direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28⋊2Q8, C42.272D14, C28⋊5(C2×Q8), C14⋊1(C4⋊Q8), (C2×C28)⋊12Q8, C4⋊3(C2×Dic14), (C2×C4)⋊9Dic14, C4.42(C2×D28), (C2×C4).97D28, (C2×C28).388D4, C28.285(C2×D4), C14.1(C22×D4), (C2×C42).19D7, C2.4(C22×D28), C14.2(C22×Q8), (C2×C14).12C24, C22.62(C2×D28), (C2×C28).778C23, (C4×C28).312C22, (C22×C4).434D14, (C2×Dic7).1C23, C2.4(C22×Dic14), C22.59(C23×D7), C4⋊Dic7.286C22, (C22×Dic14).7C2, C22.34(C2×Dic14), C23.310(C22×D7), (C22×C14).374C23, (C22×C28).521C22, (C2×Dic14).222C22, (C22×Dic7).72C22, C7⋊1(C2×C4⋊Q8), (C2×C4×C28).13C2, (C2×C14).46(C2×Q8), (C2×C14).168(C2×D4), (C2×C4⋊Dic7).24C2, (C2×C4).727(C22×D7), SmallGroup(448,921)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28⋊2Q8
G = < a,b,c,d | a2=b28=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 1156 in 290 conjugacy classes, 159 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C2×C4⋊Q8, C4⋊Dic7, C4×C28, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C22×C28, C28⋊2Q8, C2×C4⋊Dic7, C2×C4×C28, C22×Dic14, C2×C28⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C24, D14, C4⋊Q8, C22×D4, C22×Q8, Dic14, D28, C22×D7, C2×C4⋊Q8, C2×Dic14, C2×D28, C23×D7, C28⋊2Q8, C22×Dic14, C22×D28, C2×C28⋊2Q8
►Regular action on 448 points
Generators in S448
(1 313)(2 314)(3 315)(4 316)(5 317)(6 318)(7 319)(8 320)(9 321)(10 322)(11 323)(12 324)(13 325)(14 326)(15 327)(16 328)(17 329)(18 330)(19 331)(20 332)(21 333)(22 334)(23 335)(24 336)(25 309)(26 310)(27 311)(28 312)(29 448)(30 421)(31 422)(32 423)(33 424)(34 425)(35 426)(36 427)(37 428)(38 429)(39 430)(40 431)(41 432)(42 433)(43 434)(44 435)(45 436)(46 437)(47 438)(48 439)(49 440)(50 441)(51 442)(52 443)(53 444)(54 445)(55 446)(56 447)(57 368)(58 369)(59 370)(60 371)(61 372)(62 373)(63 374)(64 375)(65 376)(66 377)(67 378)(68 379)(69 380)(70 381)(71 382)(72 383)(73 384)(74 385)(75 386)(76 387)(77 388)(78 389)(79 390)(80 391)(81 392)(82 365)(83 366)(84 367)(85 208)(86 209)(87 210)(88 211)(89 212)(90 213)(91 214)(92 215)(93 216)(94 217)(95 218)(96 219)(97 220)(98 221)(99 222)(100 223)(101 224)(102 197)(103 198)(104 199)(105 200)(106 201)(107 202)(108 203)(109 204)(110 205)(111 206)(112 207)(113 282)(114 283)(115 284)(116 285)(117 286)(118 287)(119 288)(120 289)(121 290)(122 291)(123 292)(124 293)(125 294)(126 295)(127 296)(128 297)(129 298)(130 299)(131 300)(132 301)(133 302)(134 303)(135 304)(136 305)(137 306)(138 307)(139 308)(140 281)(141 181)(142 182)(143 183)(144 184)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(151 191)(152 192)(153 193)(154 194)(155 195)(156 196)(157 169)(158 170)(159 171)(160 172)(161 173)(162 174)(163 175)(164 176)(165 177)(166 178)(167 179)(168 180)(225 402)(226 403)(227 404)(228 405)(229 406)(230 407)(231 408)(232 409)(233 410)(234 411)(235 412)(236 413)(237 414)(238 415)(239 416)(240 417)(241 418)(242 419)(243 420)(244 393)(245 394)(246 395)(247 396)(248 397)(249 398)(250 399)(251 400)(252 401)(253 346)(254 347)(255 348)(256 349)(257 350)(258 351)(259 352)(260 353)(261 354)(262 355)(263 356)(264 357)(265 358)(266 359)(267 360)(268 361)(269 362)(270 363)(271 364)(272 337)(273 338)(274 339)(275 340)(276 341)(277 342)(278 343)(279 344)(280 345)
(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)
(1 239 435 58)(2 240 436 59)(3 241 437 60)(4 242 438 61)(5 243 439 62)(6 244 440 63)(7 245 441 64)(8 246 442 65)(9 247 443 66)(10 248 444 67)(11 249 445 68)(12 250 446 69)(13 251 447 70)(14 252 448 71)(15 225 421 72)(16 226 422 73)(17 227 423 74)(18 228 424 75)(19 229 425 76)(20 230 426 77)(21 231 427 78)(22 232 428 79)(23 233 429 80)(24 234 430 81)(25 235 431 82)(26 236 432 83)(27 237 433 84)(28 238 434 57)(29 382 326 401)(30 383 327 402)(31 384 328 403)(32 385 329 404)(33 386 330 405)(34 387 331 406)(35 388 332 407)(36 389 333 408)(37 390 334 409)(38 391 335 410)(39 392 336 411)(40 365 309 412)(41 366 310 413)(42 367 311 414)(43 368 312 415)(44 369 313 416)(45 370 314 417)(46 371 315 418)(47 372 316 419)(48 373 317 420)(49 374 318 393)(50 375 319 394)(51 376 320 395)(52 377 321 396)(53 378 322 397)(54 379 323 398)(55 380 324 399)(56 381 325 400)(85 306 187 266)(86 307 188 267)(87 308 189 268)(88 281 190 269)(89 282 191 270)(90 283 192 271)(91 284 193 272)(92 285 194 273)(93 286 195 274)(94 287 196 275)(95 288 169 276)(96 289 170 277)(97 290 171 278)(98 291 172 279)(99 292 173 280)(100 293 174 253)(101 294 175 254)(102 295 176 255)(103 296 177 256)(104 297 178 257)(105 298 179 258)(106 299 180 259)(107 300 181 260)(108 301 182 261)(109 302 183 262)(110 303 184 263)(111 304 185 264)(112 305 186 265)(113 151 363 212)(114 152 364 213)(115 153 337 214)(116 154 338 215)(117 155 339 216)(118 156 340 217)(119 157 341 218)(120 158 342 219)(121 159 343 220)(122 160 344 221)(123 161 345 222)(124 162 346 223)(125 163 347 224)(126 164 348 197)(127 165 349 198)(128 166 350 199)(129 167 351 200)(130 168 352 201)(131 141 353 202)(132 142 354 203)(133 143 355 204)(134 144 356 205)(135 145 357 206)(136 146 358 207)(137 147 359 208)(138 148 360 209)(139 149 361 210)(140 150 362 211)
(1 145 435 206)(2 144 436 205)(3 143 437 204)(4 142 438 203)(5 141 439 202)(6 168 440 201)(7 167 441 200)(8 166 442 199)(9 165 443 198)(10 164 444 197)(11 163 445 224)(12 162 446 223)(13 161 447 222)(14 160 448 221)(15 159 421 220)(16 158 422 219)(17 157 423 218)(18 156 424 217)(19 155 425 216)(20 154 426 215)(21 153 427 214)(22 152 428 213)(23 151 429 212)(24 150 430 211)(25 149 431 210)(26 148 432 209)(27 147 433 208)(28 146 434 207)(29 98 326 172)(30 97 327 171)(31 96 328 170)(32 95 329 169)(33 94 330 196)(34 93 331 195)(35 92 332 194)(36 91 333 193)(37 90 334 192)(38 89 335 191)(39 88 336 190)(40 87 309 189)(41 86 310 188)(42 85 311 187)(43 112 312 186)(44 111 313 185)(45 110 314 184)(46 109 315 183)(47 108 316 182)(48 107 317 181)(49 106 318 180)(50 105 319 179)(51 104 320 178)(52 103 321 177)(53 102 322 176)(54 101 323 175)(55 100 324 174)(56 99 325 173)(57 358 238 136)(58 357 239 135)(59 356 240 134)(60 355 241 133)(61 354 242 132)(62 353 243 131)(63 352 244 130)(64 351 245 129)(65 350 246 128)(66 349 247 127)(67 348 248 126)(68 347 249 125)(69 346 250 124)(70 345 251 123)(71 344 252 122)(72 343 225 121)(73 342 226 120)(74 341 227 119)(75 340 228 118)(76 339 229 117)(77 338 230 116)(78 337 231 115)(79 364 232 114)(80 363 233 113)(81 362 234 140)(82 361 235 139)(83 360 236 138)(84 359 237 137)(253 399 293 380)(254 398 294 379)(255 397 295 378)(256 396 296 377)(257 395 297 376)(258 394 298 375)(259 393 299 374)(260 420 300 373)(261 419 301 372)(262 418 302 371)(263 417 303 370)(264 416 304 369)(265 415 305 368)(266 414 306 367)(267 413 307 366)(268 412 308 365)(269 411 281 392)(270 410 282 391)(271 409 283 390)(272 408 284 389)(273 407 285 388)(274 406 286 387)(275 405 287 386)(276 404 288 385)(277 403 289 384)(278 402 290 383)(279 401 291 382)(280 400 292 381)
G:=sub<Sym(448)| (1,313)(2,314)(3,315)(4,316)(5,317)(6,318)(7,319)(8,320)(9,321)(10,322)(11,323)(12,324)(13,325)(14,326)(15,327)(16,328)(17,329)(18,330)(19,331)(20,332)(21,333)(22,334)(23,335)(24,336)(25,309)(26,310)(27,311)(28,312)(29,448)(30,421)(31,422)(32,423)(33,424)(34,425)(35,426)(36,427)(37,428)(38,429)(39,430)(40,431)(41,432)(42,433)(43,434)(44,435)(45,436)(46,437)(47,438)(48,439)(49,440)(50,441)(51,442)(52,443)(53,444)(54,445)(55,446)(56,447)(57,368)(58,369)(59,370)(60,371)(61,372)(62,373)(63,374)(64,375)(65,376)(66,377)(67,378)(68,379)(69,380)(70,381)(71,382)(72,383)(73,384)(74,385)(75,386)(76,387)(77,388)(78,389)(79,390)(80,391)(81,392)(82,365)(83,366)(84,367)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,217)(95,218)(96,219)(97,220)(98,221)(99,222)(100,223)(101,224)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(113,282)(114,283)(115,284)(116,285)(117,286)(118,287)(119,288)(120,289)(121,290)(122,291)(123,292)(124,293)(125,294)(126,295)(127,296)(128,297)(129,298)(130,299)(131,300)(132,301)(133,302)(134,303)(135,304)(136,305)(137,306)(138,307)(139,308)(140,281)(141,181)(142,182)(143,183)(144,184)(145,185)(146,186)(147,187)(148,188)(149,189)(150,190)(151,191)(152,192)(153,193)(154,194)(155,195)(156,196)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)(167,179)(168,180)(225,402)(226,403)(227,404)(228,405)(229,406)(230,407)(231,408)(232,409)(233,410)(234,411)(235,412)(236,413)(237,414)(238,415)(239,416)(240,417)(241,418)(242,419)(243,420)(244,393)(245,394)(246,395)(247,396)(248,397)(249,398)(250,399)(251,400)(252,401)(253,346)(254,347)(255,348)(256,349)(257,350)(258,351)(259,352)(260,353)(261,354)(262,355)(263,356)(264,357)(265,358)(266,359)(267,360)(268,361)(269,362)(270,363)(271,364)(272,337)(273,338)(274,339)(275,340)(276,341)(277,342)(278,343)(279,344)(280,345), (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), (1,239,435,58)(2,240,436,59)(3,241,437,60)(4,242,438,61)(5,243,439,62)(6,244,440,63)(7,245,441,64)(8,246,442,65)(9,247,443,66)(10,248,444,67)(11,249,445,68)(12,250,446,69)(13,251,447,70)(14,252,448,71)(15,225,421,72)(16,226,422,73)(17,227,423,74)(18,228,424,75)(19,229,425,76)(20,230,426,77)(21,231,427,78)(22,232,428,79)(23,233,429,80)(24,234,430,81)(25,235,431,82)(26,236,432,83)(27,237,433,84)(28,238,434,57)(29,382,326,401)(30,383,327,402)(31,384,328,403)(32,385,329,404)(33,386,330,405)(34,387,331,406)(35,388,332,407)(36,389,333,408)(37,390,334,409)(38,391,335,410)(39,392,336,411)(40,365,309,412)(41,366,310,413)(42,367,311,414)(43,368,312,415)(44,369,313,416)(45,370,314,417)(46,371,315,418)(47,372,316,419)(48,373,317,420)(49,374,318,393)(50,375,319,394)(51,376,320,395)(52,377,321,396)(53,378,322,397)(54,379,323,398)(55,380,324,399)(56,381,325,400)(85,306,187,266)(86,307,188,267)(87,308,189,268)(88,281,190,269)(89,282,191,270)(90,283,192,271)(91,284,193,272)(92,285,194,273)(93,286,195,274)(94,287,196,275)(95,288,169,276)(96,289,170,277)(97,290,171,278)(98,291,172,279)(99,292,173,280)(100,293,174,253)(101,294,175,254)(102,295,176,255)(103,296,177,256)(104,297,178,257)(105,298,179,258)(106,299,180,259)(107,300,181,260)(108,301,182,261)(109,302,183,262)(110,303,184,263)(111,304,185,264)(112,305,186,265)(113,151,363,212)(114,152,364,213)(115,153,337,214)(116,154,338,215)(117,155,339,216)(118,156,340,217)(119,157,341,218)(120,158,342,219)(121,159,343,220)(122,160,344,221)(123,161,345,222)(124,162,346,223)(125,163,347,224)(126,164,348,197)(127,165,349,198)(128,166,350,199)(129,167,351,200)(130,168,352,201)(131,141,353,202)(132,142,354,203)(133,143,355,204)(134,144,356,205)(135,145,357,206)(136,146,358,207)(137,147,359,208)(138,148,360,209)(139,149,361,210)(140,150,362,211), (1,145,435,206)(2,144,436,205)(3,143,437,204)(4,142,438,203)(5,141,439,202)(6,168,440,201)(7,167,441,200)(8,166,442,199)(9,165,443,198)(10,164,444,197)(11,163,445,224)(12,162,446,223)(13,161,447,222)(14,160,448,221)(15,159,421,220)(16,158,422,219)(17,157,423,218)(18,156,424,217)(19,155,425,216)(20,154,426,215)(21,153,427,214)(22,152,428,213)(23,151,429,212)(24,150,430,211)(25,149,431,210)(26,148,432,209)(27,147,433,208)(28,146,434,207)(29,98,326,172)(30,97,327,171)(31,96,328,170)(32,95,329,169)(33,94,330,196)(34,93,331,195)(35,92,332,194)(36,91,333,193)(37,90,334,192)(38,89,335,191)(39,88,336,190)(40,87,309,189)(41,86,310,188)(42,85,311,187)(43,112,312,186)(44,111,313,185)(45,110,314,184)(46,109,315,183)(47,108,316,182)(48,107,317,181)(49,106,318,180)(50,105,319,179)(51,104,320,178)(52,103,321,177)(53,102,322,176)(54,101,323,175)(55,100,324,174)(56,99,325,173)(57,358,238,136)(58,357,239,135)(59,356,240,134)(60,355,241,133)(61,354,242,132)(62,353,243,131)(63,352,244,130)(64,351,245,129)(65,350,246,128)(66,349,247,127)(67,348,248,126)(68,347,249,125)(69,346,250,124)(70,345,251,123)(71,344,252,122)(72,343,225,121)(73,342,226,120)(74,341,227,119)(75,340,228,118)(76,339,229,117)(77,338,230,116)(78,337,231,115)(79,364,232,114)(80,363,233,113)(81,362,234,140)(82,361,235,139)(83,360,236,138)(84,359,237,137)(253,399,293,380)(254,398,294,379)(255,397,295,378)(256,396,296,377)(257,395,297,376)(258,394,298,375)(259,393,299,374)(260,420,300,373)(261,419,301,372)(262,418,302,371)(263,417,303,370)(264,416,304,369)(265,415,305,368)(266,414,306,367)(267,413,307,366)(268,412,308,365)(269,411,281,392)(270,410,282,391)(271,409,283,390)(272,408,284,389)(273,407,285,388)(274,406,286,387)(275,405,287,386)(276,404,288,385)(277,403,289,384)(278,402,290,383)(279,401,291,382)(280,400,292,381)>;
G:=Group( (1,313)(2,314)(3,315)(4,316)(5,317)(6,318)(7,319)(8,320)(9,321)(10,322)(11,323)(12,324)(13,325)(14,326)(15,327)(16,328)(17,329)(18,330)(19,331)(20,332)(21,333)(22,334)(23,335)(24,336)(25,309)(26,310)(27,311)(28,312)(29,448)(30,421)(31,422)(32,423)(33,424)(34,425)(35,426)(36,427)(37,428)(38,429)(39,430)(40,431)(41,432)(42,433)(43,434)(44,435)(45,436)(46,437)(47,438)(48,439)(49,440)(50,441)(51,442)(52,443)(53,444)(54,445)(55,446)(56,447)(57,368)(58,369)(59,370)(60,371)(61,372)(62,373)(63,374)(64,375)(65,376)(66,377)(67,378)(68,379)(69,380)(70,381)(71,382)(72,383)(73,384)(74,385)(75,386)(76,387)(77,388)(78,389)(79,390)(80,391)(81,392)(82,365)(83,366)(84,367)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,217)(95,218)(96,219)(97,220)(98,221)(99,222)(100,223)(101,224)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(113,282)(114,283)(115,284)(116,285)(117,286)(118,287)(119,288)(120,289)(121,290)(122,291)(123,292)(124,293)(125,294)(126,295)(127,296)(128,297)(129,298)(130,299)(131,300)(132,301)(133,302)(134,303)(135,304)(136,305)(137,306)(138,307)(139,308)(140,281)(141,181)(142,182)(143,183)(144,184)(145,185)(146,186)(147,187)(148,188)(149,189)(150,190)(151,191)(152,192)(153,193)(154,194)(155,195)(156,196)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)(167,179)(168,180)(225,402)(226,403)(227,404)(228,405)(229,406)(230,407)(231,408)(232,409)(233,410)(234,411)(235,412)(236,413)(237,414)(238,415)(239,416)(240,417)(241,418)(242,419)(243,420)(244,393)(245,394)(246,395)(247,396)(248,397)(249,398)(250,399)(251,400)(252,401)(253,346)(254,347)(255,348)(256,349)(257,350)(258,351)(259,352)(260,353)(261,354)(262,355)(263,356)(264,357)(265,358)(266,359)(267,360)(268,361)(269,362)(270,363)(271,364)(272,337)(273,338)(274,339)(275,340)(276,341)(277,342)(278,343)(279,344)(280,345), (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), (1,239,435,58)(2,240,436,59)(3,241,437,60)(4,242,438,61)(5,243,439,62)(6,244,440,63)(7,245,441,64)(8,246,442,65)(9,247,443,66)(10,248,444,67)(11,249,445,68)(12,250,446,69)(13,251,447,70)(14,252,448,71)(15,225,421,72)(16,226,422,73)(17,227,423,74)(18,228,424,75)(19,229,425,76)(20,230,426,77)(21,231,427,78)(22,232,428,79)(23,233,429,80)(24,234,430,81)(25,235,431,82)(26,236,432,83)(27,237,433,84)(28,238,434,57)(29,382,326,401)(30,383,327,402)(31,384,328,403)(32,385,329,404)(33,386,330,405)(34,387,331,406)(35,388,332,407)(36,389,333,408)(37,390,334,409)(38,391,335,410)(39,392,336,411)(40,365,309,412)(41,366,310,413)(42,367,311,414)(43,368,312,415)(44,369,313,416)(45,370,314,417)(46,371,315,418)(47,372,316,419)(48,373,317,420)(49,374,318,393)(50,375,319,394)(51,376,320,395)(52,377,321,396)(53,378,322,397)(54,379,323,398)(55,380,324,399)(56,381,325,400)(85,306,187,266)(86,307,188,267)(87,308,189,268)(88,281,190,269)(89,282,191,270)(90,283,192,271)(91,284,193,272)(92,285,194,273)(93,286,195,274)(94,287,196,275)(95,288,169,276)(96,289,170,277)(97,290,171,278)(98,291,172,279)(99,292,173,280)(100,293,174,253)(101,294,175,254)(102,295,176,255)(103,296,177,256)(104,297,178,257)(105,298,179,258)(106,299,180,259)(107,300,181,260)(108,301,182,261)(109,302,183,262)(110,303,184,263)(111,304,185,264)(112,305,186,265)(113,151,363,212)(114,152,364,213)(115,153,337,214)(116,154,338,215)(117,155,339,216)(118,156,340,217)(119,157,341,218)(120,158,342,219)(121,159,343,220)(122,160,344,221)(123,161,345,222)(124,162,346,223)(125,163,347,224)(126,164,348,197)(127,165,349,198)(128,166,350,199)(129,167,351,200)(130,168,352,201)(131,141,353,202)(132,142,354,203)(133,143,355,204)(134,144,356,205)(135,145,357,206)(136,146,358,207)(137,147,359,208)(138,148,360,209)(139,149,361,210)(140,150,362,211), (1,145,435,206)(2,144,436,205)(3,143,437,204)(4,142,438,203)(5,141,439,202)(6,168,440,201)(7,167,441,200)(8,166,442,199)(9,165,443,198)(10,164,444,197)(11,163,445,224)(12,162,446,223)(13,161,447,222)(14,160,448,221)(15,159,421,220)(16,158,422,219)(17,157,423,218)(18,156,424,217)(19,155,425,216)(20,154,426,215)(21,153,427,214)(22,152,428,213)(23,151,429,212)(24,150,430,211)(25,149,431,210)(26,148,432,209)(27,147,433,208)(28,146,434,207)(29,98,326,172)(30,97,327,171)(31,96,328,170)(32,95,329,169)(33,94,330,196)(34,93,331,195)(35,92,332,194)(36,91,333,193)(37,90,334,192)(38,89,335,191)(39,88,336,190)(40,87,309,189)(41,86,310,188)(42,85,311,187)(43,112,312,186)(44,111,313,185)(45,110,314,184)(46,109,315,183)(47,108,316,182)(48,107,317,181)(49,106,318,180)(50,105,319,179)(51,104,320,178)(52,103,321,177)(53,102,322,176)(54,101,323,175)(55,100,324,174)(56,99,325,173)(57,358,238,136)(58,357,239,135)(59,356,240,134)(60,355,241,133)(61,354,242,132)(62,353,243,131)(63,352,244,130)(64,351,245,129)(65,350,246,128)(66,349,247,127)(67,348,248,126)(68,347,249,125)(69,346,250,124)(70,345,251,123)(71,344,252,122)(72,343,225,121)(73,342,226,120)(74,341,227,119)(75,340,228,118)(76,339,229,117)(77,338,230,116)(78,337,231,115)(79,364,232,114)(80,363,233,113)(81,362,234,140)(82,361,235,139)(83,360,236,138)(84,359,237,137)(253,399,293,380)(254,398,294,379)(255,397,295,378)(256,396,296,377)(257,395,297,376)(258,394,298,375)(259,393,299,374)(260,420,300,373)(261,419,301,372)(262,418,302,371)(263,417,303,370)(264,416,304,369)(265,415,305,368)(266,414,306,367)(267,413,307,366)(268,412,308,365)(269,411,281,392)(270,410,282,391)(271,409,283,390)(272,408,284,389)(273,407,285,388)(274,406,286,387)(275,405,287,386)(276,404,288,385)(277,403,289,384)(278,402,290,383)(279,401,291,382)(280,400,292,381) );
G=PermutationGroup([[(1,313),(2,314),(3,315),(4,316),(5,317),(6,318),(7,319),(8,320),(9,321),(10,322),(11,323),(12,324),(13,325),(14,326),(15,327),(16,328),(17,329),(18,330),(19,331),(20,332),(21,333),(22,334),(23,335),(24,336),(25,309),(26,310),(27,311),(28,312),(29,448),(30,421),(31,422),(32,423),(33,424),(34,425),(35,426),(36,427),(37,428),(38,429),(39,430),(40,431),(41,432),(42,433),(43,434),(44,435),(45,436),(46,437),(47,438),(48,439),(49,440),(50,441),(51,442),(52,443),(53,444),(54,445),(55,446),(56,447),(57,368),(58,369),(59,370),(60,371),(61,372),(62,373),(63,374),(64,375),(65,376),(66,377),(67,378),(68,379),(69,380),(70,381),(71,382),(72,383),(73,384),(74,385),(75,386),(76,387),(77,388),(78,389),(79,390),(80,391),(81,392),(82,365),(83,366),(84,367),(85,208),(86,209),(87,210),(88,211),(89,212),(90,213),(91,214),(92,215),(93,216),(94,217),(95,218),(96,219),(97,220),(98,221),(99,222),(100,223),(101,224),(102,197),(103,198),(104,199),(105,200),(106,201),(107,202),(108,203),(109,204),(110,205),(111,206),(112,207),(113,282),(114,283),(115,284),(116,285),(117,286),(118,287),(119,288),(120,289),(121,290),(122,291),(123,292),(124,293),(125,294),(126,295),(127,296),(128,297),(129,298),(130,299),(131,300),(132,301),(133,302),(134,303),(135,304),(136,305),(137,306),(138,307),(139,308),(140,281),(141,181),(142,182),(143,183),(144,184),(145,185),(146,186),(147,187),(148,188),(149,189),(150,190),(151,191),(152,192),(153,193),(154,194),(155,195),(156,196),(157,169),(158,170),(159,171),(160,172),(161,173),(162,174),(163,175),(164,176),(165,177),(166,178),(167,179),(168,180),(225,402),(226,403),(227,404),(228,405),(229,406),(230,407),(231,408),(232,409),(233,410),(234,411),(235,412),(236,413),(237,414),(238,415),(239,416),(240,417),(241,418),(242,419),(243,420),(244,393),(245,394),(246,395),(247,396),(248,397),(249,398),(250,399),(251,400),(252,401),(253,346),(254,347),(255,348),(256,349),(257,350),(258,351),(259,352),(260,353),(261,354),(262,355),(263,356),(264,357),(265,358),(266,359),(267,360),(268,361),(269,362),(270,363),(271,364),(272,337),(273,338),(274,339),(275,340),(276,341),(277,342),(278,343),(279,344),(280,345)], [(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)], [(1,239,435,58),(2,240,436,59),(3,241,437,60),(4,242,438,61),(5,243,439,62),(6,244,440,63),(7,245,441,64),(8,246,442,65),(9,247,443,66),(10,248,444,67),(11,249,445,68),(12,250,446,69),(13,251,447,70),(14,252,448,71),(15,225,421,72),(16,226,422,73),(17,227,423,74),(18,228,424,75),(19,229,425,76),(20,230,426,77),(21,231,427,78),(22,232,428,79),(23,233,429,80),(24,234,430,81),(25,235,431,82),(26,236,432,83),(27,237,433,84),(28,238,434,57),(29,382,326,401),(30,383,327,402),(31,384,328,403),(32,385,329,404),(33,386,330,405),(34,387,331,406),(35,388,332,407),(36,389,333,408),(37,390,334,409),(38,391,335,410),(39,392,336,411),(40,365,309,412),(41,366,310,413),(42,367,311,414),(43,368,312,415),(44,369,313,416),(45,370,314,417),(46,371,315,418),(47,372,316,419),(48,373,317,420),(49,374,318,393),(50,375,319,394),(51,376,320,395),(52,377,321,396),(53,378,322,397),(54,379,323,398),(55,380,324,399),(56,381,325,400),(85,306,187,266),(86,307,188,267),(87,308,189,268),(88,281,190,269),(89,282,191,270),(90,283,192,271),(91,284,193,272),(92,285,194,273),(93,286,195,274),(94,287,196,275),(95,288,169,276),(96,289,170,277),(97,290,171,278),(98,291,172,279),(99,292,173,280),(100,293,174,253),(101,294,175,254),(102,295,176,255),(103,296,177,256),(104,297,178,257),(105,298,179,258),(106,299,180,259),(107,300,181,260),(108,301,182,261),(109,302,183,262),(110,303,184,263),(111,304,185,264),(112,305,186,265),(113,151,363,212),(114,152,364,213),(115,153,337,214),(116,154,338,215),(117,155,339,216),(118,156,340,217),(119,157,341,218),(120,158,342,219),(121,159,343,220),(122,160,344,221),(123,161,345,222),(124,162,346,223),(125,163,347,224),(126,164,348,197),(127,165,349,198),(128,166,350,199),(129,167,351,200),(130,168,352,201),(131,141,353,202),(132,142,354,203),(133,143,355,204),(134,144,356,205),(135,145,357,206),(136,146,358,207),(137,147,359,208),(138,148,360,209),(139,149,361,210),(140,150,362,211)], [(1,145,435,206),(2,144,436,205),(3,143,437,204),(4,142,438,203),(5,141,439,202),(6,168,440,201),(7,167,441,200),(8,166,442,199),(9,165,443,198),(10,164,444,197),(11,163,445,224),(12,162,446,223),(13,161,447,222),(14,160,448,221),(15,159,421,220),(16,158,422,219),(17,157,423,218),(18,156,424,217),(19,155,425,216),(20,154,426,215),(21,153,427,214),(22,152,428,213),(23,151,429,212),(24,150,430,211),(25,149,431,210),(26,148,432,209),(27,147,433,208),(28,146,434,207),(29,98,326,172),(30,97,327,171),(31,96,328,170),(32,95,329,169),(33,94,330,196),(34,93,331,195),(35,92,332,194),(36,91,333,193),(37,90,334,192),(38,89,335,191),(39,88,336,190),(40,87,309,189),(41,86,310,188),(42,85,311,187),(43,112,312,186),(44,111,313,185),(45,110,314,184),(46,109,315,183),(47,108,316,182),(48,107,317,181),(49,106,318,180),(50,105,319,179),(51,104,320,178),(52,103,321,177),(53,102,322,176),(54,101,323,175),(55,100,324,174),(56,99,325,173),(57,358,238,136),(58,357,239,135),(59,356,240,134),(60,355,241,133),(61,354,242,132),(62,353,243,131),(63,352,244,130),(64,351,245,129),(65,350,246,128),(66,349,247,127),(67,348,248,126),(68,347,249,125),(69,346,250,124),(70,345,251,123),(71,344,252,122),(72,343,225,121),(73,342,226,120),(74,341,227,119),(75,340,228,118),(76,339,229,117),(77,338,230,116),(78,337,231,115),(79,364,232,114),(80,363,233,113),(81,362,234,140),(82,361,235,139),(83,360,236,138),(84,359,237,137),(253,399,293,380),(254,398,294,379),(255,397,295,378),(256,396,296,377),(257,395,297,376),(258,394,298,375),(259,393,299,374),(260,420,300,373),(261,419,301,372),(262,418,302,371),(263,417,303,370),(264,416,304,369),(265,415,305,368),(266,414,306,367),(267,413,307,366),(268,412,308,365),(269,411,281,392),(270,410,282,391),(271,409,283,390),(272,408,284,389),(273,407,285,388),(274,406,286,387),(275,405,287,386),(276,404,288,385),(277,403,289,384),(278,402,290,383),(279,401,291,382),(280,400,292,381)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | D14 | D14 | Dic14 | D28 |
| kernel | C2×C28⋊2Q8 | C28⋊2Q8 | C2×C4⋊Dic7 | C2×C4×C28 | C22×Dic14 | C2×C28 | C2×C28 | C2×C42 | C42 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 8 | 4 | 1 | 2 | 4 | 8 | 3 | 12 | 9 | 48 | 24 |
Matrix representation of C2×C28⋊2Q8 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 19 | 0 | 0 | 0 |
| 0 | 5 | 26 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 13 | 12 | 0 | 0 |
| 0 | 0 | 0 | 28 | 2 |
| 0 | 0 | 0 | 28 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 6 | 20 | 0 | 0 |
| 0 | 17 | 23 | 0 | 0 |
| 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,19,5,0,0,0,0,26,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,17,13,0,0,0,0,12,0,0,0,0,0,28,28,0,0,0,2,1],[1,0,0,0,0,0,6,17,0,0,0,20,23,0,0,0,0,0,17,0,0,0,0,24,12] >;
C2×C28⋊2Q8 in GAP, Magma, Sage, TeX
C_2\times C_{28}\rtimes_2Q_8 % in TeX
G:=Group("C2xC28:2Q8"); // GroupNames label
G:=SmallGroup(448,921);
// by ID
G=gap.SmallGroup(448,921);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,184,675,80,18822]);
// 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,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations