metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).44D28, (C2×C28).53D4, C14.27(C4⋊Q8), (C2×Dic7).7Q8, C22.48(Q8×D7), C2.9(C28⋊7D4), C14.61(C4⋊D4), (C22×C4).100D14, C22.128(C2×D28), C14.47(C22⋊Q8), C2.16(D14⋊2Q8), C2.7(Dic7⋊Q8), C2.13(Dic7.Q8), (C22×C28).64C22, C14.22(C42.C2), C23.377(C22×D7), C22.105(C4○D28), C14.C42.28C2, (C22×C14).347C23, C7⋊4(C23.81C23), C22.101(D4⋊2D7), C14.75(C22.D4), C2.9(C23.18D14), (C22×Dic7).55C22, (C2×C4⋊C4).20D7, (C14×C4⋊C4).22C2, (C2×C14).82(C2×Q8), (C2×C14).151(C2×D4), (C2×C4).38(C7⋊D4), (C2×C4⋊Dic7).18C2, (C2×C14).84(C4○D4), (C2×Dic7⋊C4).32C2, C22.137(C2×C7⋊D4), SmallGroup(448,517)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).44D28
G = < a,b,c,d | a2=b28=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 612 in 150 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.81C23, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×C28, C14.C42, C14.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C14×C4⋊C4, (C2×C4).44D28
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, D28, C7⋊D4, C22×D7, C23.81C23, C2×D28, C4○D28, D4⋊2D7, Q8×D7, C2×C7⋊D4, Dic7.Q8, D14⋊2Q8, C28⋊7D4, C23.18D14, Dic7⋊Q8, (C2×C4).44D28
►Regular action on 448 points
Generators in S448
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 193)(25 194)(26 195)(27 196)(28 169)(29 378)(30 379)(31 380)(32 381)(33 382)(34 383)(35 384)(36 385)(37 386)(38 387)(39 388)(40 389)(41 390)(42 391)(43 392)(44 365)(45 366)(46 367)(47 368)(48 369)(49 370)(50 371)(51 372)(52 373)(53 374)(54 375)(55 376)(56 377)(57 209)(58 210)(59 211)(60 212)(61 213)(62 214)(63 215)(64 216)(65 217)(66 218)(67 219)(68 220)(69 221)(70 222)(71 223)(72 224)(73 197)(74 198)(75 199)(76 200)(77 201)(78 202)(79 203)(80 204)(81 205)(82 206)(83 207)(84 208)(85 412)(86 413)(87 414)(88 415)(89 416)(90 417)(91 418)(92 419)(93 420)(94 393)(95 394)(96 395)(97 396)(98 397)(99 398)(100 399)(101 400)(102 401)(103 402)(104 403)(105 404)(106 405)(107 406)(108 407)(109 408)(110 409)(111 410)(112 411)(113 304)(114 305)(115 306)(116 307)(117 308)(118 281)(119 282)(120 283)(121 284)(122 285)(123 286)(124 287)(125 288)(126 289)(127 290)(128 291)(129 292)(130 293)(131 294)(132 295)(133 296)(134 297)(135 298)(136 299)(137 300)(138 301)(139 302)(140 303)(141 446)(142 447)(143 448)(144 421)(145 422)(146 423)(147 424)(148 425)(149 426)(150 427)(151 428)(152 429)(153 430)(154 431)(155 432)(156 433)(157 434)(158 435)(159 436)(160 437)(161 438)(162 439)(163 440)(164 441)(165 442)(166 443)(167 444)(168 445)(225 280)(226 253)(227 254)(228 255)(229 256)(230 257)(231 258)(232 259)(233 260)(234 261)(235 262)(236 263)(237 264)(238 265)(239 266)(240 267)(241 268)(242 269)(243 270)(244 271)(245 272)(246 273)(247 274)(248 275)(249 276)(250 277)(251 278)(252 279)(309 345)(310 346)(311 347)(312 348)(313 349)(314 350)(315 351)(316 352)(317 353)(318 354)(319 355)(320 356)(321 357)(322 358)(323 359)(324 360)(325 361)(326 362)(327 363)(328 364)(329 337)(330 338)(331 339)(332 340)(333 341)(334 342)(335 343)(336 344)
(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)
(1 212 41 395)(2 59 42 95)(3 210 43 393)(4 57 44 93)(5 208 45 419)(6 83 46 91)(7 206 47 417)(8 81 48 89)(9 204 49 415)(10 79 50 87)(11 202 51 413)(12 77 52 85)(13 200 53 411)(14 75 54 111)(15 198 55 409)(16 73 56 109)(17 224 29 407)(18 71 30 107)(19 222 31 405)(20 69 32 105)(21 220 33 403)(22 67 34 103)(23 218 35 401)(24 65 36 101)(25 216 37 399)(26 63 38 99)(27 214 39 397)(28 61 40 97)(58 392 94 172)(60 390 96 170)(62 388 98 196)(64 386 100 194)(66 384 102 192)(68 382 104 190)(70 380 106 188)(72 378 108 186)(74 376 110 184)(76 374 112 182)(78 372 86 180)(80 370 88 178)(82 368 90 176)(84 366 92 174)(113 422 279 323)(114 144 280 358)(115 448 253 321)(116 142 254 356)(117 446 255 319)(118 168 256 354)(119 444 257 317)(120 166 258 352)(121 442 259 315)(122 164 260 350)(123 440 261 313)(124 162 262 348)(125 438 263 311)(126 160 264 346)(127 436 265 309)(128 158 266 344)(129 434 267 335)(130 156 268 342)(131 432 269 333)(132 154 270 340)(133 430 271 331)(134 152 272 338)(135 428 273 329)(136 150 274 364)(137 426 275 327)(138 148 276 362)(139 424 277 325)(140 146 278 360)(141 228 355 308)(143 226 357 306)(145 252 359 304)(147 250 361 302)(149 248 363 300)(151 246 337 298)(153 244 339 296)(155 242 341 294)(157 240 343 292)(159 238 345 290)(161 236 347 288)(163 234 349 286)(165 232 351 284)(167 230 353 282)(169 213 389 396)(171 211 391 394)(173 209 365 420)(175 207 367 418)(177 205 369 416)(179 203 371 414)(181 201 373 412)(183 199 375 410)(185 197 377 408)(187 223 379 406)(189 221 381 404)(191 219 383 402)(193 217 385 400)(195 215 387 398)(225 322 305 421)(227 320 307 447)(229 318 281 445)(231 316 283 443)(233 314 285 441)(235 312 287 439)(237 310 289 437)(239 336 291 435)(241 334 293 433)(243 332 295 431)(245 330 297 429)(247 328 299 427)(249 326 301 425)(251 324 303 423)
(1 225 170 280)(2 252 171 279)(3 251 172 278)(4 250 173 277)(5 249 174 276)(6 248 175 275)(7 247 176 274)(8 246 177 273)(9 245 178 272)(10 244 179 271)(11 243 180 270)(12 242 181 269)(13 241 182 268)(14 240 183 267)(15 239 184 266)(16 238 185 265)(17 237 186 264)(18 236 187 263)(19 235 188 262)(20 234 189 261)(21 233 190 260)(22 232 191 259)(23 231 192 258)(24 230 193 257)(25 229 194 256)(26 228 195 255)(27 227 196 254)(28 226 169 253)(29 289 378 126)(30 288 379 125)(31 287 380 124)(32 286 381 123)(33 285 382 122)(34 284 383 121)(35 283 384 120)(36 282 385 119)(37 281 386 118)(38 308 387 117)(39 307 388 116)(40 306 389 115)(41 305 390 114)(42 304 391 113)(43 303 392 140)(44 302 365 139)(45 301 366 138)(46 300 367 137)(47 299 368 136)(48 298 369 135)(49 297 370 134)(50 296 371 133)(51 295 372 132)(52 294 373 131)(53 293 374 130)(54 292 375 129)(55 291 376 128)(56 290 377 127)(57 147 209 424)(58 146 210 423)(59 145 211 422)(60 144 212 421)(61 143 213 448)(62 142 214 447)(63 141 215 446)(64 168 216 445)(65 167 217 444)(66 166 218 443)(67 165 219 442)(68 164 220 441)(69 163 221 440)(70 162 222 439)(71 161 223 438)(72 160 224 437)(73 159 197 436)(74 158 198 435)(75 157 199 434)(76 156 200 433)(77 155 201 432)(78 154 202 431)(79 153 203 430)(80 152 204 429)(81 151 205 428)(82 150 206 427)(83 149 207 426)(84 148 208 425)(85 341 412 333)(86 340 413 332)(87 339 414 331)(88 338 415 330)(89 337 416 329)(90 364 417 328)(91 363 418 327)(92 362 419 326)(93 361 420 325)(94 360 393 324)(95 359 394 323)(96 358 395 322)(97 357 396 321)(98 356 397 320)(99 355 398 319)(100 354 399 318)(101 353 400 317)(102 352 401 316)(103 351 402 315)(104 350 403 314)(105 349 404 313)(106 348 405 312)(107 347 406 311)(108 346 407 310)(109 345 408 309)(110 344 409 336)(111 343 410 335)(112 342 411 334)
G:=sub<Sym(448)| (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,193)(25,194)(26,195)(27,196)(28,169)(29,378)(30,379)(31,380)(32,381)(33,382)(34,383)(35,384)(36,385)(37,386)(38,387)(39,388)(40,389)(41,390)(42,391)(43,392)(44,365)(45,366)(46,367)(47,368)(48,369)(49,370)(50,371)(51,372)(52,373)(53,374)(54,375)(55,376)(56,377)(57,209)(58,210)(59,211)(60,212)(61,213)(62,214)(63,215)(64,216)(65,217)(66,218)(67,219)(68,220)(69,221)(70,222)(71,223)(72,224)(73,197)(74,198)(75,199)(76,200)(77,201)(78,202)(79,203)(80,204)(81,205)(82,206)(83,207)(84,208)(85,412)(86,413)(87,414)(88,415)(89,416)(90,417)(91,418)(92,419)(93,420)(94,393)(95,394)(96,395)(97,396)(98,397)(99,398)(100,399)(101,400)(102,401)(103,402)(104,403)(105,404)(106,405)(107,406)(108,407)(109,408)(110,409)(111,410)(112,411)(113,304)(114,305)(115,306)(116,307)(117,308)(118,281)(119,282)(120,283)(121,284)(122,285)(123,286)(124,287)(125,288)(126,289)(127,290)(128,291)(129,292)(130,293)(131,294)(132,295)(133,296)(134,297)(135,298)(136,299)(137,300)(138,301)(139,302)(140,303)(141,446)(142,447)(143,448)(144,421)(145,422)(146,423)(147,424)(148,425)(149,426)(150,427)(151,428)(152,429)(153,430)(154,431)(155,432)(156,433)(157,434)(158,435)(159,436)(160,437)(161,438)(162,439)(163,440)(164,441)(165,442)(166,443)(167,444)(168,445)(225,280)(226,253)(227,254)(228,255)(229,256)(230,257)(231,258)(232,259)(233,260)(234,261)(235,262)(236,263)(237,264)(238,265)(239,266)(240,267)(241,268)(242,269)(243,270)(244,271)(245,272)(246,273)(247,274)(248,275)(249,276)(250,277)(251,278)(252,279)(309,345)(310,346)(311,347)(312,348)(313,349)(314,350)(315,351)(316,352)(317,353)(318,354)(319,355)(320,356)(321,357)(322,358)(323,359)(324,360)(325,361)(326,362)(327,363)(328,364)(329,337)(330,338)(331,339)(332,340)(333,341)(334,342)(335,343)(336,344), (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), (1,212,41,395)(2,59,42,95)(3,210,43,393)(4,57,44,93)(5,208,45,419)(6,83,46,91)(7,206,47,417)(8,81,48,89)(9,204,49,415)(10,79,50,87)(11,202,51,413)(12,77,52,85)(13,200,53,411)(14,75,54,111)(15,198,55,409)(16,73,56,109)(17,224,29,407)(18,71,30,107)(19,222,31,405)(20,69,32,105)(21,220,33,403)(22,67,34,103)(23,218,35,401)(24,65,36,101)(25,216,37,399)(26,63,38,99)(27,214,39,397)(28,61,40,97)(58,392,94,172)(60,390,96,170)(62,388,98,196)(64,386,100,194)(66,384,102,192)(68,382,104,190)(70,380,106,188)(72,378,108,186)(74,376,110,184)(76,374,112,182)(78,372,86,180)(80,370,88,178)(82,368,90,176)(84,366,92,174)(113,422,279,323)(114,144,280,358)(115,448,253,321)(116,142,254,356)(117,446,255,319)(118,168,256,354)(119,444,257,317)(120,166,258,352)(121,442,259,315)(122,164,260,350)(123,440,261,313)(124,162,262,348)(125,438,263,311)(126,160,264,346)(127,436,265,309)(128,158,266,344)(129,434,267,335)(130,156,268,342)(131,432,269,333)(132,154,270,340)(133,430,271,331)(134,152,272,338)(135,428,273,329)(136,150,274,364)(137,426,275,327)(138,148,276,362)(139,424,277,325)(140,146,278,360)(141,228,355,308)(143,226,357,306)(145,252,359,304)(147,250,361,302)(149,248,363,300)(151,246,337,298)(153,244,339,296)(155,242,341,294)(157,240,343,292)(159,238,345,290)(161,236,347,288)(163,234,349,286)(165,232,351,284)(167,230,353,282)(169,213,389,396)(171,211,391,394)(173,209,365,420)(175,207,367,418)(177,205,369,416)(179,203,371,414)(181,201,373,412)(183,199,375,410)(185,197,377,408)(187,223,379,406)(189,221,381,404)(191,219,383,402)(193,217,385,400)(195,215,387,398)(225,322,305,421)(227,320,307,447)(229,318,281,445)(231,316,283,443)(233,314,285,441)(235,312,287,439)(237,310,289,437)(239,336,291,435)(241,334,293,433)(243,332,295,431)(245,330,297,429)(247,328,299,427)(249,326,301,425)(251,324,303,423), (1,225,170,280)(2,252,171,279)(3,251,172,278)(4,250,173,277)(5,249,174,276)(6,248,175,275)(7,247,176,274)(8,246,177,273)(9,245,178,272)(10,244,179,271)(11,243,180,270)(12,242,181,269)(13,241,182,268)(14,240,183,267)(15,239,184,266)(16,238,185,265)(17,237,186,264)(18,236,187,263)(19,235,188,262)(20,234,189,261)(21,233,190,260)(22,232,191,259)(23,231,192,258)(24,230,193,257)(25,229,194,256)(26,228,195,255)(27,227,196,254)(28,226,169,253)(29,289,378,126)(30,288,379,125)(31,287,380,124)(32,286,381,123)(33,285,382,122)(34,284,383,121)(35,283,384,120)(36,282,385,119)(37,281,386,118)(38,308,387,117)(39,307,388,116)(40,306,389,115)(41,305,390,114)(42,304,391,113)(43,303,392,140)(44,302,365,139)(45,301,366,138)(46,300,367,137)(47,299,368,136)(48,298,369,135)(49,297,370,134)(50,296,371,133)(51,295,372,132)(52,294,373,131)(53,293,374,130)(54,292,375,129)(55,291,376,128)(56,290,377,127)(57,147,209,424)(58,146,210,423)(59,145,211,422)(60,144,212,421)(61,143,213,448)(62,142,214,447)(63,141,215,446)(64,168,216,445)(65,167,217,444)(66,166,218,443)(67,165,219,442)(68,164,220,441)(69,163,221,440)(70,162,222,439)(71,161,223,438)(72,160,224,437)(73,159,197,436)(74,158,198,435)(75,157,199,434)(76,156,200,433)(77,155,201,432)(78,154,202,431)(79,153,203,430)(80,152,204,429)(81,151,205,428)(82,150,206,427)(83,149,207,426)(84,148,208,425)(85,341,412,333)(86,340,413,332)(87,339,414,331)(88,338,415,330)(89,337,416,329)(90,364,417,328)(91,363,418,327)(92,362,419,326)(93,361,420,325)(94,360,393,324)(95,359,394,323)(96,358,395,322)(97,357,396,321)(98,356,397,320)(99,355,398,319)(100,354,399,318)(101,353,400,317)(102,352,401,316)(103,351,402,315)(104,350,403,314)(105,349,404,313)(106,348,405,312)(107,347,406,311)(108,346,407,310)(109,345,408,309)(110,344,409,336)(111,343,410,335)(112,342,411,334)>;
G:=Group( (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,193)(25,194)(26,195)(27,196)(28,169)(29,378)(30,379)(31,380)(32,381)(33,382)(34,383)(35,384)(36,385)(37,386)(38,387)(39,388)(40,389)(41,390)(42,391)(43,392)(44,365)(45,366)(46,367)(47,368)(48,369)(49,370)(50,371)(51,372)(52,373)(53,374)(54,375)(55,376)(56,377)(57,209)(58,210)(59,211)(60,212)(61,213)(62,214)(63,215)(64,216)(65,217)(66,218)(67,219)(68,220)(69,221)(70,222)(71,223)(72,224)(73,197)(74,198)(75,199)(76,200)(77,201)(78,202)(79,203)(80,204)(81,205)(82,206)(83,207)(84,208)(85,412)(86,413)(87,414)(88,415)(89,416)(90,417)(91,418)(92,419)(93,420)(94,393)(95,394)(96,395)(97,396)(98,397)(99,398)(100,399)(101,400)(102,401)(103,402)(104,403)(105,404)(106,405)(107,406)(108,407)(109,408)(110,409)(111,410)(112,411)(113,304)(114,305)(115,306)(116,307)(117,308)(118,281)(119,282)(120,283)(121,284)(122,285)(123,286)(124,287)(125,288)(126,289)(127,290)(128,291)(129,292)(130,293)(131,294)(132,295)(133,296)(134,297)(135,298)(136,299)(137,300)(138,301)(139,302)(140,303)(141,446)(142,447)(143,448)(144,421)(145,422)(146,423)(147,424)(148,425)(149,426)(150,427)(151,428)(152,429)(153,430)(154,431)(155,432)(156,433)(157,434)(158,435)(159,436)(160,437)(161,438)(162,439)(163,440)(164,441)(165,442)(166,443)(167,444)(168,445)(225,280)(226,253)(227,254)(228,255)(229,256)(230,257)(231,258)(232,259)(233,260)(234,261)(235,262)(236,263)(237,264)(238,265)(239,266)(240,267)(241,268)(242,269)(243,270)(244,271)(245,272)(246,273)(247,274)(248,275)(249,276)(250,277)(251,278)(252,279)(309,345)(310,346)(311,347)(312,348)(313,349)(314,350)(315,351)(316,352)(317,353)(318,354)(319,355)(320,356)(321,357)(322,358)(323,359)(324,360)(325,361)(326,362)(327,363)(328,364)(329,337)(330,338)(331,339)(332,340)(333,341)(334,342)(335,343)(336,344), (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), (1,212,41,395)(2,59,42,95)(3,210,43,393)(4,57,44,93)(5,208,45,419)(6,83,46,91)(7,206,47,417)(8,81,48,89)(9,204,49,415)(10,79,50,87)(11,202,51,413)(12,77,52,85)(13,200,53,411)(14,75,54,111)(15,198,55,409)(16,73,56,109)(17,224,29,407)(18,71,30,107)(19,222,31,405)(20,69,32,105)(21,220,33,403)(22,67,34,103)(23,218,35,401)(24,65,36,101)(25,216,37,399)(26,63,38,99)(27,214,39,397)(28,61,40,97)(58,392,94,172)(60,390,96,170)(62,388,98,196)(64,386,100,194)(66,384,102,192)(68,382,104,190)(70,380,106,188)(72,378,108,186)(74,376,110,184)(76,374,112,182)(78,372,86,180)(80,370,88,178)(82,368,90,176)(84,366,92,174)(113,422,279,323)(114,144,280,358)(115,448,253,321)(116,142,254,356)(117,446,255,319)(118,168,256,354)(119,444,257,317)(120,166,258,352)(121,442,259,315)(122,164,260,350)(123,440,261,313)(124,162,262,348)(125,438,263,311)(126,160,264,346)(127,436,265,309)(128,158,266,344)(129,434,267,335)(130,156,268,342)(131,432,269,333)(132,154,270,340)(133,430,271,331)(134,152,272,338)(135,428,273,329)(136,150,274,364)(137,426,275,327)(138,148,276,362)(139,424,277,325)(140,146,278,360)(141,228,355,308)(143,226,357,306)(145,252,359,304)(147,250,361,302)(149,248,363,300)(151,246,337,298)(153,244,339,296)(155,242,341,294)(157,240,343,292)(159,238,345,290)(161,236,347,288)(163,234,349,286)(165,232,351,284)(167,230,353,282)(169,213,389,396)(171,211,391,394)(173,209,365,420)(175,207,367,418)(177,205,369,416)(179,203,371,414)(181,201,373,412)(183,199,375,410)(185,197,377,408)(187,223,379,406)(189,221,381,404)(191,219,383,402)(193,217,385,400)(195,215,387,398)(225,322,305,421)(227,320,307,447)(229,318,281,445)(231,316,283,443)(233,314,285,441)(235,312,287,439)(237,310,289,437)(239,336,291,435)(241,334,293,433)(243,332,295,431)(245,330,297,429)(247,328,299,427)(249,326,301,425)(251,324,303,423), (1,225,170,280)(2,252,171,279)(3,251,172,278)(4,250,173,277)(5,249,174,276)(6,248,175,275)(7,247,176,274)(8,246,177,273)(9,245,178,272)(10,244,179,271)(11,243,180,270)(12,242,181,269)(13,241,182,268)(14,240,183,267)(15,239,184,266)(16,238,185,265)(17,237,186,264)(18,236,187,263)(19,235,188,262)(20,234,189,261)(21,233,190,260)(22,232,191,259)(23,231,192,258)(24,230,193,257)(25,229,194,256)(26,228,195,255)(27,227,196,254)(28,226,169,253)(29,289,378,126)(30,288,379,125)(31,287,380,124)(32,286,381,123)(33,285,382,122)(34,284,383,121)(35,283,384,120)(36,282,385,119)(37,281,386,118)(38,308,387,117)(39,307,388,116)(40,306,389,115)(41,305,390,114)(42,304,391,113)(43,303,392,140)(44,302,365,139)(45,301,366,138)(46,300,367,137)(47,299,368,136)(48,298,369,135)(49,297,370,134)(50,296,371,133)(51,295,372,132)(52,294,373,131)(53,293,374,130)(54,292,375,129)(55,291,376,128)(56,290,377,127)(57,147,209,424)(58,146,210,423)(59,145,211,422)(60,144,212,421)(61,143,213,448)(62,142,214,447)(63,141,215,446)(64,168,216,445)(65,167,217,444)(66,166,218,443)(67,165,219,442)(68,164,220,441)(69,163,221,440)(70,162,222,439)(71,161,223,438)(72,160,224,437)(73,159,197,436)(74,158,198,435)(75,157,199,434)(76,156,200,433)(77,155,201,432)(78,154,202,431)(79,153,203,430)(80,152,204,429)(81,151,205,428)(82,150,206,427)(83,149,207,426)(84,148,208,425)(85,341,412,333)(86,340,413,332)(87,339,414,331)(88,338,415,330)(89,337,416,329)(90,364,417,328)(91,363,418,327)(92,362,419,326)(93,361,420,325)(94,360,393,324)(95,359,394,323)(96,358,395,322)(97,357,396,321)(98,356,397,320)(99,355,398,319)(100,354,399,318)(101,353,400,317)(102,352,401,316)(103,351,402,315)(104,350,403,314)(105,349,404,313)(106,348,405,312)(107,347,406,311)(108,346,407,310)(109,345,408,309)(110,344,409,336)(111,343,410,335)(112,342,411,334) );
G=PermutationGroup([[(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,181),(13,182),(14,183),(15,184),(16,185),(17,186),(18,187),(19,188),(20,189),(21,190),(22,191),(23,192),(24,193),(25,194),(26,195),(27,196),(28,169),(29,378),(30,379),(31,380),(32,381),(33,382),(34,383),(35,384),(36,385),(37,386),(38,387),(39,388),(40,389),(41,390),(42,391),(43,392),(44,365),(45,366),(46,367),(47,368),(48,369),(49,370),(50,371),(51,372),(52,373),(53,374),(54,375),(55,376),(56,377),(57,209),(58,210),(59,211),(60,212),(61,213),(62,214),(63,215),(64,216),(65,217),(66,218),(67,219),(68,220),(69,221),(70,222),(71,223),(72,224),(73,197),(74,198),(75,199),(76,200),(77,201),(78,202),(79,203),(80,204),(81,205),(82,206),(83,207),(84,208),(85,412),(86,413),(87,414),(88,415),(89,416),(90,417),(91,418),(92,419),(93,420),(94,393),(95,394),(96,395),(97,396),(98,397),(99,398),(100,399),(101,400),(102,401),(103,402),(104,403),(105,404),(106,405),(107,406),(108,407),(109,408),(110,409),(111,410),(112,411),(113,304),(114,305),(115,306),(116,307),(117,308),(118,281),(119,282),(120,283),(121,284),(122,285),(123,286),(124,287),(125,288),(126,289),(127,290),(128,291),(129,292),(130,293),(131,294),(132,295),(133,296),(134,297),(135,298),(136,299),(137,300),(138,301),(139,302),(140,303),(141,446),(142,447),(143,448),(144,421),(145,422),(146,423),(147,424),(148,425),(149,426),(150,427),(151,428),(152,429),(153,430),(154,431),(155,432),(156,433),(157,434),(158,435),(159,436),(160,437),(161,438),(162,439),(163,440),(164,441),(165,442),(166,443),(167,444),(168,445),(225,280),(226,253),(227,254),(228,255),(229,256),(230,257),(231,258),(232,259),(233,260),(234,261),(235,262),(236,263),(237,264),(238,265),(239,266),(240,267),(241,268),(242,269),(243,270),(244,271),(245,272),(246,273),(247,274),(248,275),(249,276),(250,277),(251,278),(252,279),(309,345),(310,346),(311,347),(312,348),(313,349),(314,350),(315,351),(316,352),(317,353),(318,354),(319,355),(320,356),(321,357),(322,358),(323,359),(324,360),(325,361),(326,362),(327,363),(328,364),(329,337),(330,338),(331,339),(332,340),(333,341),(334,342),(335,343),(336,344)], [(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)], [(1,212,41,395),(2,59,42,95),(3,210,43,393),(4,57,44,93),(5,208,45,419),(6,83,46,91),(7,206,47,417),(8,81,48,89),(9,204,49,415),(10,79,50,87),(11,202,51,413),(12,77,52,85),(13,200,53,411),(14,75,54,111),(15,198,55,409),(16,73,56,109),(17,224,29,407),(18,71,30,107),(19,222,31,405),(20,69,32,105),(21,220,33,403),(22,67,34,103),(23,218,35,401),(24,65,36,101),(25,216,37,399),(26,63,38,99),(27,214,39,397),(28,61,40,97),(58,392,94,172),(60,390,96,170),(62,388,98,196),(64,386,100,194),(66,384,102,192),(68,382,104,190),(70,380,106,188),(72,378,108,186),(74,376,110,184),(76,374,112,182),(78,372,86,180),(80,370,88,178),(82,368,90,176),(84,366,92,174),(113,422,279,323),(114,144,280,358),(115,448,253,321),(116,142,254,356),(117,446,255,319),(118,168,256,354),(119,444,257,317),(120,166,258,352),(121,442,259,315),(122,164,260,350),(123,440,261,313),(124,162,262,348),(125,438,263,311),(126,160,264,346),(127,436,265,309),(128,158,266,344),(129,434,267,335),(130,156,268,342),(131,432,269,333),(132,154,270,340),(133,430,271,331),(134,152,272,338),(135,428,273,329),(136,150,274,364),(137,426,275,327),(138,148,276,362),(139,424,277,325),(140,146,278,360),(141,228,355,308),(143,226,357,306),(145,252,359,304),(147,250,361,302),(149,248,363,300),(151,246,337,298),(153,244,339,296),(155,242,341,294),(157,240,343,292),(159,238,345,290),(161,236,347,288),(163,234,349,286),(165,232,351,284),(167,230,353,282),(169,213,389,396),(171,211,391,394),(173,209,365,420),(175,207,367,418),(177,205,369,416),(179,203,371,414),(181,201,373,412),(183,199,375,410),(185,197,377,408),(187,223,379,406),(189,221,381,404),(191,219,383,402),(193,217,385,400),(195,215,387,398),(225,322,305,421),(227,320,307,447),(229,318,281,445),(231,316,283,443),(233,314,285,441),(235,312,287,439),(237,310,289,437),(239,336,291,435),(241,334,293,433),(243,332,295,431),(245,330,297,429),(247,328,299,427),(249,326,301,425),(251,324,303,423)], [(1,225,170,280),(2,252,171,279),(3,251,172,278),(4,250,173,277),(5,249,174,276),(6,248,175,275),(7,247,176,274),(8,246,177,273),(9,245,178,272),(10,244,179,271),(11,243,180,270),(12,242,181,269),(13,241,182,268),(14,240,183,267),(15,239,184,266),(16,238,185,265),(17,237,186,264),(18,236,187,263),(19,235,188,262),(20,234,189,261),(21,233,190,260),(22,232,191,259),(23,231,192,258),(24,230,193,257),(25,229,194,256),(26,228,195,255),(27,227,196,254),(28,226,169,253),(29,289,378,126),(30,288,379,125),(31,287,380,124),(32,286,381,123),(33,285,382,122),(34,284,383,121),(35,283,384,120),(36,282,385,119),(37,281,386,118),(38,308,387,117),(39,307,388,116),(40,306,389,115),(41,305,390,114),(42,304,391,113),(43,303,392,140),(44,302,365,139),(45,301,366,138),(46,300,367,137),(47,299,368,136),(48,298,369,135),(49,297,370,134),(50,296,371,133),(51,295,372,132),(52,294,373,131),(53,293,374,130),(54,292,375,129),(55,291,376,128),(56,290,377,127),(57,147,209,424),(58,146,210,423),(59,145,211,422),(60,144,212,421),(61,143,213,448),(62,142,214,447),(63,141,215,446),(64,168,216,445),(65,167,217,444),(66,166,218,443),(67,165,219,442),(68,164,220,441),(69,163,221,440),(70,162,222,439),(71,161,223,438),(72,160,224,437),(73,159,197,436),(74,158,198,435),(75,157,199,434),(76,156,200,433),(77,155,201,432),(78,154,202,431),(79,153,203,430),(80,152,204,429),(81,151,205,428),(82,150,206,427),(83,149,207,426),(84,148,208,425),(85,341,412,333),(86,340,413,332),(87,339,414,331),(88,338,415,330),(89,337,416,329),(90,364,417,328),(91,363,418,327),(92,362,419,326),(93,361,420,325),(94,360,393,324),(95,359,394,323),(96,358,395,322),(97,357,396,321),(98,356,397,320),(99,355,398,319),(100,354,399,318),(101,353,400,317),(102,352,401,316),(103,351,402,315),(104,350,403,314),(105,349,404,313),(106,348,405,312),(107,347,406,311),(108,346,407,310),(109,345,408,309),(110,344,409,336),(111,343,410,335),(112,342,411,334)]])
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 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | C4○D4 | D14 | D28 | C7⋊D4 | C4○D28 | D4⋊2D7 | Q8×D7 |
| kernel | (C2×C4).44D28 | C14.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×Dic7 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 4 | 4 | 3 | 6 | 9 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of (C2×C4).44D28 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 24 | 0 | 0 |
| 0 | 0 | 26 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,0,0,0,0,0,0,22,0,0,0,0,0,0,8,26,0,0,0,0,24,2,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,24,5,0,0,0,0,1,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,24,5,0,0,0,0,1,5,0,0,0,0,0,0,12,0,0,0,0,0,0,17] >;
(C2×C4).44D28 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{44}D_{28} % in TeX
G:=Group("(C2xC4).44D28"); // GroupNames label
G:=SmallGroup(448,517);
// by ID
G=gap.SmallGroup(448,517);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,387,184,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations