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## G = C2×C63order 432 = 24·33

### Abelian group of type [2,6,6,6]

Aliases: C2×C63, SmallGroup(432,775)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C63
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — C63 — C2×C63
 Lower central C1 — C2×C63
 Upper central C1 — C2×C63

Generators and relations for C2×C63
G = < a,b,c,d | a2=b6=c6=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 1876, all normal (4 characteristic)
C1, C2, C3, C22, C6, C23, C32, C2×C6, C24, C3×C6, C22×C6, C33, C62, C23×C6, C32×C6, C2×C62, C3×C62, C22×C62, C63, C2×C63
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C24, C3×C6, C22×C6, C33, C62, C23×C6, C32×C6, C2×C62, C3×C62, C22×C62, C63, C2×C63

Smallest permutation representation of C2×C63
Regular action on 432 points
Generators in S432
(1 347)(2 348)(3 343)(4 344)(5 345)(6 346)(7 336)(8 331)(9 332)(10 333)(11 334)(12 335)(13 220)(14 221)(15 222)(16 217)(17 218)(18 219)(19 231)(20 232)(21 233)(22 234)(23 229)(24 230)(25 241)(26 242)(27 243)(28 244)(29 245)(30 246)(31 247)(32 248)(33 249)(34 250)(35 251)(36 252)(37 253)(38 254)(39 255)(40 256)(41 257)(42 258)(43 259)(44 260)(45 261)(46 262)(47 263)(48 264)(49 265)(50 266)(51 267)(52 268)(53 269)(54 270)(55 271)(56 272)(57 273)(58 274)(59 275)(60 276)(61 277)(62 278)(63 279)(64 280)(65 281)(66 282)(67 283)(68 284)(69 285)(70 286)(71 287)(72 288)(73 289)(74 290)(75 291)(76 292)(77 293)(78 294)(79 295)(80 296)(81 297)(82 298)(83 299)(84 300)(85 301)(86 302)(87 303)(88 304)(89 305)(90 306)(91 307)(92 308)(93 309)(94 310)(95 311)(96 312)(97 313)(98 314)(99 315)(100 316)(101 317)(102 318)(103 319)(104 320)(105 321)(106 322)(107 323)(108 324)(109 330)(110 325)(111 326)(112 327)(113 328)(114 329)(115 342)(116 337)(117 338)(118 339)(119 340)(120 341)(121 239)(122 240)(123 235)(124 236)(125 237)(126 238)(127 225)(128 226)(129 227)(130 228)(131 223)(132 224)(133 349)(134 350)(135 351)(136 352)(137 353)(138 354)(139 355)(140 356)(141 357)(142 358)(143 359)(144 360)(145 361)(146 362)(147 363)(148 364)(149 365)(150 366)(151 367)(152 368)(153 369)(154 370)(155 371)(156 372)(157 373)(158 374)(159 375)(160 376)(161 377)(162 378)(163 379)(164 380)(165 381)(166 382)(167 383)(168 384)(169 385)(170 386)(171 387)(172 388)(173 389)(174 390)(175 391)(176 392)(177 393)(178 394)(179 395)(180 396)(181 397)(182 398)(183 399)(184 400)(185 401)(186 402)(187 403)(188 404)(189 405)(190 406)(191 407)(192 408)(193 409)(194 410)(195 411)(196 412)(197 413)(198 414)(199 415)(200 416)(201 417)(202 418)(203 419)(204 420)(205 421)(206 422)(207 423)(208 424)(209 425)(210 426)(211 427)(212 428)(213 429)(214 430)(215 431)(216 432)
(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)
(1 51 38 35 18 161)(2 52 39 36 13 162)(3 53 40 31 14 157)(4 54 41 32 15 158)(5 49 42 33 16 159)(6 50 37 34 17 160)(7 405 282 420 130 351)(8 406 277 415 131 352)(9 407 278 416 132 353)(10 408 279 417 127 354)(11 403 280 418 128 349)(12 404 281 419 129 350)(19 410 111 300 120 317)(20 411 112 295 115 318)(21 412 113 296 116 313)(22 413 114 297 117 314)(23 414 109 298 118 315)(24 409 110 299 119 316)(25 176 48 70 143 165)(26 177 43 71 144 166)(27 178 44 72 139 167)(28 179 45 67 140 168)(29 180 46 68 141 163)(30 175 47 69 142 164)(55 213 181 103 96 85)(56 214 182 104 91 86)(57 215 183 105 92 87)(58 216 184 106 93 88)(59 211 185 107 94 89)(60 212 186 108 95 90)(61 199 223 136 331 190)(62 200 224 137 332 191)(63 201 225 138 333 192)(64 202 226 133 334 187)(65 203 227 134 335 188)(66 204 228 135 336 189)(73 154 150 208 123 174)(74 155 145 209 124 169)(75 156 146 210 125 170)(76 151 147 205 126 171)(77 152 148 206 121 172)(78 153 149 207 122 173)(79 342 102 232 195 327)(80 337 97 233 196 328)(81 338 98 234 197 329)(82 339 99 229 198 330)(83 340 100 230 193 325)(84 341 101 231 194 326)(217 375 345 265 258 249)(218 376 346 266 253 250)(219 377 347 267 254 251)(220 378 348 268 255 252)(221 373 343 269 256 247)(222 374 344 270 257 248)(235 390 289 370 366 424)(236 385 290 371 361 425)(237 386 291 372 362 426)(238 387 292 367 363 421)(239 388 293 368 364 422)(240 389 294 369 365 423)(241 392 264 286 359 381)(242 393 259 287 360 382)(243 394 260 288 355 383)(244 395 261 283 356 384)(245 396 262 284 357 379)(246 391 263 285 358 380)(271 429 397 319 312 301)(272 430 398 320 307 302)(273 431 399 321 308 303)(274 432 400 322 309 304)(275 427 401 323 310 305)(276 428 402 324 311 306)
(1 191 59 139 148 100)(2 192 60 140 149 101)(3 187 55 141 150 102)(4 188 56 142 145 97)(5 189 57 143 146 98)(6 190 58 144 147 99)(7 303 286 372 117 375)(8 304 287 367 118 376)(9 305 288 368 119 377)(10 306 283 369 120 378)(11 301 284 370 115 373)(12 302 285 371 116 374)(13 138 95 45 78 84)(14 133 96 46 73 79)(15 134 91 47 74 80)(16 135 92 48 75 81)(17 136 93 43 76 82)(18 137 94 44 77 83)(19 268 279 428 384 423)(20 269 280 429 379 424)(21 270 281 430 380 425)(22 265 282 431 381 426)(23 266 277 432 382 421)(24 267 278 427 383 422)(25 125 197 42 204 183)(26 126 198 37 199 184)(27 121 193 38 200 185)(28 122 194 39 201 186)(29 123 195 40 202 181)(30 124 196 41 203 182)(31 226 103 180 174 327)(32 227 104 175 169 328)(33 228 105 176 170 329)(34 223 106 177 171 330)(35 224 107 178 172 325)(36 225 108 179 173 326)(49 66 215 165 210 234)(50 61 216 166 205 229)(51 62 211 167 206 230)(52 63 212 168 207 231)(53 64 213 163 208 232)(54 65 214 164 209 233)(67 153 341 162 333 90)(68 154 342 157 334 85)(69 155 337 158 335 86)(70 156 338 159 336 87)(71 151 339 160 331 88)(72 152 340 161 332 89)(109 250 131 322 393 387)(110 251 132 323 394 388)(111 252 127 324 395 389)(112 247 128 319 396 390)(113 248 129 320 391 385)(114 249 130 321 392 386)(217 351 308 264 291 297)(218 352 309 259 292 298)(219 353 310 260 293 299)(220 354 311 261 294 300)(221 349 312 262 289 295)(222 350 307 263 290 296)(235 411 256 418 397 245)(236 412 257 419 398 246)(237 413 258 420 399 241)(238 414 253 415 400 242)(239 409 254 416 401 243)(240 410 255 417 402 244)(271 357 366 318 343 403)(272 358 361 313 344 404)(273 359 362 314 345 405)(274 360 363 315 346 406)(275 355 364 316 347 407)(276 356 365 317 348 408)

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

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

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

432 conjugacy classes

 class 1 2A ··· 2O 3A ··· 3Z 6A ··· 6NZ order 1 2 ··· 2 3 ··· 3 6 ··· 6 size 1 1 ··· 1 1 ··· 1 1 ··· 1

432 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C2×C63 C63 C22×C62 C2×C62 # reps 1 15 26 390

Matrix representation of C2×C63 in GL4(𝔽7) generated by

 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 1
,
 6 0 0 0 0 6 0 0 0 0 4 0 0 0 0 4
,
 6 0 0 0 0 5 0 0 0 0 2 0 0 0 0 5
,
 3 0 0 0 0 2 0 0 0 0 6 0 0 0 0 5
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,1],[6,0,0,0,0,6,0,0,0,0,4,0,0,0,0,4],[6,0,0,0,0,5,0,0,0,0,2,0,0,0,0,5],[3,0,0,0,0,2,0,0,0,0,6,0,0,0,0,5] >;

C2×C63 in GAP, Magma, Sage, TeX

C_2\times C_6^3
% in TeX

G:=Group("C2xC6^3");
// GroupNames label

G:=SmallGroup(432,775);
// by ID

G=gap.SmallGroup(432,775);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽