direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C2.Q32, Q16⋊1C28, C56.95D4, C14.6Q32, C14.10SD32, C28.31SD16, C8.8(C2×C28), (C7×Q16)⋊7C4, C8.15(C7×D4), C2.1(C7×Q32), (C2×C112).3C2, (C2×C16).1C14, C56.64(C2×C4), (C2×C14).50D8, C2.D8.1C14, C4.2(C7×SD16), C2.2(C7×SD32), C22.9(C7×D8), (C2×C28).406D4, (C14×Q16).7C2, (C2×Q16).1C14, C28.70(C22⋊C4), (C2×C56).417C22, C14.38(D4⋊C4), (C2×C4).60(C7×D4), C4.2(C7×C22⋊C4), (C7×C2.D8).8C2, (C2×C8).72(C2×C14), C2.7(C7×D4⋊C4), SmallGroup(448,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C2.Q32
G = < a,b,c,d | a7=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 114 in 58 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, C28, C28, C2×C14, C2.D8, C2×C16, C2×Q16, C56, C2×C28, C2×C28, C7×Q8, C2.Q32, C112, C7×C4⋊C4, C2×C56, C7×Q16, C7×Q16, Q8×C14, C7×C2.D8, C2×C112, C14×Q16, C7×C2.Q32
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, C28, C2×C14, D4⋊C4, SD32, Q32, C2×C28, C7×D4, C2.Q32, C7×C22⋊C4, C7×D8, C7×SD16, C7×D4⋊C4, C7×SD32, C7×Q32, C7×C2.Q32
►Regular action on 448 points
Generators in S448
(1 157 193 364 277 216 122)(2 158 194 365 278 217 123)(3 159 195 366 279 218 124)(4 160 196 367 280 219 125)(5 145 197 368 281 220 126)(6 146 198 353 282 221 127)(7 147 199 354 283 222 128)(8 148 200 355 284 223 113)(9 149 201 356 285 224 114)(10 150 202 357 286 209 115)(11 151 203 358 287 210 116)(12 152 204 359 288 211 117)(13 153 205 360 273 212 118)(14 154 206 361 274 213 119)(15 155 207 362 275 214 120)(16 156 208 363 276 215 121)(17 92 296 401 387 64 111)(18 93 297 402 388 49 112)(19 94 298 403 389 50 97)(20 95 299 404 390 51 98)(21 96 300 405 391 52 99)(22 81 301 406 392 53 100)(23 82 302 407 393 54 101)(24 83 303 408 394 55 102)(25 84 304 409 395 56 103)(26 85 289 410 396 57 104)(27 86 290 411 397 58 105)(28 87 291 412 398 59 106)(29 88 292 413 399 60 107)(30 89 293 414 400 61 108)(31 90 294 415 385 62 109)(32 91 295 416 386 63 110)(33 311 74 444 263 336 421)(34 312 75 445 264 321 422)(35 313 76 446 265 322 423)(36 314 77 447 266 323 424)(37 315 78 448 267 324 425)(38 316 79 433 268 325 426)(39 317 80 434 269 326 427)(40 318 65 435 270 327 428)(41 319 66 436 271 328 429)(42 320 67 437 272 329 430)(43 305 68 438 257 330 431)(44 306 69 439 258 331 432)(45 307 70 440 259 332 417)(46 308 71 441 260 333 418)(47 309 72 442 261 334 419)(48 310 73 443 262 335 420)(129 233 380 256 188 346 169)(130 234 381 241 189 347 170)(131 235 382 242 190 348 171)(132 236 383 243 191 349 172)(133 237 384 244 192 350 173)(134 238 369 245 177 351 174)(135 239 370 246 178 352 175)(136 240 371 247 179 337 176)(137 225 372 248 180 338 161)(138 226 373 249 181 339 162)(139 227 374 250 182 340 163)(140 228 375 251 183 341 164)(141 229 376 252 184 342 165)(142 230 377 253 185 343 166)(143 231 378 254 186 344 167)(144 232 379 255 187 345 168)
(1 321)(2 322)(3 323)(4 324)(5 325)(6 326)(7 327)(8 328)(9 329)(10 330)(11 331)(12 332)(13 333)(14 334)(15 335)(16 336)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 177)(25 178)(26 179)(27 180)(28 181)(29 182)(30 183)(31 184)(32 185)(33 208)(34 193)(35 194)(36 195)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 379)(50 380)(51 381)(52 382)(53 383)(54 384)(55 369)(56 370)(57 371)(58 372)(59 373)(60 374)(61 375)(62 376)(63 377)(64 378)(65 283)(66 284)(67 285)(68 286)(69 287)(70 288)(71 273)(72 274)(73 275)(74 276)(75 277)(76 278)(77 279)(78 280)(79 281)(80 282)(81 349)(82 350)(83 351)(84 352)(85 337)(86 338)(87 339)(88 340)(89 341)(90 342)(91 343)(92 344)(93 345)(94 346)(95 347)(96 348)(97 256)(98 241)(99 242)(100 243)(101 244)(102 245)(103 246)(104 247)(105 248)(106 249)(107 250)(108 251)(109 252)(110 253)(111 254)(112 255)(113 271)(114 272)(115 257)(116 258)(117 259)(118 260)(119 261)(120 262)(121 263)(122 264)(123 265)(124 266)(125 267)(126 268)(127 269)(128 270)(129 403)(130 404)(131 405)(132 406)(133 407)(134 408)(135 409)(136 410)(137 411)(138 412)(139 413)(140 414)(141 415)(142 416)(143 401)(144 402)(145 426)(146 427)(147 428)(148 429)(149 430)(150 431)(151 432)(152 417)(153 418)(154 419)(155 420)(156 421)(157 422)(158 423)(159 424)(160 425)(161 290)(162 291)(163 292)(164 293)(165 294)(166 295)(167 296)(168 297)(169 298)(170 299)(171 300)(172 301)(173 302)(174 303)(175 304)(176 289)(209 438)(210 439)(211 440)(212 441)(213 442)(214 443)(215 444)(216 445)(217 446)(218 447)(219 448)(220 433)(221 434)(222 435)(223 436)(224 437)(225 397)(226 398)(227 399)(228 400)(229 385)(230 386)(231 387)(232 388)(233 389)(234 390)(235 391)(236 392)(237 393)(238 394)(239 395)(240 396)(305 357)(306 358)(307 359)(308 360)(309 361)(310 362)(311 363)(312 364)(313 365)(314 366)(315 367)(316 368)(317 353)(318 354)(319 355)(320 356)
(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)
(1 398 9 390)(2 225 10 233)(3 396 11 388)(4 239 12 231)(5 394 13 386)(6 237 14 229)(7 392 15 400)(8 235 16 227)(17 315 25 307)(18 366 26 358)(19 313 27 305)(20 364 28 356)(21 311 29 319)(22 362 30 354)(23 309 31 317)(24 360 32 368)(33 107 41 99)(34 249 42 241)(35 105 43 97)(36 247 44 255)(37 103 45 111)(38 245 46 253)(39 101 47 109)(40 243 48 251)(49 159 57 151)(50 423 58 431)(51 157 59 149)(52 421 60 429)(53 155 61 147)(54 419 62 427)(55 153 63 145)(56 417 64 425)(65 349 73 341)(66 96 74 88)(67 347 75 339)(68 94 76 86)(69 345 77 337)(70 92 78 84)(71 343 79 351)(72 90 80 82)(81 275 89 283)(83 273 91 281)(85 287 93 279)(87 285 95 277)(98 193 106 201)(100 207 108 199)(102 205 110 197)(104 203 112 195)(113 131 121 139)(114 404 122 412)(115 129 123 137)(116 402 124 410)(117 143 125 135)(118 416 126 408)(119 141 127 133)(120 414 128 406)(130 264 138 272)(132 262 140 270)(134 260 142 268)(136 258 144 266)(146 384 154 376)(148 382 156 374)(150 380 158 372)(152 378 160 370)(161 209 169 217)(162 437 170 445)(163 223 171 215)(164 435 172 443)(165 221 173 213)(166 433 174 441)(167 219 175 211)(168 447 176 439)(177 308 185 316)(178 359 186 367)(179 306 187 314)(180 357 188 365)(181 320 189 312)(182 355 190 363)(183 318 191 310)(184 353 192 361)(194 248 202 256)(196 246 204 254)(198 244 206 252)(200 242 208 250)(210 297 218 289)(212 295 220 303)(214 293 222 301)(216 291 224 299)(226 329 234 321)(228 327 236 335)(230 325 238 333)(232 323 240 331)(257 403 265 411)(259 401 267 409)(261 415 269 407)(263 413 271 405)(274 342 282 350)(276 340 284 348)(278 338 286 346)(280 352 288 344)(290 438 298 446)(292 436 300 444)(294 434 302 442)(296 448 304 440)(322 397 330 389)(324 395 332 387)(326 393 334 385)(328 391 336 399)(369 418 377 426)(371 432 379 424)(373 430 381 422)(375 428 383 420)
G:=sub<Sym(448)| (1,157,193,364,277,216,122)(2,158,194,365,278,217,123)(3,159,195,366,279,218,124)(4,160,196,367,280,219,125)(5,145,197,368,281,220,126)(6,146,198,353,282,221,127)(7,147,199,354,283,222,128)(8,148,200,355,284,223,113)(9,149,201,356,285,224,114)(10,150,202,357,286,209,115)(11,151,203,358,287,210,116)(12,152,204,359,288,211,117)(13,153,205,360,273,212,118)(14,154,206,361,274,213,119)(15,155,207,362,275,214,120)(16,156,208,363,276,215,121)(17,92,296,401,387,64,111)(18,93,297,402,388,49,112)(19,94,298,403,389,50,97)(20,95,299,404,390,51,98)(21,96,300,405,391,52,99)(22,81,301,406,392,53,100)(23,82,302,407,393,54,101)(24,83,303,408,394,55,102)(25,84,304,409,395,56,103)(26,85,289,410,396,57,104)(27,86,290,411,397,58,105)(28,87,291,412,398,59,106)(29,88,292,413,399,60,107)(30,89,293,414,400,61,108)(31,90,294,415,385,62,109)(32,91,295,416,386,63,110)(33,311,74,444,263,336,421)(34,312,75,445,264,321,422)(35,313,76,446,265,322,423)(36,314,77,447,266,323,424)(37,315,78,448,267,324,425)(38,316,79,433,268,325,426)(39,317,80,434,269,326,427)(40,318,65,435,270,327,428)(41,319,66,436,271,328,429)(42,320,67,437,272,329,430)(43,305,68,438,257,330,431)(44,306,69,439,258,331,432)(45,307,70,440,259,332,417)(46,308,71,441,260,333,418)(47,309,72,442,261,334,419)(48,310,73,443,262,335,420)(129,233,380,256,188,346,169)(130,234,381,241,189,347,170)(131,235,382,242,190,348,171)(132,236,383,243,191,349,172)(133,237,384,244,192,350,173)(134,238,369,245,177,351,174)(135,239,370,246,178,352,175)(136,240,371,247,179,337,176)(137,225,372,248,180,338,161)(138,226,373,249,181,339,162)(139,227,374,250,182,340,163)(140,228,375,251,183,341,164)(141,229,376,252,184,342,165)(142,230,377,253,185,343,166)(143,231,378,254,186,344,167)(144,232,379,255,187,345,168), (1,321)(2,322)(3,323)(4,324)(5,325)(6,326)(7,327)(8,328)(9,329)(10,330)(11,331)(12,332)(13,333)(14,334)(15,335)(16,336)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,177)(25,178)(26,179)(27,180)(28,181)(29,182)(30,183)(31,184)(32,185)(33,208)(34,193)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,379)(50,380)(51,381)(52,382)(53,383)(54,384)(55,369)(56,370)(57,371)(58,372)(59,373)(60,374)(61,375)(62,376)(63,377)(64,378)(65,283)(66,284)(67,285)(68,286)(69,287)(70,288)(71,273)(72,274)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,281)(80,282)(81,349)(82,350)(83,351)(84,352)(85,337)(86,338)(87,339)(88,340)(89,341)(90,342)(91,343)(92,344)(93,345)(94,346)(95,347)(96,348)(97,256)(98,241)(99,242)(100,243)(101,244)(102,245)(103,246)(104,247)(105,248)(106,249)(107,250)(108,251)(109,252)(110,253)(111,254)(112,255)(113,271)(114,272)(115,257)(116,258)(117,259)(118,260)(119,261)(120,262)(121,263)(122,264)(123,265)(124,266)(125,267)(126,268)(127,269)(128,270)(129,403)(130,404)(131,405)(132,406)(133,407)(134,408)(135,409)(136,410)(137,411)(138,412)(139,413)(140,414)(141,415)(142,416)(143,401)(144,402)(145,426)(146,427)(147,428)(148,429)(149,430)(150,431)(151,432)(152,417)(153,418)(154,419)(155,420)(156,421)(157,422)(158,423)(159,424)(160,425)(161,290)(162,291)(163,292)(164,293)(165,294)(166,295)(167,296)(168,297)(169,298)(170,299)(171,300)(172,301)(173,302)(174,303)(175,304)(176,289)(209,438)(210,439)(211,440)(212,441)(213,442)(214,443)(215,444)(216,445)(217,446)(218,447)(219,448)(220,433)(221,434)(222,435)(223,436)(224,437)(225,397)(226,398)(227,399)(228,400)(229,385)(230,386)(231,387)(232,388)(233,389)(234,390)(235,391)(236,392)(237,393)(238,394)(239,395)(240,396)(305,357)(306,358)(307,359)(308,360)(309,361)(310,362)(311,363)(312,364)(313,365)(314,366)(315,367)(316,368)(317,353)(318,354)(319,355)(320,356), (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), (1,398,9,390)(2,225,10,233)(3,396,11,388)(4,239,12,231)(5,394,13,386)(6,237,14,229)(7,392,15,400)(8,235,16,227)(17,315,25,307)(18,366,26,358)(19,313,27,305)(20,364,28,356)(21,311,29,319)(22,362,30,354)(23,309,31,317)(24,360,32,368)(33,107,41,99)(34,249,42,241)(35,105,43,97)(36,247,44,255)(37,103,45,111)(38,245,46,253)(39,101,47,109)(40,243,48,251)(49,159,57,151)(50,423,58,431)(51,157,59,149)(52,421,60,429)(53,155,61,147)(54,419,62,427)(55,153,63,145)(56,417,64,425)(65,349,73,341)(66,96,74,88)(67,347,75,339)(68,94,76,86)(69,345,77,337)(70,92,78,84)(71,343,79,351)(72,90,80,82)(81,275,89,283)(83,273,91,281)(85,287,93,279)(87,285,95,277)(98,193,106,201)(100,207,108,199)(102,205,110,197)(104,203,112,195)(113,131,121,139)(114,404,122,412)(115,129,123,137)(116,402,124,410)(117,143,125,135)(118,416,126,408)(119,141,127,133)(120,414,128,406)(130,264,138,272)(132,262,140,270)(134,260,142,268)(136,258,144,266)(146,384,154,376)(148,382,156,374)(150,380,158,372)(152,378,160,370)(161,209,169,217)(162,437,170,445)(163,223,171,215)(164,435,172,443)(165,221,173,213)(166,433,174,441)(167,219,175,211)(168,447,176,439)(177,308,185,316)(178,359,186,367)(179,306,187,314)(180,357,188,365)(181,320,189,312)(182,355,190,363)(183,318,191,310)(184,353,192,361)(194,248,202,256)(196,246,204,254)(198,244,206,252)(200,242,208,250)(210,297,218,289)(212,295,220,303)(214,293,222,301)(216,291,224,299)(226,329,234,321)(228,327,236,335)(230,325,238,333)(232,323,240,331)(257,403,265,411)(259,401,267,409)(261,415,269,407)(263,413,271,405)(274,342,282,350)(276,340,284,348)(278,338,286,346)(280,352,288,344)(290,438,298,446)(292,436,300,444)(294,434,302,442)(296,448,304,440)(322,397,330,389)(324,395,332,387)(326,393,334,385)(328,391,336,399)(369,418,377,426)(371,432,379,424)(373,430,381,422)(375,428,383,420)>;
G:=Group( (1,157,193,364,277,216,122)(2,158,194,365,278,217,123)(3,159,195,366,279,218,124)(4,160,196,367,280,219,125)(5,145,197,368,281,220,126)(6,146,198,353,282,221,127)(7,147,199,354,283,222,128)(8,148,200,355,284,223,113)(9,149,201,356,285,224,114)(10,150,202,357,286,209,115)(11,151,203,358,287,210,116)(12,152,204,359,288,211,117)(13,153,205,360,273,212,118)(14,154,206,361,274,213,119)(15,155,207,362,275,214,120)(16,156,208,363,276,215,121)(17,92,296,401,387,64,111)(18,93,297,402,388,49,112)(19,94,298,403,389,50,97)(20,95,299,404,390,51,98)(21,96,300,405,391,52,99)(22,81,301,406,392,53,100)(23,82,302,407,393,54,101)(24,83,303,408,394,55,102)(25,84,304,409,395,56,103)(26,85,289,410,396,57,104)(27,86,290,411,397,58,105)(28,87,291,412,398,59,106)(29,88,292,413,399,60,107)(30,89,293,414,400,61,108)(31,90,294,415,385,62,109)(32,91,295,416,386,63,110)(33,311,74,444,263,336,421)(34,312,75,445,264,321,422)(35,313,76,446,265,322,423)(36,314,77,447,266,323,424)(37,315,78,448,267,324,425)(38,316,79,433,268,325,426)(39,317,80,434,269,326,427)(40,318,65,435,270,327,428)(41,319,66,436,271,328,429)(42,320,67,437,272,329,430)(43,305,68,438,257,330,431)(44,306,69,439,258,331,432)(45,307,70,440,259,332,417)(46,308,71,441,260,333,418)(47,309,72,442,261,334,419)(48,310,73,443,262,335,420)(129,233,380,256,188,346,169)(130,234,381,241,189,347,170)(131,235,382,242,190,348,171)(132,236,383,243,191,349,172)(133,237,384,244,192,350,173)(134,238,369,245,177,351,174)(135,239,370,246,178,352,175)(136,240,371,247,179,337,176)(137,225,372,248,180,338,161)(138,226,373,249,181,339,162)(139,227,374,250,182,340,163)(140,228,375,251,183,341,164)(141,229,376,252,184,342,165)(142,230,377,253,185,343,166)(143,231,378,254,186,344,167)(144,232,379,255,187,345,168), (1,321)(2,322)(3,323)(4,324)(5,325)(6,326)(7,327)(8,328)(9,329)(10,330)(11,331)(12,332)(13,333)(14,334)(15,335)(16,336)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,177)(25,178)(26,179)(27,180)(28,181)(29,182)(30,183)(31,184)(32,185)(33,208)(34,193)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,379)(50,380)(51,381)(52,382)(53,383)(54,384)(55,369)(56,370)(57,371)(58,372)(59,373)(60,374)(61,375)(62,376)(63,377)(64,378)(65,283)(66,284)(67,285)(68,286)(69,287)(70,288)(71,273)(72,274)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,281)(80,282)(81,349)(82,350)(83,351)(84,352)(85,337)(86,338)(87,339)(88,340)(89,341)(90,342)(91,343)(92,344)(93,345)(94,346)(95,347)(96,348)(97,256)(98,241)(99,242)(100,243)(101,244)(102,245)(103,246)(104,247)(105,248)(106,249)(107,250)(108,251)(109,252)(110,253)(111,254)(112,255)(113,271)(114,272)(115,257)(116,258)(117,259)(118,260)(119,261)(120,262)(121,263)(122,264)(123,265)(124,266)(125,267)(126,268)(127,269)(128,270)(129,403)(130,404)(131,405)(132,406)(133,407)(134,408)(135,409)(136,410)(137,411)(138,412)(139,413)(140,414)(141,415)(142,416)(143,401)(144,402)(145,426)(146,427)(147,428)(148,429)(149,430)(150,431)(151,432)(152,417)(153,418)(154,419)(155,420)(156,421)(157,422)(158,423)(159,424)(160,425)(161,290)(162,291)(163,292)(164,293)(165,294)(166,295)(167,296)(168,297)(169,298)(170,299)(171,300)(172,301)(173,302)(174,303)(175,304)(176,289)(209,438)(210,439)(211,440)(212,441)(213,442)(214,443)(215,444)(216,445)(217,446)(218,447)(219,448)(220,433)(221,434)(222,435)(223,436)(224,437)(225,397)(226,398)(227,399)(228,400)(229,385)(230,386)(231,387)(232,388)(233,389)(234,390)(235,391)(236,392)(237,393)(238,394)(239,395)(240,396)(305,357)(306,358)(307,359)(308,360)(309,361)(310,362)(311,363)(312,364)(313,365)(314,366)(315,367)(316,368)(317,353)(318,354)(319,355)(320,356), (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), (1,398,9,390)(2,225,10,233)(3,396,11,388)(4,239,12,231)(5,394,13,386)(6,237,14,229)(7,392,15,400)(8,235,16,227)(17,315,25,307)(18,366,26,358)(19,313,27,305)(20,364,28,356)(21,311,29,319)(22,362,30,354)(23,309,31,317)(24,360,32,368)(33,107,41,99)(34,249,42,241)(35,105,43,97)(36,247,44,255)(37,103,45,111)(38,245,46,253)(39,101,47,109)(40,243,48,251)(49,159,57,151)(50,423,58,431)(51,157,59,149)(52,421,60,429)(53,155,61,147)(54,419,62,427)(55,153,63,145)(56,417,64,425)(65,349,73,341)(66,96,74,88)(67,347,75,339)(68,94,76,86)(69,345,77,337)(70,92,78,84)(71,343,79,351)(72,90,80,82)(81,275,89,283)(83,273,91,281)(85,287,93,279)(87,285,95,277)(98,193,106,201)(100,207,108,199)(102,205,110,197)(104,203,112,195)(113,131,121,139)(114,404,122,412)(115,129,123,137)(116,402,124,410)(117,143,125,135)(118,416,126,408)(119,141,127,133)(120,414,128,406)(130,264,138,272)(132,262,140,270)(134,260,142,268)(136,258,144,266)(146,384,154,376)(148,382,156,374)(150,380,158,372)(152,378,160,370)(161,209,169,217)(162,437,170,445)(163,223,171,215)(164,435,172,443)(165,221,173,213)(166,433,174,441)(167,219,175,211)(168,447,176,439)(177,308,185,316)(178,359,186,367)(179,306,187,314)(180,357,188,365)(181,320,189,312)(182,355,190,363)(183,318,191,310)(184,353,192,361)(194,248,202,256)(196,246,204,254)(198,244,206,252)(200,242,208,250)(210,297,218,289)(212,295,220,303)(214,293,222,301)(216,291,224,299)(226,329,234,321)(228,327,236,335)(230,325,238,333)(232,323,240,331)(257,403,265,411)(259,401,267,409)(261,415,269,407)(263,413,271,405)(274,342,282,350)(276,340,284,348)(278,338,286,346)(280,352,288,344)(290,438,298,446)(292,436,300,444)(294,434,302,442)(296,448,304,440)(322,397,330,389)(324,395,332,387)(326,393,334,385)(328,391,336,399)(369,418,377,426)(371,432,379,424)(373,430,381,422)(375,428,383,420) );
G=PermutationGroup([[(1,157,193,364,277,216,122),(2,158,194,365,278,217,123),(3,159,195,366,279,218,124),(4,160,196,367,280,219,125),(5,145,197,368,281,220,126),(6,146,198,353,282,221,127),(7,147,199,354,283,222,128),(8,148,200,355,284,223,113),(9,149,201,356,285,224,114),(10,150,202,357,286,209,115),(11,151,203,358,287,210,116),(12,152,204,359,288,211,117),(13,153,205,360,273,212,118),(14,154,206,361,274,213,119),(15,155,207,362,275,214,120),(16,156,208,363,276,215,121),(17,92,296,401,387,64,111),(18,93,297,402,388,49,112),(19,94,298,403,389,50,97),(20,95,299,404,390,51,98),(21,96,300,405,391,52,99),(22,81,301,406,392,53,100),(23,82,302,407,393,54,101),(24,83,303,408,394,55,102),(25,84,304,409,395,56,103),(26,85,289,410,396,57,104),(27,86,290,411,397,58,105),(28,87,291,412,398,59,106),(29,88,292,413,399,60,107),(30,89,293,414,400,61,108),(31,90,294,415,385,62,109),(32,91,295,416,386,63,110),(33,311,74,444,263,336,421),(34,312,75,445,264,321,422),(35,313,76,446,265,322,423),(36,314,77,447,266,323,424),(37,315,78,448,267,324,425),(38,316,79,433,268,325,426),(39,317,80,434,269,326,427),(40,318,65,435,270,327,428),(41,319,66,436,271,328,429),(42,320,67,437,272,329,430),(43,305,68,438,257,330,431),(44,306,69,439,258,331,432),(45,307,70,440,259,332,417),(46,308,71,441,260,333,418),(47,309,72,442,261,334,419),(48,310,73,443,262,335,420),(129,233,380,256,188,346,169),(130,234,381,241,189,347,170),(131,235,382,242,190,348,171),(132,236,383,243,191,349,172),(133,237,384,244,192,350,173),(134,238,369,245,177,351,174),(135,239,370,246,178,352,175),(136,240,371,247,179,337,176),(137,225,372,248,180,338,161),(138,226,373,249,181,339,162),(139,227,374,250,182,340,163),(140,228,375,251,183,341,164),(141,229,376,252,184,342,165),(142,230,377,253,185,343,166),(143,231,378,254,186,344,167),(144,232,379,255,187,345,168)], [(1,321),(2,322),(3,323),(4,324),(5,325),(6,326),(7,327),(8,328),(9,329),(10,330),(11,331),(12,332),(13,333),(14,334),(15,335),(16,336),(17,186),(18,187),(19,188),(20,189),(21,190),(22,191),(23,192),(24,177),(25,178),(26,179),(27,180),(28,181),(29,182),(30,183),(31,184),(32,185),(33,208),(34,193),(35,194),(36,195),(37,196),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,379),(50,380),(51,381),(52,382),(53,383),(54,384),(55,369),(56,370),(57,371),(58,372),(59,373),(60,374),(61,375),(62,376),(63,377),(64,378),(65,283),(66,284),(67,285),(68,286),(69,287),(70,288),(71,273),(72,274),(73,275),(74,276),(75,277),(76,278),(77,279),(78,280),(79,281),(80,282),(81,349),(82,350),(83,351),(84,352),(85,337),(86,338),(87,339),(88,340),(89,341),(90,342),(91,343),(92,344),(93,345),(94,346),(95,347),(96,348),(97,256),(98,241),(99,242),(100,243),(101,244),(102,245),(103,246),(104,247),(105,248),(106,249),(107,250),(108,251),(109,252),(110,253),(111,254),(112,255),(113,271),(114,272),(115,257),(116,258),(117,259),(118,260),(119,261),(120,262),(121,263),(122,264),(123,265),(124,266),(125,267),(126,268),(127,269),(128,270),(129,403),(130,404),(131,405),(132,406),(133,407),(134,408),(135,409),(136,410),(137,411),(138,412),(139,413),(140,414),(141,415),(142,416),(143,401),(144,402),(145,426),(146,427),(147,428),(148,429),(149,430),(150,431),(151,432),(152,417),(153,418),(154,419),(155,420),(156,421),(157,422),(158,423),(159,424),(160,425),(161,290),(162,291),(163,292),(164,293),(165,294),(166,295),(167,296),(168,297),(169,298),(170,299),(171,300),(172,301),(173,302),(174,303),(175,304),(176,289),(209,438),(210,439),(211,440),(212,441),(213,442),(214,443),(215,444),(216,445),(217,446),(218,447),(219,448),(220,433),(221,434),(222,435),(223,436),(224,437),(225,397),(226,398),(227,399),(228,400),(229,385),(230,386),(231,387),(232,388),(233,389),(234,390),(235,391),(236,392),(237,393),(238,394),(239,395),(240,396),(305,357),(306,358),(307,359),(308,360),(309,361),(310,362),(311,363),(312,364),(313,365),(314,366),(315,367),(316,368),(317,353),(318,354),(319,355),(320,356)], [(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)], [(1,398,9,390),(2,225,10,233),(3,396,11,388),(4,239,12,231),(5,394,13,386),(6,237,14,229),(7,392,15,400),(8,235,16,227),(17,315,25,307),(18,366,26,358),(19,313,27,305),(20,364,28,356),(21,311,29,319),(22,362,30,354),(23,309,31,317),(24,360,32,368),(33,107,41,99),(34,249,42,241),(35,105,43,97),(36,247,44,255),(37,103,45,111),(38,245,46,253),(39,101,47,109),(40,243,48,251),(49,159,57,151),(50,423,58,431),(51,157,59,149),(52,421,60,429),(53,155,61,147),(54,419,62,427),(55,153,63,145),(56,417,64,425),(65,349,73,341),(66,96,74,88),(67,347,75,339),(68,94,76,86),(69,345,77,337),(70,92,78,84),(71,343,79,351),(72,90,80,82),(81,275,89,283),(83,273,91,281),(85,287,93,279),(87,285,95,277),(98,193,106,201),(100,207,108,199),(102,205,110,197),(104,203,112,195),(113,131,121,139),(114,404,122,412),(115,129,123,137),(116,402,124,410),(117,143,125,135),(118,416,126,408),(119,141,127,133),(120,414,128,406),(130,264,138,272),(132,262,140,270),(134,260,142,268),(136,258,144,266),(146,384,154,376),(148,382,156,374),(150,380,158,372),(152,378,160,370),(161,209,169,217),(162,437,170,445),(163,223,171,215),(164,435,172,443),(165,221,173,213),(166,433,174,441),(167,219,175,211),(168,447,176,439),(177,308,185,316),(178,359,186,367),(179,306,187,314),(180,357,188,365),(181,320,189,312),(182,355,190,363),(183,318,191,310),(184,353,192,361),(194,248,202,256),(196,246,204,254),(198,244,206,252),(200,242,208,250),(210,297,218,289),(212,295,220,303),(214,293,222,301),(216,291,224,299),(226,329,234,321),(228,327,236,335),(230,325,238,333),(232,323,240,331),(257,403,265,411),(259,401,267,409),(261,415,269,407),(263,413,271,405),(274,342,282,350),(276,340,284,348),(278,338,286,346),(280,352,288,344),(290,438,298,446),(292,436,300,444),(294,434,302,442),(296,448,304,440),(322,397,330,389),(324,395,332,387),(326,393,334,385),(328,391,336,399),(369,418,377,426),(371,432,379,424),(373,430,381,422),(375,428,383,420)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28AJ | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | D4 | SD16 | D8 | SD32 | Q32 | C7×D4 | C7×D4 | C7×SD16 | C7×D8 | C7×SD32 | C7×Q32 |
| kernel | C7×C2.Q32 | C7×C2.D8 | C2×C112 | C14×Q16 | C7×Q16 | C2.Q32 | C2.D8 | C2×C16 | C2×Q16 | Q16 | C56 | C2×C28 | C28 | C2×C14 | C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 24 | 24 |
Matrix representation of C7×C2.Q32 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 112 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 98 | 0 | 0 |
| 0 | 53 | 69 |
| 0 | 44 | 53 |
| 112 | 0 | 0 |
| 0 | 84 | 47 |
| 0 | 47 | 29 |
G:=sub<GL(3,GF(113))| [1,0,0,0,49,0,0,0,49],[112,0,0,0,112,0,0,0,112],[98,0,0,0,53,44,0,69,53],[112,0,0,0,84,47,0,47,29] >;
C7×C2.Q32 in GAP, Magma, Sage, TeX
C_7\times C_2.Q_{32} % in TeX
G:=Group("C7xC2.Q32"); // GroupNames label
G:=SmallGroup(448,162);
// by ID
G=gap.SmallGroup(448,162);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,1576,3923,1970,360,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations