direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C7⋊C32, C14⋊C32, C56.5C8, C112.3C4, C28.3C16, C16.21D14, C16.4Dic7, C112.26C22, C7⋊2(C2×C32), C8.6(C7⋊C8), C4.3(C7⋊C16), (C2×C16).9D7, C56.75(C2×C4), (C2×C14).2C16, C14.8(C2×C16), (C2×C28).11C8, (C2×C56).24C4, C28.42(C2×C8), (C2×C112).12C2, C22.2(C7⋊C16), C8.21(C2×Dic7), (C2×C8).18Dic7, C4.14(C2×C7⋊C8), C2.2(C2×C7⋊C16), (C2×C4).8(C7⋊C8), SmallGroup(448,55)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C32 |
Generators and relations for C2×C7⋊C32
G = < a,b,c | a2=b7=c32=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C7⋊C32
►Regular action on 448 points
Generators in S448
(1 434)(2 435)(3 436)(4 437)(5 438)(6 439)(7 440)(8 441)(9 442)(10 443)(11 444)(12 445)(13 446)(14 447)(15 448)(16 417)(17 418)(18 419)(19 420)(20 421)(21 422)(22 423)(23 424)(24 425)(25 426)(26 427)(27 428)(28 429)(29 430)(30 431)(31 432)(32 433)(33 167)(34 168)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 191)(58 192)(59 161)(60 162)(61 163)(62 164)(63 165)(64 166)(65 344)(66 345)(67 346)(68 347)(69 348)(70 349)(71 350)(72 351)(73 352)(74 321)(75 322)(76 323)(77 324)(78 325)(79 326)(80 327)(81 328)(82 329)(83 330)(84 331)(85 332)(86 333)(87 334)(88 335)(89 336)(90 337)(91 338)(92 339)(93 340)(94 341)(95 342)(96 343)(97 202)(98 203)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 193)(121 194)(122 195)(123 196)(124 197)(125 198)(126 199)(127 200)(128 201)(129 234)(130 235)(131 236)(132 237)(133 238)(134 239)(135 240)(136 241)(137 242)(138 243)(139 244)(140 245)(141 246)(142 247)(143 248)(144 249)(145 250)(146 251)(147 252)(148 253)(149 254)(150 255)(151 256)(152 225)(153 226)(154 227)(155 228)(156 229)(157 230)(158 231)(159 232)(160 233)(257 395)(258 396)(259 397)(260 398)(261 399)(262 400)(263 401)(264 402)(265 403)(266 404)(267 405)(268 406)(269 407)(270 408)(271 409)(272 410)(273 411)(274 412)(275 413)(276 414)(277 415)(278 416)(279 385)(280 386)(281 387)(282 388)(283 389)(284 390)(285 391)(286 392)(287 393)(288 394)(289 363)(290 364)(291 365)(292 366)(293 367)(294 368)(295 369)(296 370)(297 371)(298 372)(299 373)(300 374)(301 375)(302 376)(303 377)(304 378)(305 379)(306 380)(307 381)(308 382)(309 383)(310 384)(311 353)(312 354)(313 355)(314 356)(315 357)(316 358)(317 359)(318 360)(319 361)(320 362)
(1 191 368 137 344 405 103)(2 104 406 345 138 369 192)(3 161 370 139 346 407 105)(4 106 408 347 140 371 162)(5 163 372 141 348 409 107)(6 108 410 349 142 373 164)(7 165 374 143 350 411 109)(8 110 412 351 144 375 166)(9 167 376 145 352 413 111)(10 112 414 321 146 377 168)(11 169 378 147 322 415 113)(12 114 416 323 148 379 170)(13 171 380 149 324 385 115)(14 116 386 325 150 381 172)(15 173 382 151 326 387 117)(16 118 388 327 152 383 174)(17 175 384 153 328 389 119)(18 120 390 329 154 353 176)(19 177 354 155 330 391 121)(20 122 392 331 156 355 178)(21 179 356 157 332 393 123)(22 124 394 333 158 357 180)(23 181 358 159 334 395 125)(24 126 396 335 160 359 182)(25 183 360 129 336 397 127)(26 128 398 337 130 361 184)(27 185 362 131 338 399 97)(28 98 400 339 132 363 186)(29 187 364 133 340 401 99)(30 100 402 341 134 365 188)(31 189 366 135 342 403 101)(32 102 404 343 136 367 190)(33 302 250 73 275 216 442)(34 443 217 276 74 251 303)(35 304 252 75 277 218 444)(36 445 219 278 76 253 305)(37 306 254 77 279 220 446)(38 447 221 280 78 255 307)(39 308 256 79 281 222 448)(40 417 223 282 80 225 309)(41 310 226 81 283 224 418)(42 419 193 284 82 227 311)(43 312 228 83 285 194 420)(44 421 195 286 84 229 313)(45 314 230 85 287 196 422)(46 423 197 288 86 231 315)(47 316 232 87 257 198 424)(48 425 199 258 88 233 317)(49 318 234 89 259 200 426)(50 427 201 260 90 235 319)(51 320 236 91 261 202 428)(52 429 203 262 92 237 289)(53 290 238 93 263 204 430)(54 431 205 264 94 239 291)(55 292 240 95 265 206 432)(56 433 207 266 96 241 293)(57 294 242 65 267 208 434)(58 435 209 268 66 243 295)(59 296 244 67 269 210 436)(60 437 211 270 68 245 297)(61 298 246 69 271 212 438)(62 439 213 272 70 247 299)(63 300 248 71 273 214 440)(64 441 215 274 72 249 301)
(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)| (1,434)(2,435)(3,436)(4,437)(5,438)(6,439)(7,440)(8,441)(9,442)(10,443)(11,444)(12,445)(13,446)(14,447)(15,448)(16,417)(17,418)(18,419)(19,420)(20,421)(21,422)(22,423)(23,424)(24,425)(25,426)(26,427)(27,428)(28,429)(29,430)(30,431)(31,432)(32,433)(33,167)(34,168)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,191)(58,192)(59,161)(60,162)(61,163)(62,164)(63,165)(64,166)(65,344)(66,345)(67,346)(68,347)(69,348)(70,349)(71,350)(72,351)(73,352)(74,321)(75,322)(76,323)(77,324)(78,325)(79,326)(80,327)(81,328)(82,329)(83,330)(84,331)(85,332)(86,333)(87,334)(88,335)(89,336)(90,337)(91,338)(92,339)(93,340)(94,341)(95,342)(96,343)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,193)(121,194)(122,195)(123,196)(124,197)(125,198)(126,199)(127,200)(128,201)(129,234)(130,235)(131,236)(132,237)(133,238)(134,239)(135,240)(136,241)(137,242)(138,243)(139,244)(140,245)(141,246)(142,247)(143,248)(144,249)(145,250)(146,251)(147,252)(148,253)(149,254)(150,255)(151,256)(152,225)(153,226)(154,227)(155,228)(156,229)(157,230)(158,231)(159,232)(160,233)(257,395)(258,396)(259,397)(260,398)(261,399)(262,400)(263,401)(264,402)(265,403)(266,404)(267,405)(268,406)(269,407)(270,408)(271,409)(272,410)(273,411)(274,412)(275,413)(276,414)(277,415)(278,416)(279,385)(280,386)(281,387)(282,388)(283,389)(284,390)(285,391)(286,392)(287,393)(288,394)(289,363)(290,364)(291,365)(292,366)(293,367)(294,368)(295,369)(296,370)(297,371)(298,372)(299,373)(300,374)(301,375)(302,376)(303,377)(304,378)(305,379)(306,380)(307,381)(308,382)(309,383)(310,384)(311,353)(312,354)(313,355)(314,356)(315,357)(316,358)(317,359)(318,360)(319,361)(320,362), (1,191,368,137,344,405,103)(2,104,406,345,138,369,192)(3,161,370,139,346,407,105)(4,106,408,347,140,371,162)(5,163,372,141,348,409,107)(6,108,410,349,142,373,164)(7,165,374,143,350,411,109)(8,110,412,351,144,375,166)(9,167,376,145,352,413,111)(10,112,414,321,146,377,168)(11,169,378,147,322,415,113)(12,114,416,323,148,379,170)(13,171,380,149,324,385,115)(14,116,386,325,150,381,172)(15,173,382,151,326,387,117)(16,118,388,327,152,383,174)(17,175,384,153,328,389,119)(18,120,390,329,154,353,176)(19,177,354,155,330,391,121)(20,122,392,331,156,355,178)(21,179,356,157,332,393,123)(22,124,394,333,158,357,180)(23,181,358,159,334,395,125)(24,126,396,335,160,359,182)(25,183,360,129,336,397,127)(26,128,398,337,130,361,184)(27,185,362,131,338,399,97)(28,98,400,339,132,363,186)(29,187,364,133,340,401,99)(30,100,402,341,134,365,188)(31,189,366,135,342,403,101)(32,102,404,343,136,367,190)(33,302,250,73,275,216,442)(34,443,217,276,74,251,303)(35,304,252,75,277,218,444)(36,445,219,278,76,253,305)(37,306,254,77,279,220,446)(38,447,221,280,78,255,307)(39,308,256,79,281,222,448)(40,417,223,282,80,225,309)(41,310,226,81,283,224,418)(42,419,193,284,82,227,311)(43,312,228,83,285,194,420)(44,421,195,286,84,229,313)(45,314,230,85,287,196,422)(46,423,197,288,86,231,315)(47,316,232,87,257,198,424)(48,425,199,258,88,233,317)(49,318,234,89,259,200,426)(50,427,201,260,90,235,319)(51,320,236,91,261,202,428)(52,429,203,262,92,237,289)(53,290,238,93,263,204,430)(54,431,205,264,94,239,291)(55,292,240,95,265,206,432)(56,433,207,266,96,241,293)(57,294,242,65,267,208,434)(58,435,209,268,66,243,295)(59,296,244,67,269,210,436)(60,437,211,270,68,245,297)(61,298,246,69,271,212,438)(62,439,213,272,70,247,299)(63,300,248,71,273,214,440)(64,441,215,274,72,249,301), (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( (1,434)(2,435)(3,436)(4,437)(5,438)(6,439)(7,440)(8,441)(9,442)(10,443)(11,444)(12,445)(13,446)(14,447)(15,448)(16,417)(17,418)(18,419)(19,420)(20,421)(21,422)(22,423)(23,424)(24,425)(25,426)(26,427)(27,428)(28,429)(29,430)(30,431)(31,432)(32,433)(33,167)(34,168)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,191)(58,192)(59,161)(60,162)(61,163)(62,164)(63,165)(64,166)(65,344)(66,345)(67,346)(68,347)(69,348)(70,349)(71,350)(72,351)(73,352)(74,321)(75,322)(76,323)(77,324)(78,325)(79,326)(80,327)(81,328)(82,329)(83,330)(84,331)(85,332)(86,333)(87,334)(88,335)(89,336)(90,337)(91,338)(92,339)(93,340)(94,341)(95,342)(96,343)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,193)(121,194)(122,195)(123,196)(124,197)(125,198)(126,199)(127,200)(128,201)(129,234)(130,235)(131,236)(132,237)(133,238)(134,239)(135,240)(136,241)(137,242)(138,243)(139,244)(140,245)(141,246)(142,247)(143,248)(144,249)(145,250)(146,251)(147,252)(148,253)(149,254)(150,255)(151,256)(152,225)(153,226)(154,227)(155,228)(156,229)(157,230)(158,231)(159,232)(160,233)(257,395)(258,396)(259,397)(260,398)(261,399)(262,400)(263,401)(264,402)(265,403)(266,404)(267,405)(268,406)(269,407)(270,408)(271,409)(272,410)(273,411)(274,412)(275,413)(276,414)(277,415)(278,416)(279,385)(280,386)(281,387)(282,388)(283,389)(284,390)(285,391)(286,392)(287,393)(288,394)(289,363)(290,364)(291,365)(292,366)(293,367)(294,368)(295,369)(296,370)(297,371)(298,372)(299,373)(300,374)(301,375)(302,376)(303,377)(304,378)(305,379)(306,380)(307,381)(308,382)(309,383)(310,384)(311,353)(312,354)(313,355)(314,356)(315,357)(316,358)(317,359)(318,360)(319,361)(320,362), (1,191,368,137,344,405,103)(2,104,406,345,138,369,192)(3,161,370,139,346,407,105)(4,106,408,347,140,371,162)(5,163,372,141,348,409,107)(6,108,410,349,142,373,164)(7,165,374,143,350,411,109)(8,110,412,351,144,375,166)(9,167,376,145,352,413,111)(10,112,414,321,146,377,168)(11,169,378,147,322,415,113)(12,114,416,323,148,379,170)(13,171,380,149,324,385,115)(14,116,386,325,150,381,172)(15,173,382,151,326,387,117)(16,118,388,327,152,383,174)(17,175,384,153,328,389,119)(18,120,390,329,154,353,176)(19,177,354,155,330,391,121)(20,122,392,331,156,355,178)(21,179,356,157,332,393,123)(22,124,394,333,158,357,180)(23,181,358,159,334,395,125)(24,126,396,335,160,359,182)(25,183,360,129,336,397,127)(26,128,398,337,130,361,184)(27,185,362,131,338,399,97)(28,98,400,339,132,363,186)(29,187,364,133,340,401,99)(30,100,402,341,134,365,188)(31,189,366,135,342,403,101)(32,102,404,343,136,367,190)(33,302,250,73,275,216,442)(34,443,217,276,74,251,303)(35,304,252,75,277,218,444)(36,445,219,278,76,253,305)(37,306,254,77,279,220,446)(38,447,221,280,78,255,307)(39,308,256,79,281,222,448)(40,417,223,282,80,225,309)(41,310,226,81,283,224,418)(42,419,193,284,82,227,311)(43,312,228,83,285,194,420)(44,421,195,286,84,229,313)(45,314,230,85,287,196,422)(46,423,197,288,86,231,315)(47,316,232,87,257,198,424)(48,425,199,258,88,233,317)(49,318,234,89,259,200,426)(50,427,201,260,90,235,319)(51,320,236,91,261,202,428)(52,429,203,262,92,237,289)(53,290,238,93,263,204,430)(54,431,205,264,94,239,291)(55,292,240,95,265,206,432)(56,433,207,266,96,241,293)(57,294,242,65,267,208,434)(58,435,209,268,66,243,295)(59,296,244,67,269,210,436)(60,437,211,270,68,245,297)(61,298,246,69,271,212,438)(62,439,213,272,70,247,299)(63,300,248,71,273,214,440)(64,441,215,274,72,249,301), (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([[(1,434),(2,435),(3,436),(4,437),(5,438),(6,439),(7,440),(8,441),(9,442),(10,443),(11,444),(12,445),(13,446),(14,447),(15,448),(16,417),(17,418),(18,419),(19,420),(20,421),(21,422),(22,423),(23,424),(24,425),(25,426),(26,427),(27,428),(28,429),(29,430),(30,431),(31,432),(32,433),(33,167),(34,168),(35,169),(36,170),(37,171),(38,172),(39,173),(40,174),(41,175),(42,176),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,183),(50,184),(51,185),(52,186),(53,187),(54,188),(55,189),(56,190),(57,191),(58,192),(59,161),(60,162),(61,163),(62,164),(63,165),(64,166),(65,344),(66,345),(67,346),(68,347),(69,348),(70,349),(71,350),(72,351),(73,352),(74,321),(75,322),(76,323),(77,324),(78,325),(79,326),(80,327),(81,328),(82,329),(83,330),(84,331),(85,332),(86,333),(87,334),(88,335),(89,336),(90,337),(91,338),(92,339),(93,340),(94,341),(95,342),(96,343),(97,202),(98,203),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,193),(121,194),(122,195),(123,196),(124,197),(125,198),(126,199),(127,200),(128,201),(129,234),(130,235),(131,236),(132,237),(133,238),(134,239),(135,240),(136,241),(137,242),(138,243),(139,244),(140,245),(141,246),(142,247),(143,248),(144,249),(145,250),(146,251),(147,252),(148,253),(149,254),(150,255),(151,256),(152,225),(153,226),(154,227),(155,228),(156,229),(157,230),(158,231),(159,232),(160,233),(257,395),(258,396),(259,397),(260,398),(261,399),(262,400),(263,401),(264,402),(265,403),(266,404),(267,405),(268,406),(269,407),(270,408),(271,409),(272,410),(273,411),(274,412),(275,413),(276,414),(277,415),(278,416),(279,385),(280,386),(281,387),(282,388),(283,389),(284,390),(285,391),(286,392),(287,393),(288,394),(289,363),(290,364),(291,365),(292,366),(293,367),(294,368),(295,369),(296,370),(297,371),(298,372),(299,373),(300,374),(301,375),(302,376),(303,377),(304,378),(305,379),(306,380),(307,381),(308,382),(309,383),(310,384),(311,353),(312,354),(313,355),(314,356),(315,357),(316,358),(317,359),(318,360),(319,361),(320,362)], [(1,191,368,137,344,405,103),(2,104,406,345,138,369,192),(3,161,370,139,346,407,105),(4,106,408,347,140,371,162),(5,163,372,141,348,409,107),(6,108,410,349,142,373,164),(7,165,374,143,350,411,109),(8,110,412,351,144,375,166),(9,167,376,145,352,413,111),(10,112,414,321,146,377,168),(11,169,378,147,322,415,113),(12,114,416,323,148,379,170),(13,171,380,149,324,385,115),(14,116,386,325,150,381,172),(15,173,382,151,326,387,117),(16,118,388,327,152,383,174),(17,175,384,153,328,389,119),(18,120,390,329,154,353,176),(19,177,354,155,330,391,121),(20,122,392,331,156,355,178),(21,179,356,157,332,393,123),(22,124,394,333,158,357,180),(23,181,358,159,334,395,125),(24,126,396,335,160,359,182),(25,183,360,129,336,397,127),(26,128,398,337,130,361,184),(27,185,362,131,338,399,97),(28,98,400,339,132,363,186),(29,187,364,133,340,401,99),(30,100,402,341,134,365,188),(31,189,366,135,342,403,101),(32,102,404,343,136,367,190),(33,302,250,73,275,216,442),(34,443,217,276,74,251,303),(35,304,252,75,277,218,444),(36,445,219,278,76,253,305),(37,306,254,77,279,220,446),(38,447,221,280,78,255,307),(39,308,256,79,281,222,448),(40,417,223,282,80,225,309),(41,310,226,81,283,224,418),(42,419,193,284,82,227,311),(43,312,228,83,285,194,420),(44,421,195,286,84,229,313),(45,314,230,85,287,196,422),(46,423,197,288,86,231,315),(47,316,232,87,257,198,424),(48,425,199,258,88,233,317),(49,318,234,89,259,200,426),(50,427,201,260,90,235,319),(51,320,236,91,261,202,428),(52,429,203,262,92,237,289),(53,290,238,93,263,204,430),(54,431,205,264,94,239,291),(55,292,240,95,265,206,432),(56,433,207,266,96,241,293),(57,294,242,65,267,208,434),(58,435,209,268,66,243,295),(59,296,244,67,269,210,436),(60,437,211,270,68,245,297),(61,298,246,69,271,212,438),(62,439,213,272,70,247,299),(63,300,248,71,273,214,440),(64,441,215,274,72,249,301)], [(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 | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 16A | ··· | 16P | 28A | ··· | 28L | 32A | ··· | 32AF | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | D7 | Dic7 | D14 | Dic7 | C7⋊C8 | C7⋊C8 | C7⋊C16 | C7⋊C16 | C7⋊C32 |
| kernel | C2×C7⋊C32 | C7⋊C32 | C2×C112 | C112 | C2×C56 | C56 | C2×C28 | C28 | C2×C14 | C14 | C2×C16 | C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 3 | 3 | 3 | 3 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C2×C7⋊C32 ►in GL3(𝔽449) generated by
| 1 | 0 | 0 |
| 0 | 448 | 0 |
| 0 | 0 | 448 |
| 1 | 0 | 0 |
| 0 | 44 | 448 |
| 0 | 45 | 448 |
| 404 | 0 | 0 |
| 0 | 404 | 21 |
| 0 | 267 | 45 |
G:=sub<GL(3,GF(449))| [1,0,0,0,448,0,0,0,448],[1,0,0,0,44,45,0,448,448],[404,0,0,0,404,267,0,21,45] >;
C2×C7⋊C32 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_{32} % in TeX
G:=Group("C2xC7:C32"); // GroupNames label
G:=SmallGroup(448,55);
// by ID
G=gap.SmallGroup(448,55);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,58,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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