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## G = C22×C7⋊C16order 448 = 26·7

### Direct product of C22 and C7⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×C7⋊C16
 Chief series C1 — C7 — C14 — C28 — C56 — C7⋊C16 — C2×C7⋊C16 — C22×C7⋊C16
 Lower central C7 — C22×C7⋊C16
 Upper central C1 — C22×C8

Generators and relations for C22×C7⋊C16
G = < a,b,c,d | a2=b2=c7=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 164 in 98 conjugacy classes, 87 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C23, C14, C14, C16, C2×C8, C22×C4, C28, C28, C2×C14, C2×C16, C22×C8, C56, C56, C2×C28, C22×C14, C22×C16, C7⋊C16, C2×C56, C22×C28, C2×C7⋊C16, C22×C56, C22×C7⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C16, C2×C8, C22×C4, Dic7, D14, C2×C16, C22×C8, C7⋊C8, C2×Dic7, C22×D7, C22×C16, C7⋊C16, C2×C7⋊C8, C22×Dic7, C2×C7⋊C16, C22×C7⋊C8, C22×C7⋊C16

Smallest permutation representation of C22×C7⋊C16
Regular action on 448 points
Generators in S448
(1 61)(2 62)(3 63)(4 64)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 352)(18 337)(19 338)(20 339)(21 340)(22 341)(23 342)(24 343)(25 344)(26 345)(27 346)(28 347)(29 348)(30 349)(31 350)(32 351)(33 262)(34 263)(35 264)(36 265)(37 266)(38 267)(39 268)(40 269)(41 270)(42 271)(43 272)(44 257)(45 258)(46 259)(47 260)(48 261)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 119)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 190)(98 191)(99 192)(100 177)(101 178)(102 179)(103 180)(104 181)(105 182)(106 183)(107 184)(108 185)(109 186)(110 187)(111 188)(112 189)(129 163)(130 164)(131 165)(132 166)(133 167)(134 168)(135 169)(136 170)(137 171)(138 172)(139 173)(140 174)(141 175)(142 176)(143 161)(144 162)(193 274)(194 275)(195 276)(196 277)(197 278)(198 279)(199 280)(200 281)(201 282)(202 283)(203 284)(204 285)(205 286)(206 287)(207 288)(208 273)(209 335)(210 336)(211 321)(212 322)(213 323)(214 324)(215 325)(216 326)(217 327)(218 328)(219 329)(220 330)(221 331)(222 332)(223 333)(224 334)(225 312)(226 313)(227 314)(228 315)(229 316)(230 317)(231 318)(232 319)(233 320)(234 305)(235 306)(236 307)(237 308)(238 309)(239 310)(240 311)(241 368)(242 353)(243 354)(244 355)(245 356)(246 357)(247 358)(248 359)(249 360)(250 361)(251 362)(252 363)(253 364)(254 365)(255 366)(256 367)(289 422)(290 423)(291 424)(292 425)(293 426)(294 427)(295 428)(296 429)(297 430)(298 431)(299 432)(300 417)(301 418)(302 419)(303 420)(304 421)(369 415)(370 416)(371 401)(372 402)(373 403)(374 404)(375 405)(376 406)(377 407)(378 408)(379 409)(380 410)(381 411)(382 412)(383 413)(384 414)(385 440)(386 441)(387 442)(388 443)(389 444)(390 445)(391 446)(392 447)(393 448)(394 433)(395 434)(396 435)(397 436)(398 437)(399 438)(400 439)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 289)(18 290)(19 291)(20 292)(21 293)(22 294)(23 295)(24 296)(25 297)(26 298)(27 299)(28 300)(29 301)(30 302)(31 303)(32 304)(33 96)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 159)(62 160)(63 145)(64 146)(97 364)(98 365)(99 366)(100 367)(101 368)(102 353)(103 354)(104 355)(105 356)(106 357)(107 358)(108 359)(109 360)(110 361)(111 362)(112 363)(113 257)(114 258)(115 259)(116 260)(117 261)(118 262)(119 263)(120 264)(121 265)(122 266)(123 267)(124 268)(125 269)(126 270)(127 271)(128 272)(129 278)(130 279)(131 280)(132 281)(133 282)(134 283)(135 284)(136 285)(137 286)(138 287)(139 288)(140 273)(141 274)(142 275)(143 276)(144 277)(161 195)(162 196)(163 197)(164 198)(165 199)(166 200)(167 201)(168 202)(169 203)(170 204)(171 205)(172 206)(173 207)(174 208)(175 193)(176 194)(177 256)(178 241)(179 242)(180 243)(181 244)(182 245)(183 246)(184 247)(185 248)(186 249)(187 250)(188 251)(189 252)(190 253)(191 254)(192 255)(209 375)(210 376)(211 377)(212 378)(213 379)(214 380)(215 381)(216 382)(217 383)(218 384)(219 369)(220 370)(221 371)(222 372)(223 373)(224 374)(225 434)(226 435)(227 436)(228 437)(229 438)(230 439)(231 440)(232 441)(233 442)(234 443)(235 444)(236 445)(237 446)(238 447)(239 448)(240 433)(305 388)(306 389)(307 390)(308 391)(309 392)(310 393)(311 394)(312 395)(313 396)(314 397)(315 398)(316 399)(317 400)(318 385)(319 386)(320 387)(321 407)(322 408)(323 409)(324 410)(325 411)(326 412)(327 413)(328 414)(329 415)(330 416)(331 401)(332 402)(333 403)(334 404)(335 405)(336 406)(337 423)(338 424)(339 425)(340 426)(341 427)(342 428)(343 429)(344 430)(345 431)(346 432)(347 417)(348 418)(349 419)(350 420)(351 421)(352 422)
(1 123 143 239 221 418 367)(2 368 419 222 240 144 124)(3 125 129 225 223 420 353)(4 354 421 224 226 130 126)(5 127 131 227 209 422 355)(6 356 423 210 228 132 128)(7 113 133 229 211 424 357)(8 358 425 212 230 134 114)(9 115 135 231 213 426 359)(10 360 427 214 232 136 116)(11 117 137 233 215 428 361)(12 362 429 216 234 138 118)(13 119 139 235 217 430 363)(14 364 431 218 236 140 120)(15 121 141 237 219 432 365)(16 366 417 220 238 142 122)(17 181 147 42 199 397 405)(18 406 398 200 43 148 182)(19 183 149 44 201 399 407)(20 408 400 202 45 150 184)(21 185 151 46 203 385 409)(22 410 386 204 47 152 186)(23 187 153 48 205 387 411)(24 412 388 206 33 154 188)(25 189 155 34 207 389 413)(26 414 390 208 35 156 190)(27 191 157 36 193 391 415)(28 416 392 194 37 158 192)(29 177 159 38 195 393 401)(30 402 394 196 39 160 178)(31 179 145 40 197 395 403)(32 404 396 198 41 146 180)(49 89 165 314 335 289 244)(50 245 290 336 315 166 90)(51 91 167 316 321 291 246)(52 247 292 322 317 168 92)(53 93 169 318 323 293 248)(54 249 294 324 319 170 94)(55 95 171 320 325 295 250)(56 251 296 326 305 172 96)(57 81 173 306 327 297 252)(58 253 298 328 307 174 82)(59 83 175 308 329 299 254)(60 255 300 330 309 176 84)(61 85 161 310 331 301 256)(62 241 302 332 311 162 86)(63 87 163 312 333 303 242)(64 243 304 334 313 164 88)(65 109 341 380 441 285 260)(66 261 286 442 381 342 110)(67 111 343 382 443 287 262)(68 263 288 444 383 344 112)(69 97 345 384 445 273 264)(70 265 274 446 369 346 98)(71 99 347 370 447 275 266)(72 267 276 448 371 348 100)(73 101 349 372 433 277 268)(74 269 278 434 373 350 102)(75 103 351 374 435 279 270)(76 271 280 436 375 352 104)(77 105 337 376 437 281 272)(78 257 282 438 377 338 106)(79 107 339 378 439 283 258)(80 259 284 440 379 340 108)
(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)

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

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

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

160 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 7A 7B 7C 8A ··· 8P 14A ··· 14U 16A ··· 16AF 28A ··· 28X 56A ··· 56AV order 1 2 ··· 2 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 size 1 1 ··· 1 1 ··· 1 2 2 2 1 ··· 1 2 ··· 2 7 ··· 7 2 ··· 2 2 ··· 2

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 D7 Dic7 D14 Dic7 C7⋊C8 C7⋊C8 C7⋊C16 kernel C22×C7⋊C16 C2×C7⋊C16 C22×C56 C2×C56 C22×C28 C2×C28 C22×C14 C2×C14 C22×C8 C2×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 12 4 32 3 9 9 3 18 6 48

Matrix representation of C22×C7⋊C16 in GL4(𝔽113) generated by

 112 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 24 112 0 0 1 0
,
 15 0 0 0 0 112 0 0 0 0 45 20 0 0 83 68
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,24,1,0,0,112,0],[15,0,0,0,0,112,0,0,0,0,45,83,0,0,20,68] >;

C22×C7⋊C16 in GAP, Magma, Sage, TeX

C_2^2\times C_7\rtimes C_{16}
% in TeX

G:=Group("C2^2xC7:C16");
// GroupNames label

G:=SmallGroup(448,630);
// by ID

G=gap.SmallGroup(448,630);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,80,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^7=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽