direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28⋊Q8, C28⋊4(C2×Q8), (C2×C28)⋊2Q8, C14⋊2(C4⋊Q8), Dic7⋊1(C2×Q8), (C2×Dic7)⋊8Q8, (C2×C4)⋊6Dic14, C4⋊2(C2×Dic14), C4⋊C4.258D14, Dic7.2(C2×D4), C22.30(Q8×D7), C14.7(C22×Q8), (C2×C14).41C24, C14.39(C22×D4), C22.129(D4×D7), (C2×C28).132C23, (C2×Dic7).120D4, (C22×C4).173D14, C2.9(C22×Dic14), C22.79(C23×D7), C4⋊Dic7.356C22, (C2×Dic7).13C23, (C22×Dic14).8C2, C22.36(C2×Dic14), C23.321(C22×D7), Dic7⋊C4.102C22, (C22×C14).390C23, (C22×C28).214C22, (C4×Dic7).247C22, (C2×Dic14).226C22, (C22×Dic7).206C22, C7⋊2(C2×C4⋊Q8), C2.5(C2×Q8×D7), C2.13(C2×D4×D7), (C2×C4⋊C4).24D7, (C14×C4⋊C4).17C2, (C2×C14).50(C2×Q8), (C2×C4×Dic7).12C2, (C2×C14).385(C2×D4), (C2×C4⋊Dic7).42C2, (C7×C4⋊C4).290C22, (C2×Dic7⋊C4).21C2, (C2×C4).137(C22×D7), SmallGroup(448,950)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28⋊Q8
G = < a,b,c,d | a2=b28=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b15, dbd-1=b13, dcd-1=c-1 >
Subgroups: 1156 in 290 conjugacy classes, 143 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C4⋊Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C28⋊Q8, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C14×C4⋊C4, C22×Dic14, C2×C28⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C24, D14, C4⋊Q8, C22×D4, C22×Q8, Dic14, C22×D7, C2×C4⋊Q8, C2×Dic14, D4×D7, Q8×D7, C23×D7, C28⋊Q8, C22×Dic14, C2×D4×D7, C2×Q8×D7, C2×C28⋊Q8
►Regular action on 448 points
Generators in S448
(1 248)(2 249)(3 250)(4 251)(5 252)(6 225)(7 226)(8 227)(9 228)(10 229)(11 230)(12 231)(13 232)(14 233)(15 234)(16 235)(17 236)(18 237)(19 238)(20 239)(21 240)(22 241)(23 242)(24 243)(25 244)(26 245)(27 246)(28 247)(29 164)(30 165)(31 166)(32 167)(33 168)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 156)(50 157)(51 158)(52 159)(53 160)(54 161)(55 162)(56 163)(57 298)(58 299)(59 300)(60 301)(61 302)(62 303)(63 304)(64 305)(65 306)(66 307)(67 308)(68 281)(69 282)(70 283)(71 284)(72 285)(73 286)(74 287)(75 288)(76 289)(77 290)(78 291)(79 292)(80 293)(81 294)(82 295)(83 296)(84 297)(85 268)(86 269)(87 270)(88 271)(89 272)(90 273)(91 274)(92 275)(93 276)(94 277)(95 278)(96 279)(97 280)(98 253)(99 254)(100 255)(101 256)(102 257)(103 258)(104 259)(105 260)(106 261)(107 262)(108 263)(109 264)(110 265)(111 266)(112 267)(113 196)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)(121 176)(122 177)(123 178)(124 179)(125 180)(126 181)(127 182)(128 183)(129 184)(130 185)(131 186)(132 187)(133 188)(134 189)(135 190)(136 191)(137 192)(138 193)(139 194)(140 195)(197 387)(198 388)(199 389)(200 390)(201 391)(202 392)(203 365)(204 366)(205 367)(206 368)(207 369)(208 370)(209 371)(210 372)(211 373)(212 374)(213 375)(214 376)(215 377)(216 378)(217 379)(218 380)(219 381)(220 382)(221 383)(222 384)(223 385)(224 386)(309 394)(310 395)(311 396)(312 397)(313 398)(314 399)(315 400)(316 401)(317 402)(318 403)(319 404)(320 405)(321 406)(322 407)(323 408)(324 409)(325 410)(326 411)(327 412)(328 413)(329 414)(330 415)(331 416)(332 417)(333 418)(334 419)(335 420)(336 393)(337 443)(338 444)(339 445)(340 446)(341 447)(342 448)(343 421)(344 422)(345 423)(346 424)(347 425)(348 426)(349 427)(350 428)(351 429)(352 430)(353 431)(354 432)(355 433)(356 434)(357 435)(358 436)(359 437)(360 438)(361 439)(362 440)(363 441)(364 442)
(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)
(1 163 383 76)(2 150 384 63)(3 165 385 78)(4 152 386 65)(5 167 387 80)(6 154 388 67)(7 141 389 82)(8 156 390 69)(9 143 391 84)(10 158 392 71)(11 145 365 58)(12 160 366 73)(13 147 367 60)(14 162 368 75)(15 149 369 62)(16 164 370 77)(17 151 371 64)(18 166 372 79)(19 153 373 66)(20 168 374 81)(21 155 375 68)(22 142 376 83)(23 157 377 70)(24 144 378 57)(25 159 379 72)(26 146 380 59)(27 161 381 74)(28 148 382 61)(29 208 290 235)(30 223 291 250)(31 210 292 237)(32 197 293 252)(33 212 294 239)(34 199 295 226)(35 214 296 241)(36 201 297 228)(37 216 298 243)(38 203 299 230)(39 218 300 245)(40 205 301 232)(41 220 302 247)(42 207 303 234)(43 222 304 249)(44 209 305 236)(45 224 306 251)(46 211 307 238)(47 198 308 225)(48 213 281 240)(49 200 282 227)(50 215 283 242)(51 202 284 229)(52 217 285 244)(53 204 286 231)(54 219 287 246)(55 206 288 233)(56 221 289 248)(85 185 406 432)(86 172 407 447)(87 187 408 434)(88 174 409 421)(89 189 410 436)(90 176 411 423)(91 191 412 438)(92 178 413 425)(93 193 414 440)(94 180 415 427)(95 195 416 442)(96 182 417 429)(97 169 418 444)(98 184 419 431)(99 171 420 446)(100 186 393 433)(101 173 394 448)(102 188 395 435)(103 175 396 422)(104 190 397 437)(105 177 398 424)(106 192 399 439)(107 179 400 426)(108 194 401 441)(109 181 402 428)(110 196 403 443)(111 183 404 430)(112 170 405 445)(113 318 337 265)(114 333 338 280)(115 320 339 267)(116 335 340 254)(117 322 341 269)(118 309 342 256)(119 324 343 271)(120 311 344 258)(121 326 345 273)(122 313 346 260)(123 328 347 275)(124 315 348 262)(125 330 349 277)(126 317 350 264)(127 332 351 279)(128 319 352 266)(129 334 353 253)(130 321 354 268)(131 336 355 255)(132 323 356 270)(133 310 357 257)(134 325 358 272)(135 312 359 259)(136 327 360 274)(137 314 361 261)(138 329 362 276)(139 316 363 263)(140 331 364 278)
(1 176 383 423)(2 189 384 436)(3 174 385 421)(4 187 386 434)(5 172 387 447)(6 185 388 432)(7 170 389 445)(8 183 390 430)(9 196 391 443)(10 181 392 428)(11 194 365 441)(12 179 366 426)(13 192 367 439)(14 177 368 424)(15 190 369 437)(16 175 370 422)(17 188 371 435)(18 173 372 448)(19 186 373 433)(20 171 374 446)(21 184 375 431)(22 169 376 444)(23 182 377 429)(24 195 378 442)(25 180 379 427)(26 193 380 440)(27 178 381 425)(28 191 382 438)(29 258 290 311)(30 271 291 324)(31 256 292 309)(32 269 293 322)(33 254 294 335)(34 267 295 320)(35 280 296 333)(36 265 297 318)(37 278 298 331)(38 263 299 316)(39 276 300 329)(40 261 301 314)(41 274 302 327)(42 259 303 312)(43 272 304 325)(44 257 305 310)(45 270 306 323)(46 255 307 336)(47 268 308 321)(48 253 281 334)(49 266 282 319)(50 279 283 332)(51 264 284 317)(52 277 285 330)(53 262 286 315)(54 275 287 328)(55 260 288 313)(56 273 289 326)(57 416 144 95)(58 401 145 108)(59 414 146 93)(60 399 147 106)(61 412 148 91)(62 397 149 104)(63 410 150 89)(64 395 151 102)(65 408 152 87)(66 393 153 100)(67 406 154 85)(68 419 155 98)(69 404 156 111)(70 417 157 96)(71 402 158 109)(72 415 159 94)(73 400 160 107)(74 413 161 92)(75 398 162 105)(76 411 163 90)(77 396 164 103)(78 409 165 88)(79 394 166 101)(80 407 167 86)(81 420 168 99)(82 405 141 112)(83 418 142 97)(84 403 143 110)(113 201 337 228)(114 214 338 241)(115 199 339 226)(116 212 340 239)(117 197 341 252)(118 210 342 237)(119 223 343 250)(120 208 344 235)(121 221 345 248)(122 206 346 233)(123 219 347 246)(124 204 348 231)(125 217 349 244)(126 202 350 229)(127 215 351 242)(128 200 352 227)(129 213 353 240)(130 198 354 225)(131 211 355 238)(132 224 356 251)(133 209 357 236)(134 222 358 249)(135 207 359 234)(136 220 360 247)(137 205 361 232)(138 218 362 245)(139 203 363 230)(140 216 364 243)
G:=sub<Sym(448)| (1,248)(2,249)(3,250)(4,251)(5,252)(6,225)(7,226)(8,227)(9,228)(10,229)(11,230)(12,231)(13,232)(14,233)(15,234)(16,235)(17,236)(18,237)(19,238)(20,239)(21,240)(22,241)(23,242)(24,243)(25,244)(26,245)(27,246)(28,247)(29,164)(30,165)(31,166)(32,167)(33,168)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,298)(58,299)(59,300)(60,301)(61,302)(62,303)(63,304)(64,305)(65,306)(66,307)(67,308)(68,281)(69,282)(70,283)(71,284)(72,285)(73,286)(74,287)(75,288)(76,289)(77,290)(78,291)(79,292)(80,293)(81,294)(82,295)(83,296)(84,297)(85,268)(86,269)(87,270)(88,271)(89,272)(90,273)(91,274)(92,275)(93,276)(94,277)(95,278)(96,279)(97,280)(98,253)(99,254)(100,255)(101,256)(102,257)(103,258)(104,259)(105,260)(106,261)(107,262)(108,263)(109,264)(110,265)(111,266)(112,267)(113,196)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,176)(122,177)(123,178)(124,179)(125,180)(126,181)(127,182)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)(135,190)(136,191)(137,192)(138,193)(139,194)(140,195)(197,387)(198,388)(199,389)(200,390)(201,391)(202,392)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)(217,379)(218,380)(219,381)(220,382)(221,383)(222,384)(223,385)(224,386)(309,394)(310,395)(311,396)(312,397)(313,398)(314,399)(315,400)(316,401)(317,402)(318,403)(319,404)(320,405)(321,406)(322,407)(323,408)(324,409)(325,410)(326,411)(327,412)(328,413)(329,414)(330,415)(331,416)(332,417)(333,418)(334,419)(335,420)(336,393)(337,443)(338,444)(339,445)(340,446)(341,447)(342,448)(343,421)(344,422)(345,423)(346,424)(347,425)(348,426)(349,427)(350,428)(351,429)(352,430)(353,431)(354,432)(355,433)(356,434)(357,435)(358,436)(359,437)(360,438)(361,439)(362,440)(363,441)(364,442), (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), (1,163,383,76)(2,150,384,63)(3,165,385,78)(4,152,386,65)(5,167,387,80)(6,154,388,67)(7,141,389,82)(8,156,390,69)(9,143,391,84)(10,158,392,71)(11,145,365,58)(12,160,366,73)(13,147,367,60)(14,162,368,75)(15,149,369,62)(16,164,370,77)(17,151,371,64)(18,166,372,79)(19,153,373,66)(20,168,374,81)(21,155,375,68)(22,142,376,83)(23,157,377,70)(24,144,378,57)(25,159,379,72)(26,146,380,59)(27,161,381,74)(28,148,382,61)(29,208,290,235)(30,223,291,250)(31,210,292,237)(32,197,293,252)(33,212,294,239)(34,199,295,226)(35,214,296,241)(36,201,297,228)(37,216,298,243)(38,203,299,230)(39,218,300,245)(40,205,301,232)(41,220,302,247)(42,207,303,234)(43,222,304,249)(44,209,305,236)(45,224,306,251)(46,211,307,238)(47,198,308,225)(48,213,281,240)(49,200,282,227)(50,215,283,242)(51,202,284,229)(52,217,285,244)(53,204,286,231)(54,219,287,246)(55,206,288,233)(56,221,289,248)(85,185,406,432)(86,172,407,447)(87,187,408,434)(88,174,409,421)(89,189,410,436)(90,176,411,423)(91,191,412,438)(92,178,413,425)(93,193,414,440)(94,180,415,427)(95,195,416,442)(96,182,417,429)(97,169,418,444)(98,184,419,431)(99,171,420,446)(100,186,393,433)(101,173,394,448)(102,188,395,435)(103,175,396,422)(104,190,397,437)(105,177,398,424)(106,192,399,439)(107,179,400,426)(108,194,401,441)(109,181,402,428)(110,196,403,443)(111,183,404,430)(112,170,405,445)(113,318,337,265)(114,333,338,280)(115,320,339,267)(116,335,340,254)(117,322,341,269)(118,309,342,256)(119,324,343,271)(120,311,344,258)(121,326,345,273)(122,313,346,260)(123,328,347,275)(124,315,348,262)(125,330,349,277)(126,317,350,264)(127,332,351,279)(128,319,352,266)(129,334,353,253)(130,321,354,268)(131,336,355,255)(132,323,356,270)(133,310,357,257)(134,325,358,272)(135,312,359,259)(136,327,360,274)(137,314,361,261)(138,329,362,276)(139,316,363,263)(140,331,364,278), (1,176,383,423)(2,189,384,436)(3,174,385,421)(4,187,386,434)(5,172,387,447)(6,185,388,432)(7,170,389,445)(8,183,390,430)(9,196,391,443)(10,181,392,428)(11,194,365,441)(12,179,366,426)(13,192,367,439)(14,177,368,424)(15,190,369,437)(16,175,370,422)(17,188,371,435)(18,173,372,448)(19,186,373,433)(20,171,374,446)(21,184,375,431)(22,169,376,444)(23,182,377,429)(24,195,378,442)(25,180,379,427)(26,193,380,440)(27,178,381,425)(28,191,382,438)(29,258,290,311)(30,271,291,324)(31,256,292,309)(32,269,293,322)(33,254,294,335)(34,267,295,320)(35,280,296,333)(36,265,297,318)(37,278,298,331)(38,263,299,316)(39,276,300,329)(40,261,301,314)(41,274,302,327)(42,259,303,312)(43,272,304,325)(44,257,305,310)(45,270,306,323)(46,255,307,336)(47,268,308,321)(48,253,281,334)(49,266,282,319)(50,279,283,332)(51,264,284,317)(52,277,285,330)(53,262,286,315)(54,275,287,328)(55,260,288,313)(56,273,289,326)(57,416,144,95)(58,401,145,108)(59,414,146,93)(60,399,147,106)(61,412,148,91)(62,397,149,104)(63,410,150,89)(64,395,151,102)(65,408,152,87)(66,393,153,100)(67,406,154,85)(68,419,155,98)(69,404,156,111)(70,417,157,96)(71,402,158,109)(72,415,159,94)(73,400,160,107)(74,413,161,92)(75,398,162,105)(76,411,163,90)(77,396,164,103)(78,409,165,88)(79,394,166,101)(80,407,167,86)(81,420,168,99)(82,405,141,112)(83,418,142,97)(84,403,143,110)(113,201,337,228)(114,214,338,241)(115,199,339,226)(116,212,340,239)(117,197,341,252)(118,210,342,237)(119,223,343,250)(120,208,344,235)(121,221,345,248)(122,206,346,233)(123,219,347,246)(124,204,348,231)(125,217,349,244)(126,202,350,229)(127,215,351,242)(128,200,352,227)(129,213,353,240)(130,198,354,225)(131,211,355,238)(132,224,356,251)(133,209,357,236)(134,222,358,249)(135,207,359,234)(136,220,360,247)(137,205,361,232)(138,218,362,245)(139,203,363,230)(140,216,364,243)>;
G:=Group( (1,248)(2,249)(3,250)(4,251)(5,252)(6,225)(7,226)(8,227)(9,228)(10,229)(11,230)(12,231)(13,232)(14,233)(15,234)(16,235)(17,236)(18,237)(19,238)(20,239)(21,240)(22,241)(23,242)(24,243)(25,244)(26,245)(27,246)(28,247)(29,164)(30,165)(31,166)(32,167)(33,168)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,298)(58,299)(59,300)(60,301)(61,302)(62,303)(63,304)(64,305)(65,306)(66,307)(67,308)(68,281)(69,282)(70,283)(71,284)(72,285)(73,286)(74,287)(75,288)(76,289)(77,290)(78,291)(79,292)(80,293)(81,294)(82,295)(83,296)(84,297)(85,268)(86,269)(87,270)(88,271)(89,272)(90,273)(91,274)(92,275)(93,276)(94,277)(95,278)(96,279)(97,280)(98,253)(99,254)(100,255)(101,256)(102,257)(103,258)(104,259)(105,260)(106,261)(107,262)(108,263)(109,264)(110,265)(111,266)(112,267)(113,196)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,176)(122,177)(123,178)(124,179)(125,180)(126,181)(127,182)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)(135,190)(136,191)(137,192)(138,193)(139,194)(140,195)(197,387)(198,388)(199,389)(200,390)(201,391)(202,392)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)(217,379)(218,380)(219,381)(220,382)(221,383)(222,384)(223,385)(224,386)(309,394)(310,395)(311,396)(312,397)(313,398)(314,399)(315,400)(316,401)(317,402)(318,403)(319,404)(320,405)(321,406)(322,407)(323,408)(324,409)(325,410)(326,411)(327,412)(328,413)(329,414)(330,415)(331,416)(332,417)(333,418)(334,419)(335,420)(336,393)(337,443)(338,444)(339,445)(340,446)(341,447)(342,448)(343,421)(344,422)(345,423)(346,424)(347,425)(348,426)(349,427)(350,428)(351,429)(352,430)(353,431)(354,432)(355,433)(356,434)(357,435)(358,436)(359,437)(360,438)(361,439)(362,440)(363,441)(364,442), (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), (1,163,383,76)(2,150,384,63)(3,165,385,78)(4,152,386,65)(5,167,387,80)(6,154,388,67)(7,141,389,82)(8,156,390,69)(9,143,391,84)(10,158,392,71)(11,145,365,58)(12,160,366,73)(13,147,367,60)(14,162,368,75)(15,149,369,62)(16,164,370,77)(17,151,371,64)(18,166,372,79)(19,153,373,66)(20,168,374,81)(21,155,375,68)(22,142,376,83)(23,157,377,70)(24,144,378,57)(25,159,379,72)(26,146,380,59)(27,161,381,74)(28,148,382,61)(29,208,290,235)(30,223,291,250)(31,210,292,237)(32,197,293,252)(33,212,294,239)(34,199,295,226)(35,214,296,241)(36,201,297,228)(37,216,298,243)(38,203,299,230)(39,218,300,245)(40,205,301,232)(41,220,302,247)(42,207,303,234)(43,222,304,249)(44,209,305,236)(45,224,306,251)(46,211,307,238)(47,198,308,225)(48,213,281,240)(49,200,282,227)(50,215,283,242)(51,202,284,229)(52,217,285,244)(53,204,286,231)(54,219,287,246)(55,206,288,233)(56,221,289,248)(85,185,406,432)(86,172,407,447)(87,187,408,434)(88,174,409,421)(89,189,410,436)(90,176,411,423)(91,191,412,438)(92,178,413,425)(93,193,414,440)(94,180,415,427)(95,195,416,442)(96,182,417,429)(97,169,418,444)(98,184,419,431)(99,171,420,446)(100,186,393,433)(101,173,394,448)(102,188,395,435)(103,175,396,422)(104,190,397,437)(105,177,398,424)(106,192,399,439)(107,179,400,426)(108,194,401,441)(109,181,402,428)(110,196,403,443)(111,183,404,430)(112,170,405,445)(113,318,337,265)(114,333,338,280)(115,320,339,267)(116,335,340,254)(117,322,341,269)(118,309,342,256)(119,324,343,271)(120,311,344,258)(121,326,345,273)(122,313,346,260)(123,328,347,275)(124,315,348,262)(125,330,349,277)(126,317,350,264)(127,332,351,279)(128,319,352,266)(129,334,353,253)(130,321,354,268)(131,336,355,255)(132,323,356,270)(133,310,357,257)(134,325,358,272)(135,312,359,259)(136,327,360,274)(137,314,361,261)(138,329,362,276)(139,316,363,263)(140,331,364,278), (1,176,383,423)(2,189,384,436)(3,174,385,421)(4,187,386,434)(5,172,387,447)(6,185,388,432)(7,170,389,445)(8,183,390,430)(9,196,391,443)(10,181,392,428)(11,194,365,441)(12,179,366,426)(13,192,367,439)(14,177,368,424)(15,190,369,437)(16,175,370,422)(17,188,371,435)(18,173,372,448)(19,186,373,433)(20,171,374,446)(21,184,375,431)(22,169,376,444)(23,182,377,429)(24,195,378,442)(25,180,379,427)(26,193,380,440)(27,178,381,425)(28,191,382,438)(29,258,290,311)(30,271,291,324)(31,256,292,309)(32,269,293,322)(33,254,294,335)(34,267,295,320)(35,280,296,333)(36,265,297,318)(37,278,298,331)(38,263,299,316)(39,276,300,329)(40,261,301,314)(41,274,302,327)(42,259,303,312)(43,272,304,325)(44,257,305,310)(45,270,306,323)(46,255,307,336)(47,268,308,321)(48,253,281,334)(49,266,282,319)(50,279,283,332)(51,264,284,317)(52,277,285,330)(53,262,286,315)(54,275,287,328)(55,260,288,313)(56,273,289,326)(57,416,144,95)(58,401,145,108)(59,414,146,93)(60,399,147,106)(61,412,148,91)(62,397,149,104)(63,410,150,89)(64,395,151,102)(65,408,152,87)(66,393,153,100)(67,406,154,85)(68,419,155,98)(69,404,156,111)(70,417,157,96)(71,402,158,109)(72,415,159,94)(73,400,160,107)(74,413,161,92)(75,398,162,105)(76,411,163,90)(77,396,164,103)(78,409,165,88)(79,394,166,101)(80,407,167,86)(81,420,168,99)(82,405,141,112)(83,418,142,97)(84,403,143,110)(113,201,337,228)(114,214,338,241)(115,199,339,226)(116,212,340,239)(117,197,341,252)(118,210,342,237)(119,223,343,250)(120,208,344,235)(121,221,345,248)(122,206,346,233)(123,219,347,246)(124,204,348,231)(125,217,349,244)(126,202,350,229)(127,215,351,242)(128,200,352,227)(129,213,353,240)(130,198,354,225)(131,211,355,238)(132,224,356,251)(133,209,357,236)(134,222,358,249)(135,207,359,234)(136,220,360,247)(137,205,361,232)(138,218,362,245)(139,203,363,230)(140,216,364,243) );
G=PermutationGroup([[(1,248),(2,249),(3,250),(4,251),(5,252),(6,225),(7,226),(8,227),(9,228),(10,229),(11,230),(12,231),(13,232),(14,233),(15,234),(16,235),(17,236),(18,237),(19,238),(20,239),(21,240),(22,241),(23,242),(24,243),(25,244),(26,245),(27,246),(28,247),(29,164),(30,165),(31,166),(32,167),(33,168),(34,141),(35,142),(36,143),(37,144),(38,145),(39,146),(40,147),(41,148),(42,149),(43,150),(44,151),(45,152),(46,153),(47,154),(48,155),(49,156),(50,157),(51,158),(52,159),(53,160),(54,161),(55,162),(56,163),(57,298),(58,299),(59,300),(60,301),(61,302),(62,303),(63,304),(64,305),(65,306),(66,307),(67,308),(68,281),(69,282),(70,283),(71,284),(72,285),(73,286),(74,287),(75,288),(76,289),(77,290),(78,291),(79,292),(80,293),(81,294),(82,295),(83,296),(84,297),(85,268),(86,269),(87,270),(88,271),(89,272),(90,273),(91,274),(92,275),(93,276),(94,277),(95,278),(96,279),(97,280),(98,253),(99,254),(100,255),(101,256),(102,257),(103,258),(104,259),(105,260),(106,261),(107,262),(108,263),(109,264),(110,265),(111,266),(112,267),(113,196),(114,169),(115,170),(116,171),(117,172),(118,173),(119,174),(120,175),(121,176),(122,177),(123,178),(124,179),(125,180),(126,181),(127,182),(128,183),(129,184),(130,185),(131,186),(132,187),(133,188),(134,189),(135,190),(136,191),(137,192),(138,193),(139,194),(140,195),(197,387),(198,388),(199,389),(200,390),(201,391),(202,392),(203,365),(204,366),(205,367),(206,368),(207,369),(208,370),(209,371),(210,372),(211,373),(212,374),(213,375),(214,376),(215,377),(216,378),(217,379),(218,380),(219,381),(220,382),(221,383),(222,384),(223,385),(224,386),(309,394),(310,395),(311,396),(312,397),(313,398),(314,399),(315,400),(316,401),(317,402),(318,403),(319,404),(320,405),(321,406),(322,407),(323,408),(324,409),(325,410),(326,411),(327,412),(328,413),(329,414),(330,415),(331,416),(332,417),(333,418),(334,419),(335,420),(336,393),(337,443),(338,444),(339,445),(340,446),(341,447),(342,448),(343,421),(344,422),(345,423),(346,424),(347,425),(348,426),(349,427),(350,428),(351,429),(352,430),(353,431),(354,432),(355,433),(356,434),(357,435),(358,436),(359,437),(360,438),(361,439),(362,440),(363,441),(364,442)], [(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)], [(1,163,383,76),(2,150,384,63),(3,165,385,78),(4,152,386,65),(5,167,387,80),(6,154,388,67),(7,141,389,82),(8,156,390,69),(9,143,391,84),(10,158,392,71),(11,145,365,58),(12,160,366,73),(13,147,367,60),(14,162,368,75),(15,149,369,62),(16,164,370,77),(17,151,371,64),(18,166,372,79),(19,153,373,66),(20,168,374,81),(21,155,375,68),(22,142,376,83),(23,157,377,70),(24,144,378,57),(25,159,379,72),(26,146,380,59),(27,161,381,74),(28,148,382,61),(29,208,290,235),(30,223,291,250),(31,210,292,237),(32,197,293,252),(33,212,294,239),(34,199,295,226),(35,214,296,241),(36,201,297,228),(37,216,298,243),(38,203,299,230),(39,218,300,245),(40,205,301,232),(41,220,302,247),(42,207,303,234),(43,222,304,249),(44,209,305,236),(45,224,306,251),(46,211,307,238),(47,198,308,225),(48,213,281,240),(49,200,282,227),(50,215,283,242),(51,202,284,229),(52,217,285,244),(53,204,286,231),(54,219,287,246),(55,206,288,233),(56,221,289,248),(85,185,406,432),(86,172,407,447),(87,187,408,434),(88,174,409,421),(89,189,410,436),(90,176,411,423),(91,191,412,438),(92,178,413,425),(93,193,414,440),(94,180,415,427),(95,195,416,442),(96,182,417,429),(97,169,418,444),(98,184,419,431),(99,171,420,446),(100,186,393,433),(101,173,394,448),(102,188,395,435),(103,175,396,422),(104,190,397,437),(105,177,398,424),(106,192,399,439),(107,179,400,426),(108,194,401,441),(109,181,402,428),(110,196,403,443),(111,183,404,430),(112,170,405,445),(113,318,337,265),(114,333,338,280),(115,320,339,267),(116,335,340,254),(117,322,341,269),(118,309,342,256),(119,324,343,271),(120,311,344,258),(121,326,345,273),(122,313,346,260),(123,328,347,275),(124,315,348,262),(125,330,349,277),(126,317,350,264),(127,332,351,279),(128,319,352,266),(129,334,353,253),(130,321,354,268),(131,336,355,255),(132,323,356,270),(133,310,357,257),(134,325,358,272),(135,312,359,259),(136,327,360,274),(137,314,361,261),(138,329,362,276),(139,316,363,263),(140,331,364,278)], [(1,176,383,423),(2,189,384,436),(3,174,385,421),(4,187,386,434),(5,172,387,447),(6,185,388,432),(7,170,389,445),(8,183,390,430),(9,196,391,443),(10,181,392,428),(11,194,365,441),(12,179,366,426),(13,192,367,439),(14,177,368,424),(15,190,369,437),(16,175,370,422),(17,188,371,435),(18,173,372,448),(19,186,373,433),(20,171,374,446),(21,184,375,431),(22,169,376,444),(23,182,377,429),(24,195,378,442),(25,180,379,427),(26,193,380,440),(27,178,381,425),(28,191,382,438),(29,258,290,311),(30,271,291,324),(31,256,292,309),(32,269,293,322),(33,254,294,335),(34,267,295,320),(35,280,296,333),(36,265,297,318),(37,278,298,331),(38,263,299,316),(39,276,300,329),(40,261,301,314),(41,274,302,327),(42,259,303,312),(43,272,304,325),(44,257,305,310),(45,270,306,323),(46,255,307,336),(47,268,308,321),(48,253,281,334),(49,266,282,319),(50,279,283,332),(51,264,284,317),(52,277,285,330),(53,262,286,315),(54,275,287,328),(55,260,288,313),(56,273,289,326),(57,416,144,95),(58,401,145,108),(59,414,146,93),(60,399,147,106),(61,412,148,91),(62,397,149,104),(63,410,150,89),(64,395,151,102),(65,408,152,87),(66,393,153,100),(67,406,154,85),(68,419,155,98),(69,404,156,111),(70,417,157,96),(71,402,158,109),(72,415,159,94),(73,400,160,107),(74,413,161,92),(75,398,162,105),(76,411,163,90),(77,396,164,103),(78,409,165,88),(79,394,166,101),(80,407,167,86),(81,420,168,99),(82,405,141,112),(83,418,142,97),(84,403,143,110),(113,201,337,228),(114,214,338,241),(115,199,339,226),(116,212,340,239),(117,197,341,252),(118,210,342,237),(119,223,343,250),(120,208,344,235),(121,221,345,248),(122,206,346,233),(123,219,347,246),(124,204,348,231),(125,217,349,244),(126,202,350,229),(127,215,351,242),(128,200,352,227),(129,213,353,240),(130,198,354,225),(131,211,355,238),(132,224,356,251),(133,209,357,236),(134,222,358,249),(135,207,359,234),(136,220,360,247),(137,205,361,232),(138,218,362,245),(139,203,363,230),(140,216,364,243)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D7 | D14 | D14 | Dic14 | D4×D7 | Q8×D7 |
| kernel | C2×C28⋊Q8 | C28⋊Q8 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | C22×Dic14 | C2×Dic7 | C2×Dic7 | C2×C28 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 3 | 12 | 9 | 24 | 6 | 6 |
Matrix representation of C2×C28⋊Q8 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 14 | 21 | 0 | 0 | 0 | 0 |
| 21 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 24 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 22 | 1 |
| 8 | 14 | 0 | 0 | 0 | 0 |
| 14 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 23 |
| 0 | 0 | 0 | 0 | 6 | 8 |
| 14 | 21 | 0 | 0 | 0 | 0 |
| 21 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 0 | 8 | 21 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[14,21,0,0,0,0,21,15,0,0,0,0,0,0,3,24,0,0,0,0,7,8,0,0,0,0,0,0,10,22,0,0,0,0,7,1],[8,14,0,0,0,0,14,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,21,6,0,0,0,0,23,8],[14,21,0,0,0,0,21,15,0,0,0,0,0,0,0,3,0,0,0,0,10,0,0,0,0,0,0,0,8,8,0,0,0,0,10,21] >;
C2×C28⋊Q8 in GAP, Magma, Sage, TeX
C_2\times C_{28}\rtimes Q_8 % in TeX
G:=Group("C2xC28:Q8"); // GroupNames label
G:=SmallGroup(448,950);
// by ID
G=gap.SmallGroup(448,950);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,184,675,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^15,d*b*d^-1=b^13,d*c*d^-1=c^-1>;
// generators/relations