direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C7⋊Q32, C14⋊2Q32, C56.25D4, C28.25D8, Q16.8D14, C56.28C23, Dic28.12C22, C7⋊3(C2×Q32), C14.67(C2×D8), (C2×C14).46D8, C7⋊C16.9C22, (C2×Q16).2D7, C4.11(D4⋊D7), (C2×C8).240D14, (C2×C28).184D4, C28.183(C2×D4), C8.17(C7⋊D4), (C14×Q16).3C2, C8.34(C22×D7), (C2×C56).92C22, (C7×Q16).8C22, C22.24(D4⋊D7), (C2×Dic28).10C2, (C2×C7⋊C16).6C2, C2.22(C2×D4⋊D7), C4.13(C2×C7⋊D4), (C2×C4).145(C7⋊D4), SmallGroup(448,714)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C7⋊Q32
G = < a,b,c,d | a2=b7=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 356 in 82 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C14, C16, C2×C8, Q16, Q16, C2×Q8, Dic7, C28, C28, C2×C14, C2×C16, Q32, C2×Q16, C2×Q16, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C2×Q32, C7⋊C16, Dic28, Dic28, C2×C56, C7×Q16, C7×Q16, C2×Dic14, Q8×C14, C2×C7⋊C16, C7⋊Q32, C2×Dic28, C14×Q16, C2×C7⋊Q32
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, Q32, C2×D8, C7⋊D4, C22×D7, C2×Q32, D4⋊D7, C2×C7⋊D4, C7⋊Q32, C2×D4⋊D7, C2×C7⋊Q32
►Regular action on 448 points
Generators in S448
(1 371)(2 372)(3 373)(4 374)(5 375)(6 376)(7 377)(8 378)(9 379)(10 380)(11 381)(12 382)(13 383)(14 384)(15 369)(16 370)(17 413)(18 414)(19 415)(20 416)(21 401)(22 402)(23 403)(24 404)(25 405)(26 406)(27 407)(28 408)(29 409)(30 410)(31 411)(32 412)(33 278)(34 279)(35 280)(36 281)(37 282)(38 283)(39 284)(40 285)(41 286)(42 287)(43 288)(44 273)(45 274)(46 275)(47 276)(48 277)(49 140)(50 141)(51 142)(52 143)(53 144)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 214)(66 215)(67 216)(68 217)(69 218)(70 219)(71 220)(72 221)(73 222)(74 223)(75 224)(76 209)(77 210)(78 211)(79 212)(80 213)(81 299)(82 300)(83 301)(84 302)(85 303)(86 304)(87 289)(88 290)(89 291)(90 292)(91 293)(92 294)(93 295)(94 296)(95 297)(96 298)(97 347)(98 348)(99 349)(100 350)(101 351)(102 352)(103 337)(104 338)(105 339)(106 340)(107 341)(108 342)(109 343)(110 344)(111 345)(112 346)(113 160)(114 145)(115 146)(116 147)(117 148)(118 149)(119 150)(120 151)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(161 208)(162 193)(163 194)(164 195)(165 196)(166 197)(167 198)(168 199)(169 200)(170 201)(171 202)(172 203)(173 204)(174 205)(175 206)(176 207)(177 357)(178 358)(179 359)(180 360)(181 361)(182 362)(183 363)(184 364)(185 365)(186 366)(187 367)(188 368)(189 353)(190 354)(191 355)(192 356)(225 391)(226 392)(227 393)(228 394)(229 395)(230 396)(231 397)(232 398)(233 399)(234 400)(235 385)(236 386)(237 387)(238 388)(239 389)(240 390)(241 436)(242 437)(243 438)(244 439)(245 440)(246 441)(247 442)(248 443)(249 444)(250 445)(251 446)(252 447)(253 448)(254 433)(255 434)(256 435)(257 305)(258 306)(259 307)(260 308)(261 309)(262 310)(263 311)(264 312)(265 313)(266 314)(267 315)(268 316)(269 317)(270 318)(271 319)(272 320)(321 425)(322 426)(323 427)(324 428)(325 429)(326 430)(327 431)(328 432)(329 417)(330 418)(331 419)(332 420)(333 421)(334 422)(335 423)(336 424)
(1 351 318 158 328 228 185)(2 186 229 329 159 319 352)(3 337 320 160 330 230 187)(4 188 231 331 145 305 338)(5 339 306 146 332 232 189)(6 190 233 333 147 307 340)(7 341 308 148 334 234 191)(8 192 235 335 149 309 342)(9 343 310 150 336 236 177)(10 178 237 321 151 311 344)(11 345 312 152 322 238 179)(12 180 239 323 153 313 346)(13 347 314 154 324 240 181)(14 182 225 325 155 315 348)(15 349 316 156 326 226 183)(16 184 227 327 157 317 350)(17 141 213 205 85 241 43)(18 44 242 86 206 214 142)(19 143 215 207 87 243 45)(20 46 244 88 208 216 144)(21 129 217 193 89 245 47)(22 48 246 90 194 218 130)(23 131 219 195 91 247 33)(24 34 248 92 196 220 132)(25 133 221 197 93 249 35)(26 36 250 94 198 222 134)(27 135 223 199 95 251 37)(28 38 252 96 200 224 136)(29 137 209 201 81 253 39)(30 40 254 82 202 210 138)(31 139 211 203 83 255 41)(32 42 256 84 204 212 140)(49 412 287 435 302 173 79)(50 80 174 303 436 288 413)(51 414 273 437 304 175 65)(52 66 176 289 438 274 415)(53 416 275 439 290 161 67)(54 68 162 291 440 276 401)(55 402 277 441 292 163 69)(56 70 164 293 442 278 403)(57 404 279 443 294 165 71)(58 72 166 295 444 280 405)(59 406 281 445 296 167 73)(60 74 168 297 446 282 407)(61 408 283 447 298 169 75)(62 76 170 299 448 284 409)(63 410 285 433 300 171 77)(64 78 172 301 434 286 411)(97 266 123 428 390 361 383)(98 384 362 391 429 124 267)(99 268 125 430 392 363 369)(100 370 364 393 431 126 269)(101 270 127 432 394 365 371)(102 372 366 395 417 128 271)(103 272 113 418 396 367 373)(104 374 368 397 419 114 257)(105 258 115 420 398 353 375)(106 376 354 399 421 116 259)(107 260 117 422 400 355 377)(108 378 356 385 423 118 261)(109 262 119 424 386 357 379)(110 380 358 387 425 120 263)(111 264 121 426 388 359 381)(112 382 360 389 427 122 265)
(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)
(1 135 9 143)(2 134 10 142)(3 133 11 141)(4 132 12 140)(5 131 13 139)(6 130 14 138)(7 129 15 137)(8 144 16 136)(17 187 25 179)(18 186 26 178)(19 185 27 177)(20 184 28 192)(21 183 29 191)(22 182 30 190)(23 181 31 189)(24 180 32 188)(33 240 41 232)(34 239 42 231)(35 238 43 230)(36 237 44 229)(37 236 45 228)(38 235 46 227)(39 234 47 226)(40 233 48 225)(49 374 57 382)(50 373 58 381)(51 372 59 380)(52 371 60 379)(53 370 61 378)(54 369 62 377)(55 384 63 376)(56 383 64 375)(65 102 73 110)(66 101 74 109)(67 100 75 108)(68 99 76 107)(69 98 77 106)(70 97 78 105)(71 112 79 104)(72 111 80 103)(81 148 89 156)(82 147 90 155)(83 146 91 154)(84 145 92 153)(85 160 93 152)(86 159 94 151)(87 158 95 150)(88 157 96 149)(113 295 121 303)(114 294 122 302)(115 293 123 301)(116 292 124 300)(117 291 125 299)(118 290 126 298)(119 289 127 297)(120 304 128 296)(161 269 169 261)(162 268 170 260)(163 267 171 259)(164 266 172 258)(165 265 173 257)(166 264 174 272)(167 263 175 271)(168 262 176 270)(193 316 201 308)(194 315 202 307)(195 314 203 306)(196 313 204 305)(197 312 205 320)(198 311 206 319)(199 310 207 318)(200 309 208 317)(209 341 217 349)(210 340 218 348)(211 339 219 347)(212 338 220 346)(213 337 221 345)(214 352 222 344)(215 351 223 343)(216 350 224 342)(241 330 249 322)(242 329 250 321)(243 328 251 336)(244 327 252 335)(245 326 253 334)(246 325 254 333)(247 324 255 332)(248 323 256 331)(273 395 281 387)(274 394 282 386)(275 393 283 385)(276 392 284 400)(277 391 285 399)(278 390 286 398)(279 389 287 397)(280 388 288 396)(353 403 361 411)(354 402 362 410)(355 401 363 409)(356 416 364 408)(357 415 365 407)(358 414 366 406)(359 413 367 405)(360 412 368 404)(417 445 425 437)(418 444 426 436)(419 443 427 435)(420 442 428 434)(421 441 429 433)(422 440 430 448)(423 439 431 447)(424 438 432 446)
G:=sub<Sym(448)| (1,371)(2,372)(3,373)(4,374)(5,375)(6,376)(7,377)(8,378)(9,379)(10,380)(11,381)(12,382)(13,383)(14,384)(15,369)(16,370)(17,413)(18,414)(19,415)(20,416)(21,401)(22,402)(23,403)(24,404)(25,405)(26,406)(27,407)(28,408)(29,409)(30,410)(31,411)(32,412)(33,278)(34,279)(35,280)(36,281)(37,282)(38,283)(39,284)(40,285)(41,286)(42,287)(43,288)(44,273)(45,274)(46,275)(47,276)(48,277)(49,140)(50,141)(51,142)(52,143)(53,144)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,214)(66,215)(67,216)(68,217)(69,218)(70,219)(71,220)(72,221)(73,222)(74,223)(75,224)(76,209)(77,210)(78,211)(79,212)(80,213)(81,299)(82,300)(83,301)(84,302)(85,303)(86,304)(87,289)(88,290)(89,291)(90,292)(91,293)(92,294)(93,295)(94,296)(95,297)(96,298)(97,347)(98,348)(99,349)(100,350)(101,351)(102,352)(103,337)(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)(113,160)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(161,208)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,357)(178,358)(179,359)(180,360)(181,361)(182,362)(183,363)(184,364)(185,365)(186,366)(187,367)(188,368)(189,353)(190,354)(191,355)(192,356)(225,391)(226,392)(227,393)(228,394)(229,395)(230,396)(231,397)(232,398)(233,399)(234,400)(235,385)(236,386)(237,387)(238,388)(239,389)(240,390)(241,436)(242,437)(243,438)(244,439)(245,440)(246,441)(247,442)(248,443)(249,444)(250,445)(251,446)(252,447)(253,448)(254,433)(255,434)(256,435)(257,305)(258,306)(259,307)(260,308)(261,309)(262,310)(263,311)(264,312)(265,313)(266,314)(267,315)(268,316)(269,317)(270,318)(271,319)(272,320)(321,425)(322,426)(323,427)(324,428)(325,429)(326,430)(327,431)(328,432)(329,417)(330,418)(331,419)(332,420)(333,421)(334,422)(335,423)(336,424), (1,351,318,158,328,228,185)(2,186,229,329,159,319,352)(3,337,320,160,330,230,187)(4,188,231,331,145,305,338)(5,339,306,146,332,232,189)(6,190,233,333,147,307,340)(7,341,308,148,334,234,191)(8,192,235,335,149,309,342)(9,343,310,150,336,236,177)(10,178,237,321,151,311,344)(11,345,312,152,322,238,179)(12,180,239,323,153,313,346)(13,347,314,154,324,240,181)(14,182,225,325,155,315,348)(15,349,316,156,326,226,183)(16,184,227,327,157,317,350)(17,141,213,205,85,241,43)(18,44,242,86,206,214,142)(19,143,215,207,87,243,45)(20,46,244,88,208,216,144)(21,129,217,193,89,245,47)(22,48,246,90,194,218,130)(23,131,219,195,91,247,33)(24,34,248,92,196,220,132)(25,133,221,197,93,249,35)(26,36,250,94,198,222,134)(27,135,223,199,95,251,37)(28,38,252,96,200,224,136)(29,137,209,201,81,253,39)(30,40,254,82,202,210,138)(31,139,211,203,83,255,41)(32,42,256,84,204,212,140)(49,412,287,435,302,173,79)(50,80,174,303,436,288,413)(51,414,273,437,304,175,65)(52,66,176,289,438,274,415)(53,416,275,439,290,161,67)(54,68,162,291,440,276,401)(55,402,277,441,292,163,69)(56,70,164,293,442,278,403)(57,404,279,443,294,165,71)(58,72,166,295,444,280,405)(59,406,281,445,296,167,73)(60,74,168,297,446,282,407)(61,408,283,447,298,169,75)(62,76,170,299,448,284,409)(63,410,285,433,300,171,77)(64,78,172,301,434,286,411)(97,266,123,428,390,361,383)(98,384,362,391,429,124,267)(99,268,125,430,392,363,369)(100,370,364,393,431,126,269)(101,270,127,432,394,365,371)(102,372,366,395,417,128,271)(103,272,113,418,396,367,373)(104,374,368,397,419,114,257)(105,258,115,420,398,353,375)(106,376,354,399,421,116,259)(107,260,117,422,400,355,377)(108,378,356,385,423,118,261)(109,262,119,424,386,357,379)(110,380,358,387,425,120,263)(111,264,121,426,388,359,381)(112,382,360,389,427,122,265), (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), (1,135,9,143)(2,134,10,142)(3,133,11,141)(4,132,12,140)(5,131,13,139)(6,130,14,138)(7,129,15,137)(8,144,16,136)(17,187,25,179)(18,186,26,178)(19,185,27,177)(20,184,28,192)(21,183,29,191)(22,182,30,190)(23,181,31,189)(24,180,32,188)(33,240,41,232)(34,239,42,231)(35,238,43,230)(36,237,44,229)(37,236,45,228)(38,235,46,227)(39,234,47,226)(40,233,48,225)(49,374,57,382)(50,373,58,381)(51,372,59,380)(52,371,60,379)(53,370,61,378)(54,369,62,377)(55,384,63,376)(56,383,64,375)(65,102,73,110)(66,101,74,109)(67,100,75,108)(68,99,76,107)(69,98,77,106)(70,97,78,105)(71,112,79,104)(72,111,80,103)(81,148,89,156)(82,147,90,155)(83,146,91,154)(84,145,92,153)(85,160,93,152)(86,159,94,151)(87,158,95,150)(88,157,96,149)(113,295,121,303)(114,294,122,302)(115,293,123,301)(116,292,124,300)(117,291,125,299)(118,290,126,298)(119,289,127,297)(120,304,128,296)(161,269,169,261)(162,268,170,260)(163,267,171,259)(164,266,172,258)(165,265,173,257)(166,264,174,272)(167,263,175,271)(168,262,176,270)(193,316,201,308)(194,315,202,307)(195,314,203,306)(196,313,204,305)(197,312,205,320)(198,311,206,319)(199,310,207,318)(200,309,208,317)(209,341,217,349)(210,340,218,348)(211,339,219,347)(212,338,220,346)(213,337,221,345)(214,352,222,344)(215,351,223,343)(216,350,224,342)(241,330,249,322)(242,329,250,321)(243,328,251,336)(244,327,252,335)(245,326,253,334)(246,325,254,333)(247,324,255,332)(248,323,256,331)(273,395,281,387)(274,394,282,386)(275,393,283,385)(276,392,284,400)(277,391,285,399)(278,390,286,398)(279,389,287,397)(280,388,288,396)(353,403,361,411)(354,402,362,410)(355,401,363,409)(356,416,364,408)(357,415,365,407)(358,414,366,406)(359,413,367,405)(360,412,368,404)(417,445,425,437)(418,444,426,436)(419,443,427,435)(420,442,428,434)(421,441,429,433)(422,440,430,448)(423,439,431,447)(424,438,432,446)>;
G:=Group( (1,371)(2,372)(3,373)(4,374)(5,375)(6,376)(7,377)(8,378)(9,379)(10,380)(11,381)(12,382)(13,383)(14,384)(15,369)(16,370)(17,413)(18,414)(19,415)(20,416)(21,401)(22,402)(23,403)(24,404)(25,405)(26,406)(27,407)(28,408)(29,409)(30,410)(31,411)(32,412)(33,278)(34,279)(35,280)(36,281)(37,282)(38,283)(39,284)(40,285)(41,286)(42,287)(43,288)(44,273)(45,274)(46,275)(47,276)(48,277)(49,140)(50,141)(51,142)(52,143)(53,144)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,214)(66,215)(67,216)(68,217)(69,218)(70,219)(71,220)(72,221)(73,222)(74,223)(75,224)(76,209)(77,210)(78,211)(79,212)(80,213)(81,299)(82,300)(83,301)(84,302)(85,303)(86,304)(87,289)(88,290)(89,291)(90,292)(91,293)(92,294)(93,295)(94,296)(95,297)(96,298)(97,347)(98,348)(99,349)(100,350)(101,351)(102,352)(103,337)(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)(113,160)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(161,208)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,357)(178,358)(179,359)(180,360)(181,361)(182,362)(183,363)(184,364)(185,365)(186,366)(187,367)(188,368)(189,353)(190,354)(191,355)(192,356)(225,391)(226,392)(227,393)(228,394)(229,395)(230,396)(231,397)(232,398)(233,399)(234,400)(235,385)(236,386)(237,387)(238,388)(239,389)(240,390)(241,436)(242,437)(243,438)(244,439)(245,440)(246,441)(247,442)(248,443)(249,444)(250,445)(251,446)(252,447)(253,448)(254,433)(255,434)(256,435)(257,305)(258,306)(259,307)(260,308)(261,309)(262,310)(263,311)(264,312)(265,313)(266,314)(267,315)(268,316)(269,317)(270,318)(271,319)(272,320)(321,425)(322,426)(323,427)(324,428)(325,429)(326,430)(327,431)(328,432)(329,417)(330,418)(331,419)(332,420)(333,421)(334,422)(335,423)(336,424), (1,351,318,158,328,228,185)(2,186,229,329,159,319,352)(3,337,320,160,330,230,187)(4,188,231,331,145,305,338)(5,339,306,146,332,232,189)(6,190,233,333,147,307,340)(7,341,308,148,334,234,191)(8,192,235,335,149,309,342)(9,343,310,150,336,236,177)(10,178,237,321,151,311,344)(11,345,312,152,322,238,179)(12,180,239,323,153,313,346)(13,347,314,154,324,240,181)(14,182,225,325,155,315,348)(15,349,316,156,326,226,183)(16,184,227,327,157,317,350)(17,141,213,205,85,241,43)(18,44,242,86,206,214,142)(19,143,215,207,87,243,45)(20,46,244,88,208,216,144)(21,129,217,193,89,245,47)(22,48,246,90,194,218,130)(23,131,219,195,91,247,33)(24,34,248,92,196,220,132)(25,133,221,197,93,249,35)(26,36,250,94,198,222,134)(27,135,223,199,95,251,37)(28,38,252,96,200,224,136)(29,137,209,201,81,253,39)(30,40,254,82,202,210,138)(31,139,211,203,83,255,41)(32,42,256,84,204,212,140)(49,412,287,435,302,173,79)(50,80,174,303,436,288,413)(51,414,273,437,304,175,65)(52,66,176,289,438,274,415)(53,416,275,439,290,161,67)(54,68,162,291,440,276,401)(55,402,277,441,292,163,69)(56,70,164,293,442,278,403)(57,404,279,443,294,165,71)(58,72,166,295,444,280,405)(59,406,281,445,296,167,73)(60,74,168,297,446,282,407)(61,408,283,447,298,169,75)(62,76,170,299,448,284,409)(63,410,285,433,300,171,77)(64,78,172,301,434,286,411)(97,266,123,428,390,361,383)(98,384,362,391,429,124,267)(99,268,125,430,392,363,369)(100,370,364,393,431,126,269)(101,270,127,432,394,365,371)(102,372,366,395,417,128,271)(103,272,113,418,396,367,373)(104,374,368,397,419,114,257)(105,258,115,420,398,353,375)(106,376,354,399,421,116,259)(107,260,117,422,400,355,377)(108,378,356,385,423,118,261)(109,262,119,424,386,357,379)(110,380,358,387,425,120,263)(111,264,121,426,388,359,381)(112,382,360,389,427,122,265), (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), (1,135,9,143)(2,134,10,142)(3,133,11,141)(4,132,12,140)(5,131,13,139)(6,130,14,138)(7,129,15,137)(8,144,16,136)(17,187,25,179)(18,186,26,178)(19,185,27,177)(20,184,28,192)(21,183,29,191)(22,182,30,190)(23,181,31,189)(24,180,32,188)(33,240,41,232)(34,239,42,231)(35,238,43,230)(36,237,44,229)(37,236,45,228)(38,235,46,227)(39,234,47,226)(40,233,48,225)(49,374,57,382)(50,373,58,381)(51,372,59,380)(52,371,60,379)(53,370,61,378)(54,369,62,377)(55,384,63,376)(56,383,64,375)(65,102,73,110)(66,101,74,109)(67,100,75,108)(68,99,76,107)(69,98,77,106)(70,97,78,105)(71,112,79,104)(72,111,80,103)(81,148,89,156)(82,147,90,155)(83,146,91,154)(84,145,92,153)(85,160,93,152)(86,159,94,151)(87,158,95,150)(88,157,96,149)(113,295,121,303)(114,294,122,302)(115,293,123,301)(116,292,124,300)(117,291,125,299)(118,290,126,298)(119,289,127,297)(120,304,128,296)(161,269,169,261)(162,268,170,260)(163,267,171,259)(164,266,172,258)(165,265,173,257)(166,264,174,272)(167,263,175,271)(168,262,176,270)(193,316,201,308)(194,315,202,307)(195,314,203,306)(196,313,204,305)(197,312,205,320)(198,311,206,319)(199,310,207,318)(200,309,208,317)(209,341,217,349)(210,340,218,348)(211,339,219,347)(212,338,220,346)(213,337,221,345)(214,352,222,344)(215,351,223,343)(216,350,224,342)(241,330,249,322)(242,329,250,321)(243,328,251,336)(244,327,252,335)(245,326,253,334)(246,325,254,333)(247,324,255,332)(248,323,256,331)(273,395,281,387)(274,394,282,386)(275,393,283,385)(276,392,284,400)(277,391,285,399)(278,390,286,398)(279,389,287,397)(280,388,288,396)(353,403,361,411)(354,402,362,410)(355,401,363,409)(356,416,364,408)(357,415,365,407)(358,414,366,406)(359,413,367,405)(360,412,368,404)(417,445,425,437)(418,444,426,436)(419,443,427,435)(420,442,428,434)(421,441,429,433)(422,440,430,448)(423,439,431,447)(424,438,432,446) );
G=PermutationGroup([[(1,371),(2,372),(3,373),(4,374),(5,375),(6,376),(7,377),(8,378),(9,379),(10,380),(11,381),(12,382),(13,383),(14,384),(15,369),(16,370),(17,413),(18,414),(19,415),(20,416),(21,401),(22,402),(23,403),(24,404),(25,405),(26,406),(27,407),(28,408),(29,409),(30,410),(31,411),(32,412),(33,278),(34,279),(35,280),(36,281),(37,282),(38,283),(39,284),(40,285),(41,286),(42,287),(43,288),(44,273),(45,274),(46,275),(47,276),(48,277),(49,140),(50,141),(51,142),(52,143),(53,144),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,214),(66,215),(67,216),(68,217),(69,218),(70,219),(71,220),(72,221),(73,222),(74,223),(75,224),(76,209),(77,210),(78,211),(79,212),(80,213),(81,299),(82,300),(83,301),(84,302),(85,303),(86,304),(87,289),(88,290),(89,291),(90,292),(91,293),(92,294),(93,295),(94,296),(95,297),(96,298),(97,347),(98,348),(99,349),(100,350),(101,351),(102,352),(103,337),(104,338),(105,339),(106,340),(107,341),(108,342),(109,343),(110,344),(111,345),(112,346),(113,160),(114,145),(115,146),(116,147),(117,148),(118,149),(119,150),(120,151),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(161,208),(162,193),(163,194),(164,195),(165,196),(166,197),(167,198),(168,199),(169,200),(170,201),(171,202),(172,203),(173,204),(174,205),(175,206),(176,207),(177,357),(178,358),(179,359),(180,360),(181,361),(182,362),(183,363),(184,364),(185,365),(186,366),(187,367),(188,368),(189,353),(190,354),(191,355),(192,356),(225,391),(226,392),(227,393),(228,394),(229,395),(230,396),(231,397),(232,398),(233,399),(234,400),(235,385),(236,386),(237,387),(238,388),(239,389),(240,390),(241,436),(242,437),(243,438),(244,439),(245,440),(246,441),(247,442),(248,443),(249,444),(250,445),(251,446),(252,447),(253,448),(254,433),(255,434),(256,435),(257,305),(258,306),(259,307),(260,308),(261,309),(262,310),(263,311),(264,312),(265,313),(266,314),(267,315),(268,316),(269,317),(270,318),(271,319),(272,320),(321,425),(322,426),(323,427),(324,428),(325,429),(326,430),(327,431),(328,432),(329,417),(330,418),(331,419),(332,420),(333,421),(334,422),(335,423),(336,424)], [(1,351,318,158,328,228,185),(2,186,229,329,159,319,352),(3,337,320,160,330,230,187),(4,188,231,331,145,305,338),(5,339,306,146,332,232,189),(6,190,233,333,147,307,340),(7,341,308,148,334,234,191),(8,192,235,335,149,309,342),(9,343,310,150,336,236,177),(10,178,237,321,151,311,344),(11,345,312,152,322,238,179),(12,180,239,323,153,313,346),(13,347,314,154,324,240,181),(14,182,225,325,155,315,348),(15,349,316,156,326,226,183),(16,184,227,327,157,317,350),(17,141,213,205,85,241,43),(18,44,242,86,206,214,142),(19,143,215,207,87,243,45),(20,46,244,88,208,216,144),(21,129,217,193,89,245,47),(22,48,246,90,194,218,130),(23,131,219,195,91,247,33),(24,34,248,92,196,220,132),(25,133,221,197,93,249,35),(26,36,250,94,198,222,134),(27,135,223,199,95,251,37),(28,38,252,96,200,224,136),(29,137,209,201,81,253,39),(30,40,254,82,202,210,138),(31,139,211,203,83,255,41),(32,42,256,84,204,212,140),(49,412,287,435,302,173,79),(50,80,174,303,436,288,413),(51,414,273,437,304,175,65),(52,66,176,289,438,274,415),(53,416,275,439,290,161,67),(54,68,162,291,440,276,401),(55,402,277,441,292,163,69),(56,70,164,293,442,278,403),(57,404,279,443,294,165,71),(58,72,166,295,444,280,405),(59,406,281,445,296,167,73),(60,74,168,297,446,282,407),(61,408,283,447,298,169,75),(62,76,170,299,448,284,409),(63,410,285,433,300,171,77),(64,78,172,301,434,286,411),(97,266,123,428,390,361,383),(98,384,362,391,429,124,267),(99,268,125,430,392,363,369),(100,370,364,393,431,126,269),(101,270,127,432,394,365,371),(102,372,366,395,417,128,271),(103,272,113,418,396,367,373),(104,374,368,397,419,114,257),(105,258,115,420,398,353,375),(106,376,354,399,421,116,259),(107,260,117,422,400,355,377),(108,378,356,385,423,118,261),(109,262,119,424,386,357,379),(110,380,358,387,425,120,263),(111,264,121,426,388,359,381),(112,382,360,389,427,122,265)], [(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)], [(1,135,9,143),(2,134,10,142),(3,133,11,141),(4,132,12,140),(5,131,13,139),(6,130,14,138),(7,129,15,137),(8,144,16,136),(17,187,25,179),(18,186,26,178),(19,185,27,177),(20,184,28,192),(21,183,29,191),(22,182,30,190),(23,181,31,189),(24,180,32,188),(33,240,41,232),(34,239,42,231),(35,238,43,230),(36,237,44,229),(37,236,45,228),(38,235,46,227),(39,234,47,226),(40,233,48,225),(49,374,57,382),(50,373,58,381),(51,372,59,380),(52,371,60,379),(53,370,61,378),(54,369,62,377),(55,384,63,376),(56,383,64,375),(65,102,73,110),(66,101,74,109),(67,100,75,108),(68,99,76,107),(69,98,77,106),(70,97,78,105),(71,112,79,104),(72,111,80,103),(81,148,89,156),(82,147,90,155),(83,146,91,154),(84,145,92,153),(85,160,93,152),(86,159,94,151),(87,158,95,150),(88,157,96,149),(113,295,121,303),(114,294,122,302),(115,293,123,301),(116,292,124,300),(117,291,125,299),(118,290,126,298),(119,289,127,297),(120,304,128,296),(161,269,169,261),(162,268,170,260),(163,267,171,259),(164,266,172,258),(165,265,173,257),(166,264,174,272),(167,263,175,271),(168,262,176,270),(193,316,201,308),(194,315,202,307),(195,314,203,306),(196,313,204,305),(197,312,205,320),(198,311,206,319),(199,310,207,318),(200,309,208,317),(209,341,217,349),(210,340,218,348),(211,339,219,347),(212,338,220,346),(213,337,221,345),(214,352,222,344),(215,351,223,343),(216,350,224,342),(241,330,249,322),(242,329,250,321),(243,328,251,336),(244,327,252,335),(245,326,253,334),(246,325,254,333),(247,324,255,332),(248,323,256,331),(273,395,281,387),(274,394,282,386),(275,393,283,385),(276,392,284,400),(277,391,285,399),(278,390,286,398),(279,389,287,397),(280,388,288,396),(353,403,361,411),(354,402,362,410),(355,401,363,409),(356,416,364,408),(357,415,365,407),(358,414,366,406),(359,413,367,405),(360,412,368,404),(417,445,425,437),(418,444,426,436),(419,443,427,435),(420,442,428,434),(421,441,429,433),(422,440,430,448),(423,439,431,447),(424,438,432,446)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | Q32 | C7⋊D4 | C7⋊D4 | D4⋊D7 | D4⋊D7 | C7⋊Q32 |
| kernel | C2×C7⋊Q32 | C2×C7⋊C16 | C7⋊Q32 | C2×Dic28 | C14×Q16 | C56 | C2×C28 | C2×Q16 | C28 | C2×C14 | C2×C8 | Q16 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 8 | 6 | 6 | 3 | 3 | 12 |
Matrix representation of C2×C7⋊Q32 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 79 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 75 | 20 | 0 | 0 |
| 69 | 38 | 0 | 0 |
| 0 | 0 | 99 | 31 |
| 0 | 0 | 50 | 91 |
| 91 | 12 | 0 | 0 |
| 101 | 22 | 0 | 0 |
| 0 | 0 | 104 | 13 |
| 0 | 0 | 98 | 9 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[79,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[75,69,0,0,20,38,0,0,0,0,99,50,0,0,31,91],[91,101,0,0,12,22,0,0,0,0,104,98,0,0,13,9] >;
C2×C7⋊Q32 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes Q_{32} % in TeX
G:=Group("C2xC7:Q32"); // GroupNames label
G:=SmallGroup(448,714);
// by ID
G=gap.SmallGroup(448,714);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,184,675,185,192,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^7=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations