22 231 113 134)(23 232 114 135)(24 225 115 136)(25 203 91 137)(26 204 92 138)(27 205 93 139)(28 206 94 140)(29 207 95 141)(30 208 96 142)(31 201 89 143)(32 202 90 144)(41 199 87 153)(42 200 88 154)(43 193 81 155)(44 194 82 156)(45 195 83 157)(46 196 84 158)(47 197 85 159)(48 198 86 160)(57 286 398 339)(58 287 399 340)(59 288 400 341)(60 281 393 342)(61 282 394 343)(62 283 395 344)(63 284 396 337)(64 285 397 338)(65 223 111 177)(66 224 112 178)(67 217 105 179)(68 218 106 180)(69 219 107 181)(70 220 108 182)(71 221 109 183)(72 222 110 184)(73 211 99 123)(74 212 100 124)(75 213 101 125)(76 214 102 126)(77 215 103 127)(78 216 104 128)(79 209 97 121)(80 210 98 122)(233 345 261 373)(234 346 262 374)(235 347 263 375)(236 348 264 376)(237 349 257 369)(238 350 258 370)(239 351 259 371)(240 352 260 372)(241 353 269 381)(242 354 270 382)(243 355 271 383)(244 356 272 384)(245 357 265 377)(246 358 266 378)(247 359 267 379)(248 360 268 380)(249 361 277 389)(250 362 278 390)(251 363 279 391)(252 364 280 392)(253 365 273 385)(254 366 274 386)(255 367 275 387)(256 368 276 388)(289 401 317 429)(290 402 318 430)(291 403 319 431)(292 404 320 432)(293 405 313 425)(294 406 314 426)(295 407 315 427)(296 408 316 428)(297 409 325 437)(298 410 326 438)(299 411 327 439)(300 412 328 440)(301 413 321 433)(302 414 322 434)(303 415 323 435)(304 416 324 436)(305 417 333 445)(306 418 334 446)(307 419 335 447)(308 420 336 448)(309 421 329 441)(310 422 330 442)(311 423 331 443)(312 424 332 444)
(1 259 39 239)(2 262 40 234)(3 257 33 237)(4 260 34 240)(5 263 35 235)(6 258 36 238)(7 261 37 233)(8 264 38 236)(9 278 56 250)(10 273 49 253)(11 276 50 256)(12 279 51 251)(13 274 52 254)(14 277 53 249)(15 280 54 252)(16 275 55 255)(17 338 116 285)(18 341 117 288)(19 344 118 283)(20 339 119 286)(21 342 120 281)(22 337 113 284)(23 340 114 287)(24 343 115 282)(25 313 91 293)(26 316 92 296)(27 319 93 291)(28 314 94 294)(29 317 95 289)(30 320 96 292)(31 315 89 295)(32 318 90 290)(41 245 87 265)(42 248 88 268)(43 243 81 271)(44 246 82 266)(45 241 83 269)(46 244 84 272)(47 247 85 267)(48 242 86 270)(57 229 398 132)(58 232 399 135)(59 227 400 130)(60 230 393 133)(61 225 394 136)(62 228 395 131)(63 231 396 134)(64 226 397 129)(65 333 111 305)(66 336 112 308)(67 331 105 311)(68 334 106 306)(69 329 107 309)(70 332 108 312)(71 335 109 307)(72 330 110 310)(73 321 99 301)(74 324 100 304)(75 327 101 299)(76 322 102 302)(77 325 103 297)(78 328 104 300)(79 323 97 303)(80 326 98 298)(121 435 209 415)(122 438 210 410)(123 433 211 413)(124 436 212 416)(125 439 213 411)(126 434 214 414)(127 437 215 409)(128 440 216 412)(137 425 203 405)(138 428 204 408)(139 431 205 403)(140 426 206 406)(141 429 207 401)(142 432 208 404)(143 427 201 407)(144 430 202 402)(145 349 191 369)(146 352 192 372)(147 347 185 375)(148 350 186 370)(149 345 187 373)(150 348 188 376)(151 351 189 371)(152 346 190 374)(153 357 199 377)(154 360 200 380)(155 355 193 383)(156 358 194 378)(157 353 195 381)(158 356 196 384)(159 359 197 379)(160 354 198 382)(161 365 171 385)(162 368 172 388)(163 363 173 391)(164 366 174 386)(165 361 175 389)(166 364 176 392)(167 367 169 387)(168 362 170 390)(177 445 223 417)(178 448 224 420)(179 443 217 423)(180 446 218 418)(181 441 219 421)(182 444 220 424)(183 447 221 419)(184 442 222 422)
G:=sub<Sym(448)| (1,20,65,16,77,85,29)(2,21,66,9,78,86,30)(3,22,67,10,79,87,31)(4,23,68,11,80,88,32)(5,24,69,12,73,81,25)(6,17,70,13,74,82,26)(7,18,71,14,75,83,27)(8,19,72,15,76,84,28)(33,113,105,49,97,41,89)(34,114,106,50,98,42,90)(35,115,107,51,99,43,91)(36,116,108,52,100,44,92)(37,117,109,53,101,45,93)(38,118,110,54,102,46,94)(39,119,111,55,103,47,95)(40,120,112,56,104,48,96)(57,445,387,437,379,429,371)(58,446,388,438,380,430,372)(59,447,389,439,381,431,373)(60,448,390,440,382,432,374)(61,441,391,433,383,425,375)(62,442,392,434,384,426,376)(63,443,385,435,377,427,369)(64,444,386,436,378,428,370)(121,199,143,191,134,179,171)(122,200,144,192,135,180,172)(123,193,137,185,136,181,173)(124,194,138,186,129,182,174)(125,195,139,187,130,183,175)(126,196,140,188,131,184,176)(127,197,141,189,132,177,169)(128,198,142,190,133,178,170)(145,231,217,161,209,153,201)(146,232,218,162,210,154,202)(147,225,219,163,211,155,203)(148,226,220,164,212,156,204)(149,227,221,165,213,157,205)(150,228,222,166,214,158,206)(151,229,223,167,215,159,207)(152,230,224,168,216,160,208)(233,288,307,249,299,241,291)(234,281,308,250,300,242,292)(235,282,309,251,301,243,293)(236,283,310,252,302,244,294)(237,284,311,253,303,245,295)(238,285,312,254,304,246,296)(239,286,305,255,297,247,289)(240,287,306,256,298,248,290)(257,337,331,273,323,265,315)(258,338,332,274,324,266,316)(259,339,333,275,325,267,317)(260,340,334,276,326,268,318)(261,341,335,277,327,269,319)(262,342,336,278,328,270,320)(263,343,329,279,321,271,313)(264,344,330,280,322,272,314)(345,400,419,361,411,353,403)(346,393,420,362,412,354,404)(347,394,421,363,413,355,405)(348,395,422,364,414,356,406)(349,396,423,365,415,357,407)(350,397,424,366,416,358,408)(351,398,417,367,409,359,401)(352,399,418,368,410,360,402), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328)(329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344)(345,346,347,348,349,350,351,352)(353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368)(369,370,371,372,373,374,375,376)(377,378,379,380,381,382,383,384)(385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408)(409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424)(425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448), (1,151,39,189)(2,152,40,190)(3,145,33,191)(4,146,34,192)(5,147,35,185)(6,148,36,186)(7,149,37,187)(8,150,38,188)(9,168,56,170)(10,161,49,171)(11,162,50,172)(12,163,51,173)(13,164,52,174)(14,165,53,175)(15,166,54,176)(16,167,55,169)(17,226,116,129)(18,227,117,130)(19,228,118,131)(20,229,119,132)(21,230,120,133)(22,231,113,134)(23,232,114,135)(24,225,115,136)(25,203,91,137)(26,204,92,138)(27,205,93,139)(28,206,94,140)(29,207,95,141)(30,208,96,142)(31,201,89,143)(32,202,90,144)(41,199,87,153)(42,200,88,154)(43,193,81,155)(44,194,82,156)(45,195,83,157)(46,196,84,158)(47,197,85,159)(48,198,86,160)(57,286,398,339)(58,287,399,340)(59,288,400,341)(60,281,393,342)(61,282,394,343)(62,283,395,344)(63,284,396,337)(64,285,397,338)(65,223,111,177)(66,224,112,178)(67,217,105,179)(68,218,106,180)(69,219,107,181)(70,220,108,182)(71,221,109,183)(72,222,110,184)(73,211,99,123)(74,212,100,124)(75,213,101,125)(76,214,102,126)(77,215,103,127)(78,216,104,128)(79,209,97,121)(80,210,98,122)(233,345,261,373)(234,346,262,374)(235,347,263,375)(236,348,264,376)(237,349,257,369)(238,350,258,370)(239,351,259,371)(240,352,260,372)(241,353,269,381)(242,354,270,382)(243,355,271,383)(244,356,272,384)(245,357,265,377)(246,358,266,378)(247,359,267,379)(248,360,268,380)(249,361,277,389)(250,362,278,390)(251,363,279,391)(252,364,280,392)(253,365,273,385)(254,366,274,386)(255,367,275,387)(256,368,276,388)(289,401,317,429)(290,402,318,430)(291,403,319,431)(292,404,320,432)(293,405,313,425)(294,406,314,426)(295,407,315,427)(296,408,316,428)(297,409,325,437)(298,410,326,438)(299,411,327,439)(300,412,328,440)(301,413,321,433)(302,414,322,434)(303,415,323,435)(304,416,324,436)(305,417,333,445)(306,418,334,446)(307,419,335,447)(308,420,336,448)(309,421,329,441)(310,422,330,442)(311,423,331,443)(312,424,332,444), (1,259,39,239)(2,262,40,234)(3,257,33,237)(4,260,34,240)(5,263,35,235)(6,258,36,238)(7,261,37,233)(8,264,38,236)(9,278,56,250)(10,273,49,253)(11,276,50,256)(12,279,51,251)(13,274,52,254)(14,277,53,249)(15,280,54,252)(16,275,55,255)(17,338,116,285)(18,341,117,288)(19,344,118,283)(20,339,119,286)(21,342,120,281)(22,337,113,284)(23,340,114,287)(24,343,115,282)(25,313,91,293)(26,316,92,296)(27,319,93,291)(28,314,94,294)(29,317,95,289)(30,320,96,292)(31,315,89,295)(32,318,90,290)(41,245,87,265)(42,248,88,268)(43,243,81,271)(44,246,82,266)(45,241,83,269)(46,244,84,272)(47,247,85,267)(48,242,86,270)(57,229,398,132)(58,232,399,135)(59,227,400,130)(60,230,393,133)(61,225,394,136)(62,228,395,131)(63,231,396,134)(64,226,397,129)(65,333,111,305)(66,336,112,308)(67,331,105,311)(68,334,106,306)(69,329,107,309)(70,332,108,312)(71,335,109,307)(72,330,110,310)(73,321,99,301)(74,324,100,304)(75,327,101,299)(76,322,102,302)(77,325,103,297)(78,328,104,300)(79,323,97,303)(80,326,98,298)(121,435,209,415)(122,438,210,410)(123,433,211,413)(124,436,212,416)(125,439,213,411)(126,434,214,414)(127,437,215,409)(128,440,216,412)(137,425,203,405)(138,428,204,408)(139,431,205,403)(140,426,206,406)(141,429,207,401)(142,432,208,404)(143,427,201,407)(144,430,202,402)(145,349,191,369)(146,352,192,372)(147,347,185,375)(148,350,186,370)(149,345,187,373)(150,348,188,376)(151,351,189,371)(152,346,190,374)(153,357,199,377)(154,360,200,380)(155,355,193,383)(156,358,194,378)(157,353,195,381)(158,356,196,384)(159,359,197,379)(160,354,198,382)(161,365,171,385)(162,368,172,388)(163,363,173,391)(164,366,174,386)(165,361,175,389)(166,364,176,392)(167,367,169,387)(168,362,170,390)(177,445,223,417)(178,448,224,420)(179,443,217,423)(180,446,218,418)(181,441,219,421)(182,444,220,424)(183,447,221,419)(184,442,222,422)>;
G:=Group( (1,20,65,16,77,85,29)(2,21,66,9,78,86,30)(3,22,67,10,79,87,31)(4,23,68,11,80,88,32)(5,24,69,12,73,81,25)(6,17,70,13,74,82,26)(7,18,71,14,75,83,27)(8,19,72,15,76,84,28)(33,113,105,49,97,41,89)(34,114,106,50,98,42,90)(35,115,107,51,99,43,91)(36,116,108,52,100,44,92)(37,117,109,53,101,45,93)(38,118,110,54,102,46,94)(39,119,111,55,103,47,95)(40,120,112,56,104,48,96)(57,445,387,437,379,429,371)(58,446,388,438,380,430,372)(59,447,389,439,381,431,373)(60,448,390,440,382,432,374)(61,441,391,433,383,425,375)(62,442,392,434,384,426,376)(63,443,385,435,377,427,369)(64,444,386,436,378,428,370)(121,199,143,191,134,179,171)(122,200,144,192,135,180,172)(123,193,137,185,136,181,173)(124,194,138,186,129,182,174)(125,195,139,187,130,183,175)(126,196,140,188,131,184,176)(127,197,141,189,132,177,169)(128,198,142,190,133,178,170)(145,231,217,161,209,153,201)(146,232,218,162,210,154,202)(147,225,219,163,211,155,203)(148,226,220,164,212,156,204)(149,227,221,165,213,157,205)(150,228,222,166,214,158,206)(151,229,223,167,215,159,207)(152,230,224,168,216,160,208)(233,288,307,249,299,241,291)(234,281,308,250,300,242,292)(235,282,309,251,301,243,293)(236,283,310,252,302,244,294)(237,284,311,253,303,245,295)(238,285,312,254,304,246,296)(239,286,305,255,297,247,289)(240,287,306,256,298,248,290)(257,337,331,273,323,265,315)(258,338,332,274,324,266,316)(259,339,333,275,325,267,317)(260,340,334,276,326,268,318)(261,341,335,277,327,269,319)(262,342,336,278,328,270,320)(263,343,329,279,321,271,313)(264,344,330,280,322,272,314)(345,400,419,361,411,353,403)(346,393,420,362,412,354,404)(347,394,421,363,413,355,405)(348,395,422,364,414,356,406)(349,396,423,365,415,357,407)(350,397,424,366,416,358,408)(351,398,417,367,409,359,401)(352,399,418,368,410,360,402), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328)(329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344)(345,346,347,348,349,350,351,352)(353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368)(369,370,371,372,373,374,375,376)(377,378,379,380,381,382,383,384)(385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408)(409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424)(425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448), (1,151,39,189)(2,152,40,190)(3,145,33,191)(4,146,34,192)(5,147,35,185)(6,148,36,186)(7,149,37,187)(8,150,38,188)(9,168,56,170)(10,161,49,171)(11,162,50,172)(12,163,51,173)(13,164,52,174)(14,165,53,175)(15,166,54,176)(16,167,55,169)(17,226,116,129)(18,227,117,130)(19,228,118,131)(20,229,119,132)(21,230,120,133)(22,231,113,134)(23,232,114,135)(24,225,115,136)(25,203,91,137)(26,204,92,138)(27,205,93,139)(28,206,94,140)(29,207,95,141)(30,208,96,142)(31,201,89,143)(32,202,90,144)(41,199,87,153)(42,200,88,154)(43,193,81,155)(44,194,82,156)(45,195,83,157)(46,196,84,158)(47,197,85,159)(48,198,86,160)(57,286,398,339)(58,287,399,340)(59,288,400,341)(60,281,393,342)(61,282,394,343)(62,283,395,344)(63,284,396,337)(64,285,397,338)(65,223,111,177)(66,224,112,178)(67,217,105,179)(68,218,106,180)(69,219,107,181)(70,220,108,182)(71,221,109,183)(72,222,110,184)(73,211,99,123)(74,212,100,124)(75,213,101,125)(76,214,102,126)(77,215,103,127)(78,216,104,128)(79,209,97,121)(80,210,98,122)(233,345,261,373)(234,346,262,374)(235,347,263,375)(236,348,264,376)(237,349,257,369)(238,350,258,370)(239,351,259,371)(240,352,260,372)(241,353,269,381)(242,354,270,382)(243,355,271,383)(244,356,272,384)(245,357,265,377)(246,358,266,378)(247,359,267,379)(248,360,268,380)(249,361,277,389)(250,362,278,390)(251,363,279,391)(252,364,280,392)(253,365,273,385)(254,366,274,386)(255,367,275,387)(256,368,276,388)(289,401,317,429)(290,402,318,430)(291,403,319,431)(292,404,320,432)(293,405,313,425)(294,406,314,426)(295,407,315,427)(296,408,316,428)(297,409,325,437)(298,410,326,438)(299,411,327,439)(300,412,328,440)(301,413,321,433)(302,414,322,434)(303,415,323,435)(304,416,324,436)(305,417,333,445)(306,418,334,446)(307,419,335,447)(308,420,336,448)(309,421,329,441)(310,422,330,442)(311,423,331,443)(312,424,332,444), (1,259,39,239)(2,262,40,234)(3,257,33,237)(4,260,34,240)(5,263,35,235)(6,258,36,238)(7,261,37,233)(8,264,38,236)(9,278,56,250)(10,273,49,253)(11,276,50,256)(12,279,51,251)(13,274,52,254)(14,277,53,249)(15,280,54,252)(16,275,55,255)(17,338,116,285)(18,341,117,288)(19,344,118,283)(20,339,119,286)(21,342,120,281)(22,337,113,284)(23,340,114,287)(24,343,115,282)(25,313,91,293)(26,316,92,296)(27,319,93,291)(28,314,94,294)(29,317,95,289)(30,320,96,292)(31,315,89,295)(32,318,90,290)(41,245,87,265)(42,248,88,268)(43,243,81,271)(44,246,82,266)(45,241,83,269)(46,244,84,272)(47,247,85,267)(48,242,86,270)(57,229,398,132)(58,232,399,135)(59,227,400,130)(60,230,393,133)(61,225,394,136)(62,228,395,131)(63,231,396,134)(64,226,397,129)(65,333,111,305)(66,336,112,308)(67,331,105,311)(68,334,106,306)(69,329,107,309)(70,332,108,312)(71,335,109,307)(72,330,110,310)(73,321,99,301)(74,324,100,304)(75,327,101,299)(76,322,102,302)(77,325,103,297)(78,328,104,300)(79,323,97,303)(80,326,98,298)(121,435,209,415)(122,438,210,410)(123,433,211,413)(124,436,212,416)(125,439,213,411)(126,434,214,414)(127,437,215,409)(128,440,216,412)(137,425,203,405)(138,428,204,408)(139,431,205,403)(140,426,206,406)(141,429,207,401)(142,432,208,404)(143,427,201,407)(144,430,202,402)(145,349,191,369)(146,352,192,372)(147,347,185,375)(148,350,186,370)(149,345,187,373)(150,348,188,376)(151,351,189,371)(152,346,190,374)(153,357,199,377)(154,360,200,380)(155,355,193,383)(156,358,194,378)(157,353,195,381)(158,356,196,384)(159,359,197,379)(160,354,198,382)(161,365,171,385)(162,368,172,388)(163,363,173,391)(164,366,174,386)(165,361,175,389)(166,364,176,392)(167,367,169,387)(168,362,170,390)(177,445,223,417)(178,448,224,420)(179,443,217,423)(180,446,218,418)(181,441,219,421)(182,444,220,424)(183,447,221,419)(184,442,222,422) );
G=PermutationGroup([[(1,20,65,16,77,85,29),(2,21,66,9,78,86,30),(3,22,67,10,79,87,31),(4,23,68,11,80,88,32),(5,24,69,12,73,81,25),(6,17,70,13,74,82,26),(7,18,71,14,75,83,27),(8,19,72,15,76,84,28),(33,113,105,49,97,41,89),(34,114,106,50,98,42,90),(35,115,107,51,99,43,91),(36,116,108,52,100,44,92),(37,117,109,53,101,45,93),(38,118,110,54,102,46,94),(39,119,111,55,103,47,95),(40,120,112,56,104,48,96),(57,445,387,437,379,429,371),(58,446,388,438,380,430,372),(59,447,389,439,381,431,373),(60,448,390,440,382,432,374),(61,441,391,433,383,425,375),(62,442,392,434,384,426,376),(63,443,385,435,377,427,369),(64,444,386,436,378,428,370),(121,199,143,191,134,179,171),(122,200,144,192,135,180,172),(123,193,137,185,136,181,173),(124,194,138,186,129,182,174),(125,195,139,187,130,183,175),(126,196,140,188,131,184,176),(127,197,141,189,132,177,169),(128,198,142,190,133,178,170),(145,231,217,161,209,153,201),(146,232,218,162,210,154,202),(147,225,219,163,211,155,203),(148,226,220,164,212,156,204),(149,227,221,165,213,157,205),(150,228,222,166,214,158,206),(151,229,223,167,215,159,207),(152,230,224,168,216,160,208),(233,288,307,249,299,241,291),(234,281,308,250,300,242,292),(235,282,309,251,301,243,293),(236,283,310,252,302,244,294),(237,284,311,253,303,245,295),(238,285,312,254,304,246,296),(239,286,305,255,297,247,289),(240,287,306,256,298,248,290),(257,337,331,273,323,265,315),(258,338,332,274,324,266,316),(259,339,333,275,325,267,317),(260,340,334,276,326,268,318),(261,341,335,277,327,269,319),(262,342,336,278,328,270,320),(263,343,329,279,321,271,313),(264,344,330,280,322,272,314),(345,400,419,361,411,353,403),(346,393,420,362,412,354,404),(347,394,421,363,413,355,405),(348,395,422,364,414,356,406),(349,396,423,365,415,357,407),(350,397,424,366,416,358,408),(351,398,417,367,409,359,401),(352,399,418,368,410,360,402)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176),(177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232),(233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248),(249,250,251,252,253,254,255,256),(257,258,259,260,261,262,263,264),(265,266,267,268,269,270,271,272),(273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288),(289,290,291,292,293,294,295,296),(297,298,299,300,301,302,303,304),(305,306,307,308,309,310,311,312),(313,314,315,316,317,318,319,320),(321,322,323,324,325,326,327,328),(329,330,331,332,333,334,335,336),(337,338,339,340,341,342,343,344),(345,346,347,348,349,350,351,352),(353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368),(369,370,371,372,373,374,375,376),(377,378,379,380,381,382,383,384),(385,386,387,388,389,390,391,392),(393,394,395,396,397,398,399,400),(401,402,403,404,405,406,407,408),(409,410,411,412,413,414,415,416),(417,418,419,420,421,422,423,424),(425,426,427,428,429,430,431,432),(433,434,435,436,437,438,439,440),(441,442,443,444,445,446,447,448)], [(1,151,39,189),(2,152,40,190),(3,145,33,191),(4,146,34,192),(5,147,35,185),(6,148,36,186),(7,149,37,187),(8,150,38,188),(9,168,56,170),(10,161,49,171),(11,162,50,172),(12,163,51,173),(13,164,52,174),(14,165,53,175),(15,166,54,176),(16,167,55,169),(17,226,116,129),(18,227,117,130),(19,228,118,131),(20,229,119,132),(21,230,120,133),(22,231,113,134),(23,232,114,135),(24,225,115,136),(25,203,91,137),(26,204,92,138),(27,205,93,139),(28,206,94,140),(29,207,95,141),(30,208,96,142),(31,201,89,143),(32,202,90,144),(41,199,87,153),(42,200,88,154),(43,193,81,155),(44,194,82,156),(45,195,83,157),(46,196,84,158),(47,197,85,159),(48,198,86,160),(57,286,398,339),(58,287,399,340),(59,288,400,341),(60,281,393,342),(61,282,394,343),(62,283,395,344),(63,284,396,337),(64,285,397,338),(65,223,111,177),(66,224,112,178),(67,217,105,179),(68,218,106,180),(69,219,107,181),(70,220,108,182),(71,221,109,183),(72,222,110,184),(73,211,99,123),(74,212,100,124),(75,213,101,125),(76,214,102,126),(77,215,103,127),(78,216,104,128),(79,209,97,121),(80,210,98,122),(233,345,261,373),(234,346,262,374),(235,347,263,375),(236,348,264,376),(237,349,257,369),(238,350,258,370),(239,351,259,371),(240,352,260,372),(241,353,269,381),(242,354,270,382),(243,355,271,383),(244,356,272,384),(245,357,265,377),(246,358,266,378),(247,359,267,379),(248,360,268,380),(249,361,277,389),(250,362,278,390),(251,363,279,391),(252,364,280,392),(253,365,273,385),(254,366,274,386),(255,367,275,387),(256,368,276,388),(289,401,317,429),(290,402,318,430),(291,403,319,431),(292,404,320,432),(293,405,313,425),(294,406,314,426),(295,407,315,427),(296,408,316,428),(297,409,325,437),(298,410,326,438),(299,411,327,439),(300,412,328,440),(301,413,321,433),(302,414,322,434),(303,415,323,435),(304,416,324,436),(305,417,333,445),(306,418,334,446),(307,419,335,447),(308,420,336,448),(309,421,329,441),(310,422,330,442),(311,423,331,443),(312,424,332,444)], [(1,259,39,239),(2,262,40,234),(3,257,33,237),(4,260,34,240),(5,263,35,235),(6,258,36,238),(7,261,37,233),(8,264,38,236),(9,278,56,250),(10,273,49,253),(11,276,50,256),(12,279,51,251),(13,274,52,254),(14,277,53,249),(15,280,54,252),(16,275,55,255),(17,338,116,285),(18,341,117,288),(19,344,118,283),(20,339,119,286),(21,342,120,281),(22,337,113,284),(23,340,114,287),(24,343,115,282),(25,313,91,293),(26,316,92,296),(27,319,93,291),(28,314,94,294),(29,317,95,289),(30,320,96,292),(31,315,89,295),(32,318,90,290),(41,245,87,265),(42,248,88,268),(43,243,81,271),(44,246,82,266),(45,241,83,269),(46,244,84,272),(47,247,85,267),(48,242,86,270),(57,229,398,132),(58,232,399,135),(59,227,400,130),(60,230,393,133),(61,225,394,136),(62,228,395,131),(63,231,396,134),(64,226,397,129),(65,333,111,305),(66,336,112,308),(67,331,105,311),(68,334,106,306),(69,329,107,309),(70,332,108,312),(71,335,109,307),(72,330,110,310),(73,321,99,301),(74,324,100,304),(75,327,101,299),(76,322,102,302),(77,325,103,297),(78,328,104,300),(79,323,97,303),(80,326,98,298),(121,435,209,415),(122,438,210,410),(123,433,211,413),(124,436,212,416),(125,439,213,411),(126,434,214,414),(127,437,215,409),(128,440,216,412),(137,425,203,405),(138,428,204,408),(139,431,205,403),(140,426,206,406),(141,429,207,401),(142,432,208,404),(143,427,201,407),(144,430,202,402),(145,349,191,369),(146,352,192,372),(147,347,185,375),(148,350,186,370),(149,345,187,373),(150,348,188,376),(151,351,189,371),(152,346,190,374),(153,357,199,377),(154,360,200,380),(155,355,193,383),(156,358,194,378),(157,353,195,381),(158,356,196,384),(159,359,197,379),(160,354,198,382),(161,365,171,385),(162,368,172,388),(163,363,173,391),(164,366,174,386),(165,361,175,389),(166,364,176,392),(167,367,169,387),(168,362,170,390),(177,445,223,417),(178,448,224,420),(179,443,217,423),(180,446,218,418),(181,441,219,421),(182,444,220,424),(183,447,221,419),(184,442,222,422)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28AJ | 28AK | ··· | 28BH | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | Q8 | D4 | SD16 | C7×Q8 | C7×D4 | C7×SD16 |
| kernel | C7×C8⋊3Q8 | C4×C56 | C7×C4.Q8 | C7×C4⋊Q8 | C8⋊3Q8 | C4×C8 | C4.Q8 | C4⋊Q8 | C56 | C2×C28 | C28 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 4 | 2 | 8 | 24 | 12 | 48 |
Matrix representation of C7×C8⋊3Q8 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 111 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 0 | 0 | 13 | 100 |
| 0 | 0 | 13 | 13 |
| 1 | 111 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 40 | 0 | 0 |
| 27 | 106 | 0 | 0 |
| 0 | 0 | 27 | 17 |
| 0 | 0 | 17 | 86 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,111,112,0,0,0,0,13,13,0,0,100,13],[1,1,0,0,111,112,0,0,0,0,1,0,0,0,0,1],[7,27,0,0,40,106,0,0,0,0,27,17,0,0,17,86] >;
C7×C8⋊3Q8 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_3Q_8
% in TeX
G:=Group("C7xC8:3Q8"); // GroupNames label
G:=SmallGroup(448,906);
// by ID
G=gap.SmallGroup(448,906);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,400,2438,604,14117,124]);
// 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,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations