metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).1Q8, C2.7(C28⋊Q8), C14.12(C4⋊Q8), (C2×Dic7).1Q8, (C2×Dic7).8D4, (C2×C4).8Dic14, C22.39(Q8×D7), C14.2(C4⋊D4), C22.151(D4×D7), (C22×C4).68D14, C2.8(D14⋊D4), C2.8(D14⋊Q8), C2.8(Dic7.Q8), C14.21(C22⋊Q8), (C22×C28).7C22, C2.4(C28.6Q8), C2.8(D14.D4), C22.84(C4○D28), C14.C42.5C2, C14.11(C42.C2), C22.41(C2×Dic14), C2.C42.13D7, C23.353(C22×D7), C22.82(D4⋊2D7), (C22×C14).282C23, C7⋊1(C23.81C23), C2.9(C22⋊Dic14), C14.4(C22.D4), (C22×Dic7).7C22, (C2×C4⋊Dic7).7C2, (C2×C14).63(C2×Q8), (C2×C14).194(C2×D4), (C2×Dic7⋊C4).6C2, (C2×C14).57(C4○D4), (C7×C2.C42).9C2, SmallGroup(448,191)
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=ab15, dbd-1=ab13, dcd-1=ac-1 >
Subgroups: 636 in 150 conjugacy classes, 61 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.81C23, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C2.(C28⋊Q8)
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, Dic14, C22×D7, C23.81C23, C2×Dic14, C4○D28, D4×D7, D4⋊2D7, Q8×D7, C28.6Q8, C22⋊Dic14, D14.D4, D14⋊D4, C28⋊Q8, Dic7.Q8, D14⋊Q8, C2.(C28⋊Q8)
►Regular action on 448 points
Generators in S448
(1 245)(2 246)(3 247)(4 248)(5 249)(6 250)(7 251)(8 252)(9 225)(10 226)(11 227)(12 228)(13 229)(14 230)(15 231)(16 232)(17 233)(18 234)(19 235)(20 236)(21 237)(22 238)(23 239)(24 240)(25 241)(26 242)(27 243)(28 244)(29 374)(30 375)(31 376)(32 377)(33 378)(34 379)(35 380)(36 381)(37 382)(38 383)(39 384)(40 385)(41 386)(42 387)(43 388)(44 389)(45 390)(46 391)(47 392)(48 365)(49 366)(50 367)(51 368)(52 369)(53 370)(54 371)(55 372)(56 373)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 177)(64 178)(65 179)(66 180)(67 181)(68 182)(69 183)(70 184)(71 185)(72 186)(73 187)(74 188)(75 189)(76 190)(77 191)(78 192)(79 193)(80 194)(81 195)(82 196)(83 169)(84 170)(85 267)(86 268)(87 269)(88 270)(89 271)(90 272)(91 273)(92 274)(93 275)(94 276)(95 277)(96 278)(97 279)(98 280)(99 253)(100 254)(101 255)(102 256)(103 257)(104 258)(105 259)(106 260)(107 261)(108 262)(109 263)(110 264)(111 265)(112 266)(113 286)(114 287)(115 288)(116 289)(117 290)(118 291)(119 292)(120 293)(121 294)(122 295)(123 296)(124 297)(125 298)(126 299)(127 300)(128 301)(129 302)(130 303)(131 304)(132 305)(133 306)(134 307)(135 308)(136 281)(137 282)(138 283)(139 284)(140 285)(141 331)(142 332)(143 333)(144 334)(145 335)(146 336)(147 309)(148 310)(149 311)(150 312)(151 313)(152 314)(153 315)(154 316)(155 317)(156 318)(157 319)(158 320)(159 321)(160 322)(161 323)(162 324)(163 325)(164 326)(165 327)(166 328)(167 329)(168 330)(197 436)(198 437)(199 438)(200 439)(201 440)(202 441)(203 442)(204 443)(205 444)(206 445)(207 446)(208 447)(209 448)(210 421)(211 422)(212 423)(213 424)(214 425)(215 426)(216 427)(217 428)(218 429)(219 430)(220 431)(221 432)(222 433)(223 434)(224 435)(337 412)(338 413)(339 414)(340 415)(341 416)(342 417)(343 418)(344 419)(345 420)(346 393)(347 394)(348 395)(349 396)(350 397)(351 398)(352 399)(353 400)(354 401)(355 402)(356 403)(357 404)(358 405)(359 406)(360 407)(361 408)(362 409)(363 410)(364 411)
(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)
(1 190 166 131)(2 63 167 291)(3 192 168 133)(4 65 141 293)(5 194 142 135)(6 67 143 295)(7 196 144 137)(8 69 145 297)(9 170 146 139)(10 71 147 299)(11 172 148 113)(12 73 149 301)(13 174 150 115)(14 75 151 303)(15 176 152 117)(16 77 153 305)(17 178 154 119)(18 79 155 307)(19 180 156 121)(20 81 157 281)(21 182 158 123)(22 83 159 283)(23 184 160 125)(24 57 161 285)(25 186 162 127)(26 59 163 287)(27 188 164 129)(28 61 165 289)(29 201 345 97)(30 427 346 266)(31 203 347 99)(32 429 348 268)(33 205 349 101)(34 431 350 270)(35 207 351 103)(36 433 352 272)(37 209 353 105)(38 435 354 274)(39 211 355 107)(40 437 356 276)(41 213 357 109)(42 439 358 278)(43 215 359 111)(44 441 360 280)(45 217 361 85)(46 443 362 254)(47 219 363 87)(48 445 364 256)(49 221 337 89)(50 447 338 258)(51 223 339 91)(52 421 340 260)(53 197 341 93)(54 423 342 262)(55 199 343 95)(56 425 344 264)(58 310 286 227)(60 312 288 229)(62 314 290 231)(64 316 292 233)(66 318 294 235)(68 320 296 237)(70 322 298 239)(72 324 300 241)(74 326 302 243)(76 328 304 245)(78 330 306 247)(80 332 308 249)(82 334 282 251)(84 336 284 225)(86 377 218 395)(88 379 220 397)(90 381 222 399)(92 383 224 401)(94 385 198 403)(96 387 200 405)(98 389 202 407)(100 391 204 409)(102 365 206 411)(104 367 208 413)(106 369 210 415)(108 371 212 417)(110 373 214 419)(112 375 216 393)(114 242 173 325)(116 244 175 327)(118 246 177 329)(120 248 179 331)(122 250 181 333)(124 252 183 335)(126 226 185 309)(128 228 187 311)(130 230 189 313)(132 232 191 315)(134 234 193 317)(136 236 195 319)(138 238 169 321)(140 240 171 323)(253 376 442 394)(255 378 444 396)(257 380 446 398)(259 382 448 400)(261 384 422 402)(263 386 424 404)(265 388 426 406)(267 390 428 408)(269 392 430 410)(271 366 432 412)(273 368 434 414)(275 370 436 416)(277 372 438 418)(279 374 440 420)
(1 87 166 219)(2 254 167 443)(3 85 168 217)(4 280 141 441)(5 111 142 215)(6 278 143 439)(7 109 144 213)(8 276 145 437)(9 107 146 211)(10 274 147 435)(11 105 148 209)(12 272 149 433)(13 103 150 207)(14 270 151 431)(15 101 152 205)(16 268 153 429)(17 99 154 203)(18 266 155 427)(19 97 156 201)(20 264 157 425)(21 95 158 199)(22 262 159 423)(23 93 160 197)(24 260 161 421)(25 91 162 223)(26 258 163 447)(27 89 164 221)(28 256 165 445)(29 66 345 294)(30 193 346 134)(31 64 347 292)(32 191 348 132)(33 62 349 290)(34 189 350 130)(35 60 351 288)(36 187 352 128)(37 58 353 286)(38 185 354 126)(39 84 355 284)(40 183 356 124)(41 82 357 282)(42 181 358 122)(43 80 359 308)(44 179 360 120)(45 78 361 306)(46 177 362 118)(47 76 363 304)(48 175 364 116)(49 74 337 302)(50 173 338 114)(51 72 339 300)(52 171 340 140)(53 70 341 298)(54 169 342 138)(55 68 343 296)(56 195 344 136)(57 415 285 369)(59 413 287 367)(61 411 289 365)(63 409 291 391)(65 407 293 389)(67 405 295 387)(69 403 297 385)(71 401 299 383)(73 399 301 381)(75 397 303 379)(77 395 305 377)(79 393 307 375)(81 419 281 373)(83 417 283 371)(86 315 218 232)(88 313 220 230)(90 311 222 228)(92 309 224 226)(94 335 198 252)(96 333 200 250)(98 331 202 248)(100 329 204 246)(102 327 206 244)(104 325 208 242)(106 323 210 240)(108 321 212 238)(110 319 214 236)(112 317 216 234)(113 382 172 400)(115 380 174 398)(117 378 176 396)(119 376 178 394)(121 374 180 420)(123 372 182 418)(125 370 184 416)(127 368 186 414)(129 366 188 412)(131 392 190 410)(133 390 192 408)(135 388 194 406)(137 386 196 404)(139 384 170 402)(225 261 336 422)(227 259 310 448)(229 257 312 446)(231 255 314 444)(233 253 316 442)(235 279 318 440)(237 277 320 438)(239 275 322 436)(241 273 324 434)(243 271 326 432)(245 269 328 430)(247 267 330 428)(249 265 332 426)(251 263 334 424)
G:=sub<Sym(448)| (1,245)(2,246)(3,247)(4,248)(5,249)(6,250)(7,251)(8,252)(9,225)(10,226)(11,227)(12,228)(13,229)(14,230)(15,231)(16,232)(17,233)(18,234)(19,235)(20,236)(21,237)(22,238)(23,239)(24,240)(25,241)(26,242)(27,243)(28,244)(29,374)(30,375)(31,376)(32,377)(33,378)(34,379)(35,380)(36,381)(37,382)(38,383)(39,384)(40,385)(41,386)(42,387)(43,388)(44,389)(45,390)(46,391)(47,392)(48,365)(49,366)(50,367)(51,368)(52,369)(53,370)(54,371)(55,372)(56,373)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,177)(64,178)(65,179)(66,180)(67,181)(68,182)(69,183)(70,184)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,193)(80,194)(81,195)(82,196)(83,169)(84,170)(85,267)(86,268)(87,269)(88,270)(89,271)(90,272)(91,273)(92,274)(93,275)(94,276)(95,277)(96,278)(97,279)(98,280)(99,253)(100,254)(101,255)(102,256)(103,257)(104,258)(105,259)(106,260)(107,261)(108,262)(109,263)(110,264)(111,265)(112,266)(113,286)(114,287)(115,288)(116,289)(117,290)(118,291)(119,292)(120,293)(121,294)(122,295)(123,296)(124,297)(125,298)(126,299)(127,300)(128,301)(129,302)(130,303)(131,304)(132,305)(133,306)(134,307)(135,308)(136,281)(137,282)(138,283)(139,284)(140,285)(141,331)(142,332)(143,333)(144,334)(145,335)(146,336)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)(197,436)(198,437)(199,438)(200,439)(201,440)(202,441)(203,442)(204,443)(205,444)(206,445)(207,446)(208,447)(209,448)(210,421)(211,422)(212,423)(213,424)(214,425)(215,426)(216,427)(217,428)(218,429)(219,430)(220,431)(221,432)(222,433)(223,434)(224,435)(337,412)(338,413)(339,414)(340,415)(341,416)(342,417)(343,418)(344,419)(345,420)(346,393)(347,394)(348,395)(349,396)(350,397)(351,398)(352,399)(353,400)(354,401)(355,402)(356,403)(357,404)(358,405)(359,406)(360,407)(361,408)(362,409)(363,410)(364,411), (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), (1,190,166,131)(2,63,167,291)(3,192,168,133)(4,65,141,293)(5,194,142,135)(6,67,143,295)(7,196,144,137)(8,69,145,297)(9,170,146,139)(10,71,147,299)(11,172,148,113)(12,73,149,301)(13,174,150,115)(14,75,151,303)(15,176,152,117)(16,77,153,305)(17,178,154,119)(18,79,155,307)(19,180,156,121)(20,81,157,281)(21,182,158,123)(22,83,159,283)(23,184,160,125)(24,57,161,285)(25,186,162,127)(26,59,163,287)(27,188,164,129)(28,61,165,289)(29,201,345,97)(30,427,346,266)(31,203,347,99)(32,429,348,268)(33,205,349,101)(34,431,350,270)(35,207,351,103)(36,433,352,272)(37,209,353,105)(38,435,354,274)(39,211,355,107)(40,437,356,276)(41,213,357,109)(42,439,358,278)(43,215,359,111)(44,441,360,280)(45,217,361,85)(46,443,362,254)(47,219,363,87)(48,445,364,256)(49,221,337,89)(50,447,338,258)(51,223,339,91)(52,421,340,260)(53,197,341,93)(54,423,342,262)(55,199,343,95)(56,425,344,264)(58,310,286,227)(60,312,288,229)(62,314,290,231)(64,316,292,233)(66,318,294,235)(68,320,296,237)(70,322,298,239)(72,324,300,241)(74,326,302,243)(76,328,304,245)(78,330,306,247)(80,332,308,249)(82,334,282,251)(84,336,284,225)(86,377,218,395)(88,379,220,397)(90,381,222,399)(92,383,224,401)(94,385,198,403)(96,387,200,405)(98,389,202,407)(100,391,204,409)(102,365,206,411)(104,367,208,413)(106,369,210,415)(108,371,212,417)(110,373,214,419)(112,375,216,393)(114,242,173,325)(116,244,175,327)(118,246,177,329)(120,248,179,331)(122,250,181,333)(124,252,183,335)(126,226,185,309)(128,228,187,311)(130,230,189,313)(132,232,191,315)(134,234,193,317)(136,236,195,319)(138,238,169,321)(140,240,171,323)(253,376,442,394)(255,378,444,396)(257,380,446,398)(259,382,448,400)(261,384,422,402)(263,386,424,404)(265,388,426,406)(267,390,428,408)(269,392,430,410)(271,366,432,412)(273,368,434,414)(275,370,436,416)(277,372,438,418)(279,374,440,420), (1,87,166,219)(2,254,167,443)(3,85,168,217)(4,280,141,441)(5,111,142,215)(6,278,143,439)(7,109,144,213)(8,276,145,437)(9,107,146,211)(10,274,147,435)(11,105,148,209)(12,272,149,433)(13,103,150,207)(14,270,151,431)(15,101,152,205)(16,268,153,429)(17,99,154,203)(18,266,155,427)(19,97,156,201)(20,264,157,425)(21,95,158,199)(22,262,159,423)(23,93,160,197)(24,260,161,421)(25,91,162,223)(26,258,163,447)(27,89,164,221)(28,256,165,445)(29,66,345,294)(30,193,346,134)(31,64,347,292)(32,191,348,132)(33,62,349,290)(34,189,350,130)(35,60,351,288)(36,187,352,128)(37,58,353,286)(38,185,354,126)(39,84,355,284)(40,183,356,124)(41,82,357,282)(42,181,358,122)(43,80,359,308)(44,179,360,120)(45,78,361,306)(46,177,362,118)(47,76,363,304)(48,175,364,116)(49,74,337,302)(50,173,338,114)(51,72,339,300)(52,171,340,140)(53,70,341,298)(54,169,342,138)(55,68,343,296)(56,195,344,136)(57,415,285,369)(59,413,287,367)(61,411,289,365)(63,409,291,391)(65,407,293,389)(67,405,295,387)(69,403,297,385)(71,401,299,383)(73,399,301,381)(75,397,303,379)(77,395,305,377)(79,393,307,375)(81,419,281,373)(83,417,283,371)(86,315,218,232)(88,313,220,230)(90,311,222,228)(92,309,224,226)(94,335,198,252)(96,333,200,250)(98,331,202,248)(100,329,204,246)(102,327,206,244)(104,325,208,242)(106,323,210,240)(108,321,212,238)(110,319,214,236)(112,317,216,234)(113,382,172,400)(115,380,174,398)(117,378,176,396)(119,376,178,394)(121,374,180,420)(123,372,182,418)(125,370,184,416)(127,368,186,414)(129,366,188,412)(131,392,190,410)(133,390,192,408)(135,388,194,406)(137,386,196,404)(139,384,170,402)(225,261,336,422)(227,259,310,448)(229,257,312,446)(231,255,314,444)(233,253,316,442)(235,279,318,440)(237,277,320,438)(239,275,322,436)(241,273,324,434)(243,271,326,432)(245,269,328,430)(247,267,330,428)(249,265,332,426)(251,263,334,424)>;
G:=Group( (1,245)(2,246)(3,247)(4,248)(5,249)(6,250)(7,251)(8,252)(9,225)(10,226)(11,227)(12,228)(13,229)(14,230)(15,231)(16,232)(17,233)(18,234)(19,235)(20,236)(21,237)(22,238)(23,239)(24,240)(25,241)(26,242)(27,243)(28,244)(29,374)(30,375)(31,376)(32,377)(33,378)(34,379)(35,380)(36,381)(37,382)(38,383)(39,384)(40,385)(41,386)(42,387)(43,388)(44,389)(45,390)(46,391)(47,392)(48,365)(49,366)(50,367)(51,368)(52,369)(53,370)(54,371)(55,372)(56,373)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,177)(64,178)(65,179)(66,180)(67,181)(68,182)(69,183)(70,184)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,193)(80,194)(81,195)(82,196)(83,169)(84,170)(85,267)(86,268)(87,269)(88,270)(89,271)(90,272)(91,273)(92,274)(93,275)(94,276)(95,277)(96,278)(97,279)(98,280)(99,253)(100,254)(101,255)(102,256)(103,257)(104,258)(105,259)(106,260)(107,261)(108,262)(109,263)(110,264)(111,265)(112,266)(113,286)(114,287)(115,288)(116,289)(117,290)(118,291)(119,292)(120,293)(121,294)(122,295)(123,296)(124,297)(125,298)(126,299)(127,300)(128,301)(129,302)(130,303)(131,304)(132,305)(133,306)(134,307)(135,308)(136,281)(137,282)(138,283)(139,284)(140,285)(141,331)(142,332)(143,333)(144,334)(145,335)(146,336)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)(197,436)(198,437)(199,438)(200,439)(201,440)(202,441)(203,442)(204,443)(205,444)(206,445)(207,446)(208,447)(209,448)(210,421)(211,422)(212,423)(213,424)(214,425)(215,426)(216,427)(217,428)(218,429)(219,430)(220,431)(221,432)(222,433)(223,434)(224,435)(337,412)(338,413)(339,414)(340,415)(341,416)(342,417)(343,418)(344,419)(345,420)(346,393)(347,394)(348,395)(349,396)(350,397)(351,398)(352,399)(353,400)(354,401)(355,402)(356,403)(357,404)(358,405)(359,406)(360,407)(361,408)(362,409)(363,410)(364,411), (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), (1,190,166,131)(2,63,167,291)(3,192,168,133)(4,65,141,293)(5,194,142,135)(6,67,143,295)(7,196,144,137)(8,69,145,297)(9,170,146,139)(10,71,147,299)(11,172,148,113)(12,73,149,301)(13,174,150,115)(14,75,151,303)(15,176,152,117)(16,77,153,305)(17,178,154,119)(18,79,155,307)(19,180,156,121)(20,81,157,281)(21,182,158,123)(22,83,159,283)(23,184,160,125)(24,57,161,285)(25,186,162,127)(26,59,163,287)(27,188,164,129)(28,61,165,289)(29,201,345,97)(30,427,346,266)(31,203,347,99)(32,429,348,268)(33,205,349,101)(34,431,350,270)(35,207,351,103)(36,433,352,272)(37,209,353,105)(38,435,354,274)(39,211,355,107)(40,437,356,276)(41,213,357,109)(42,439,358,278)(43,215,359,111)(44,441,360,280)(45,217,361,85)(46,443,362,254)(47,219,363,87)(48,445,364,256)(49,221,337,89)(50,447,338,258)(51,223,339,91)(52,421,340,260)(53,197,341,93)(54,423,342,262)(55,199,343,95)(56,425,344,264)(58,310,286,227)(60,312,288,229)(62,314,290,231)(64,316,292,233)(66,318,294,235)(68,320,296,237)(70,322,298,239)(72,324,300,241)(74,326,302,243)(76,328,304,245)(78,330,306,247)(80,332,308,249)(82,334,282,251)(84,336,284,225)(86,377,218,395)(88,379,220,397)(90,381,222,399)(92,383,224,401)(94,385,198,403)(96,387,200,405)(98,389,202,407)(100,391,204,409)(102,365,206,411)(104,367,208,413)(106,369,210,415)(108,371,212,417)(110,373,214,419)(112,375,216,393)(114,242,173,325)(116,244,175,327)(118,246,177,329)(120,248,179,331)(122,250,181,333)(124,252,183,335)(126,226,185,309)(128,228,187,311)(130,230,189,313)(132,232,191,315)(134,234,193,317)(136,236,195,319)(138,238,169,321)(140,240,171,323)(253,376,442,394)(255,378,444,396)(257,380,446,398)(259,382,448,400)(261,384,422,402)(263,386,424,404)(265,388,426,406)(267,390,428,408)(269,392,430,410)(271,366,432,412)(273,368,434,414)(275,370,436,416)(277,372,438,418)(279,374,440,420), (1,87,166,219)(2,254,167,443)(3,85,168,217)(4,280,141,441)(5,111,142,215)(6,278,143,439)(7,109,144,213)(8,276,145,437)(9,107,146,211)(10,274,147,435)(11,105,148,209)(12,272,149,433)(13,103,150,207)(14,270,151,431)(15,101,152,205)(16,268,153,429)(17,99,154,203)(18,266,155,427)(19,97,156,201)(20,264,157,425)(21,95,158,199)(22,262,159,423)(23,93,160,197)(24,260,161,421)(25,91,162,223)(26,258,163,447)(27,89,164,221)(28,256,165,445)(29,66,345,294)(30,193,346,134)(31,64,347,292)(32,191,348,132)(33,62,349,290)(34,189,350,130)(35,60,351,288)(36,187,352,128)(37,58,353,286)(38,185,354,126)(39,84,355,284)(40,183,356,124)(41,82,357,282)(42,181,358,122)(43,80,359,308)(44,179,360,120)(45,78,361,306)(46,177,362,118)(47,76,363,304)(48,175,364,116)(49,74,337,302)(50,173,338,114)(51,72,339,300)(52,171,340,140)(53,70,341,298)(54,169,342,138)(55,68,343,296)(56,195,344,136)(57,415,285,369)(59,413,287,367)(61,411,289,365)(63,409,291,391)(65,407,293,389)(67,405,295,387)(69,403,297,385)(71,401,299,383)(73,399,301,381)(75,397,303,379)(77,395,305,377)(79,393,307,375)(81,419,281,373)(83,417,283,371)(86,315,218,232)(88,313,220,230)(90,311,222,228)(92,309,224,226)(94,335,198,252)(96,333,200,250)(98,331,202,248)(100,329,204,246)(102,327,206,244)(104,325,208,242)(106,323,210,240)(108,321,212,238)(110,319,214,236)(112,317,216,234)(113,382,172,400)(115,380,174,398)(117,378,176,396)(119,376,178,394)(121,374,180,420)(123,372,182,418)(125,370,184,416)(127,368,186,414)(129,366,188,412)(131,392,190,410)(133,390,192,408)(135,388,194,406)(137,386,196,404)(139,384,170,402)(225,261,336,422)(227,259,310,448)(229,257,312,446)(231,255,314,444)(233,253,316,442)(235,279,318,440)(237,277,320,438)(239,275,322,436)(241,273,324,434)(243,271,326,432)(245,269,328,430)(247,267,330,428)(249,265,332,426)(251,263,334,424) );
G=PermutationGroup([[(1,245),(2,246),(3,247),(4,248),(5,249),(6,250),(7,251),(8,252),(9,225),(10,226),(11,227),(12,228),(13,229),(14,230),(15,231),(16,232),(17,233),(18,234),(19,235),(20,236),(21,237),(22,238),(23,239),(24,240),(25,241),(26,242),(27,243),(28,244),(29,374),(30,375),(31,376),(32,377),(33,378),(34,379),(35,380),(36,381),(37,382),(38,383),(39,384),(40,385),(41,386),(42,387),(43,388),(44,389),(45,390),(46,391),(47,392),(48,365),(49,366),(50,367),(51,368),(52,369),(53,370),(54,371),(55,372),(56,373),(57,171),(58,172),(59,173),(60,174),(61,175),(62,176),(63,177),(64,178),(65,179),(66,180),(67,181),(68,182),(69,183),(70,184),(71,185),(72,186),(73,187),(74,188),(75,189),(76,190),(77,191),(78,192),(79,193),(80,194),(81,195),(82,196),(83,169),(84,170),(85,267),(86,268),(87,269),(88,270),(89,271),(90,272),(91,273),(92,274),(93,275),(94,276),(95,277),(96,278),(97,279),(98,280),(99,253),(100,254),(101,255),(102,256),(103,257),(104,258),(105,259),(106,260),(107,261),(108,262),(109,263),(110,264),(111,265),(112,266),(113,286),(114,287),(115,288),(116,289),(117,290),(118,291),(119,292),(120,293),(121,294),(122,295),(123,296),(124,297),(125,298),(126,299),(127,300),(128,301),(129,302),(130,303),(131,304),(132,305),(133,306),(134,307),(135,308),(136,281),(137,282),(138,283),(139,284),(140,285),(141,331),(142,332),(143,333),(144,334),(145,335),(146,336),(147,309),(148,310),(149,311),(150,312),(151,313),(152,314),(153,315),(154,316),(155,317),(156,318),(157,319),(158,320),(159,321),(160,322),(161,323),(162,324),(163,325),(164,326),(165,327),(166,328),(167,329),(168,330),(197,436),(198,437),(199,438),(200,439),(201,440),(202,441),(203,442),(204,443),(205,444),(206,445),(207,446),(208,447),(209,448),(210,421),(211,422),(212,423),(213,424),(214,425),(215,426),(216,427),(217,428),(218,429),(219,430),(220,431),(221,432),(222,433),(223,434),(224,435),(337,412),(338,413),(339,414),(340,415),(341,416),(342,417),(343,418),(344,419),(345,420),(346,393),(347,394),(348,395),(349,396),(350,397),(351,398),(352,399),(353,400),(354,401),(355,402),(356,403),(357,404),(358,405),(359,406),(360,407),(361,408),(362,409),(363,410),(364,411)], [(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)], [(1,190,166,131),(2,63,167,291),(3,192,168,133),(4,65,141,293),(5,194,142,135),(6,67,143,295),(7,196,144,137),(8,69,145,297),(9,170,146,139),(10,71,147,299),(11,172,148,113),(12,73,149,301),(13,174,150,115),(14,75,151,303),(15,176,152,117),(16,77,153,305),(17,178,154,119),(18,79,155,307),(19,180,156,121),(20,81,157,281),(21,182,158,123),(22,83,159,283),(23,184,160,125),(24,57,161,285),(25,186,162,127),(26,59,163,287),(27,188,164,129),(28,61,165,289),(29,201,345,97),(30,427,346,266),(31,203,347,99),(32,429,348,268),(33,205,349,101),(34,431,350,270),(35,207,351,103),(36,433,352,272),(37,209,353,105),(38,435,354,274),(39,211,355,107),(40,437,356,276),(41,213,357,109),(42,439,358,278),(43,215,359,111),(44,441,360,280),(45,217,361,85),(46,443,362,254),(47,219,363,87),(48,445,364,256),(49,221,337,89),(50,447,338,258),(51,223,339,91),(52,421,340,260),(53,197,341,93),(54,423,342,262),(55,199,343,95),(56,425,344,264),(58,310,286,227),(60,312,288,229),(62,314,290,231),(64,316,292,233),(66,318,294,235),(68,320,296,237),(70,322,298,239),(72,324,300,241),(74,326,302,243),(76,328,304,245),(78,330,306,247),(80,332,308,249),(82,334,282,251),(84,336,284,225),(86,377,218,395),(88,379,220,397),(90,381,222,399),(92,383,224,401),(94,385,198,403),(96,387,200,405),(98,389,202,407),(100,391,204,409),(102,365,206,411),(104,367,208,413),(106,369,210,415),(108,371,212,417),(110,373,214,419),(112,375,216,393),(114,242,173,325),(116,244,175,327),(118,246,177,329),(120,248,179,331),(122,250,181,333),(124,252,183,335),(126,226,185,309),(128,228,187,311),(130,230,189,313),(132,232,191,315),(134,234,193,317),(136,236,195,319),(138,238,169,321),(140,240,171,323),(253,376,442,394),(255,378,444,396),(257,380,446,398),(259,382,448,400),(261,384,422,402),(263,386,424,404),(265,388,426,406),(267,390,428,408),(269,392,430,410),(271,366,432,412),(273,368,434,414),(275,370,436,416),(277,372,438,418),(279,374,440,420)], [(1,87,166,219),(2,254,167,443),(3,85,168,217),(4,280,141,441),(5,111,142,215),(6,278,143,439),(7,109,144,213),(8,276,145,437),(9,107,146,211),(10,274,147,435),(11,105,148,209),(12,272,149,433),(13,103,150,207),(14,270,151,431),(15,101,152,205),(16,268,153,429),(17,99,154,203),(18,266,155,427),(19,97,156,201),(20,264,157,425),(21,95,158,199),(22,262,159,423),(23,93,160,197),(24,260,161,421),(25,91,162,223),(26,258,163,447),(27,89,164,221),(28,256,165,445),(29,66,345,294),(30,193,346,134),(31,64,347,292),(32,191,348,132),(33,62,349,290),(34,189,350,130),(35,60,351,288),(36,187,352,128),(37,58,353,286),(38,185,354,126),(39,84,355,284),(40,183,356,124),(41,82,357,282),(42,181,358,122),(43,80,359,308),(44,179,360,120),(45,78,361,306),(46,177,362,118),(47,76,363,304),(48,175,364,116),(49,74,337,302),(50,173,338,114),(51,72,339,300),(52,171,340,140),(53,70,341,298),(54,169,342,138),(55,68,343,296),(56,195,344,136),(57,415,285,369),(59,413,287,367),(61,411,289,365),(63,409,291,391),(65,407,293,389),(67,405,295,387),(69,403,297,385),(71,401,299,383),(73,399,301,381),(75,397,303,379),(77,395,305,377),(79,393,307,375),(81,419,281,373),(83,417,283,371),(86,315,218,232),(88,313,220,230),(90,311,222,228),(92,309,224,226),(94,335,198,252),(96,333,200,250),(98,331,202,248),(100,329,204,246),(102,327,206,244),(104,325,208,242),(106,323,210,240),(108,321,212,238),(110,319,214,236),(112,317,216,234),(113,382,172,400),(115,380,174,398),(117,378,176,396),(119,376,178,394),(121,374,180,420),(123,372,182,418),(125,370,184,416),(127,368,186,414),(129,366,188,412),(131,392,190,410),(133,390,192,408),(135,388,194,406),(137,386,196,404),(139,384,170,402),(225,261,336,422),(227,259,310,448),(229,257,312,446),(231,255,314,444),(233,253,316,442),(235,279,318,440),(237,277,320,438),(239,275,322,436),(241,273,324,434),(243,271,326,432),(245,269,328,430),(247,267,330,428),(249,265,332,426),(251,263,334,424)]])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | - | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 |
| kernel | C2.(C28⋊Q8) | C14.C42 | C7×C2.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 4 | 2 | 2 | 3 | 6 | 9 | 12 | 24 | 6 | 3 | 3 |
Matrix representation of C2.(C28⋊Q8) ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 22 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 24 |
| 0 | 0 | 0 | 0 | 24 | 11 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 23 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 3 | 11 | 0 | 0 | 0 | 0 |
| 7 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 25 | 0 | 0 |
| 0 | 0 | 17 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[3,22,0,0,0,0,7,13,0,0,0,0,0,0,10,22,0,0,0,0,7,1,0,0,0,0,0,0,18,24,0,0,0,0,24,11],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,8,23,0,0,0,0,6,21,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[3,7,0,0,0,0,11,26,0,0,0,0,0,0,3,17,0,0,0,0,25,26,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;
C2.(C28⋊Q8) in GAP, Magma, Sage, TeX
C_2.(C_{28}\rtimes Q_8) % in TeX
G:=Group("C2.(C28:Q8)"); // GroupNames label
G:=SmallGroup(448,191);
// by ID
G=gap.SmallGroup(448,191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,64,254,387,100,18822]);
// 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=a*b^15,d*b*d^-1=a*b^13,d*c*d^-1=a*c^-1>;
// generators/relations