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G = C3×C6×C24order 432 = 24·33

Abelian group of type [3,6,24]

direct product, abelian, monomial

Aliases: C3×C6×C24, SmallGroup(432,515)

Series: Derived Chief Lower central Upper central

C1 — C3×C6×C24
C1C2C4C12C3×C12C32×C12C32×C24 — C3×C6×C24
C1 — C3×C6×C24
C1 — C3×C6×C24

Generators and relations for C3×C6×C24
 G = < a,b,c | a3=b6=c24=1, ab=ba, ac=ca, bc=cb >

Subgroups: 308, all normal (14 characteristic)
C1, C2, C2 [×2], C3 [×13], C4 [×2], C22, C6 [×39], C8 [×2], C2×C4, C32 [×13], C12 [×26], C2×C6 [×13], C2×C8, C3×C6 [×39], C24 [×26], C2×C12 [×13], C33, C3×C12 [×26], C62 [×13], C2×C24 [×13], C32×C6, C32×C6 [×2], C3×C24 [×26], C6×C12 [×13], C32×C12 [×2], C3×C62, C6×C24 [×13], C32×C24 [×2], C3×C6×C12, C3×C6×C24
Quotients: C1, C2 [×3], C3 [×13], C4 [×2], C22, C6 [×39], C8 [×2], C2×C4, C32 [×13], C12 [×26], C2×C6 [×13], C2×C8, C3×C6 [×39], C24 [×26], C2×C12 [×13], C33, C3×C12 [×26], C62 [×13], C2×C24 [×13], C32×C6 [×3], C3×C24 [×26], C6×C12 [×13], C32×C12 [×2], C3×C62, C6×C24 [×13], C32×C24 [×2], C3×C6×C12, C3×C6×C24

Smallest permutation representation of C3×C6×C24
Regular action on 432 points
Generators in S432
(1 382 39)(2 383 40)(3 384 41)(4 361 42)(5 362 43)(6 363 44)(7 364 45)(8 365 46)(9 366 47)(10 367 48)(11 368 25)(12 369 26)(13 370 27)(14 371 28)(15 372 29)(16 373 30)(17 374 31)(18 375 32)(19 376 33)(20 377 34)(21 378 35)(22 379 36)(23 380 37)(24 381 38)(49 317 183)(50 318 184)(51 319 185)(52 320 186)(53 321 187)(54 322 188)(55 323 189)(56 324 190)(57 325 191)(58 326 192)(59 327 169)(60 328 170)(61 329 171)(62 330 172)(63 331 173)(64 332 174)(65 333 175)(66 334 176)(67 335 177)(68 336 178)(69 313 179)(70 314 180)(71 315 181)(72 316 182)(73 123 268)(74 124 269)(75 125 270)(76 126 271)(77 127 272)(78 128 273)(79 129 274)(80 130 275)(81 131 276)(82 132 277)(83 133 278)(84 134 279)(85 135 280)(86 136 281)(87 137 282)(88 138 283)(89 139 284)(90 140 285)(91 141 286)(92 142 287)(93 143 288)(94 144 265)(95 121 266)(96 122 267)(97 412 257)(98 413 258)(99 414 259)(100 415 260)(101 416 261)(102 417 262)(103 418 263)(104 419 264)(105 420 241)(106 421 242)(107 422 243)(108 423 244)(109 424 245)(110 425 246)(111 426 247)(112 427 248)(113 428 249)(114 429 250)(115 430 251)(116 431 252)(117 432 253)(118 409 254)(119 410 255)(120 411 256)(145 204 217)(146 205 218)(147 206 219)(148 207 220)(149 208 221)(150 209 222)(151 210 223)(152 211 224)(153 212 225)(154 213 226)(155 214 227)(156 215 228)(157 216 229)(158 193 230)(159 194 231)(160 195 232)(161 196 233)(162 197 234)(163 198 235)(164 199 236)(165 200 237)(166 201 238)(167 202 239)(168 203 240)(289 393 359)(290 394 360)(291 395 337)(292 396 338)(293 397 339)(294 398 340)(295 399 341)(296 400 342)(297 401 343)(298 402 344)(299 403 345)(300 404 346)(301 405 347)(302 406 348)(303 407 349)(304 408 350)(305 385 351)(306 386 352)(307 387 353)(308 388 354)(309 389 355)(310 390 356)(311 391 357)(312 392 358)
(1 141 312 161 68 111)(2 142 289 162 69 112)(3 143 290 163 70 113)(4 144 291 164 71 114)(5 121 292 165 72 115)(6 122 293 166 49 116)(7 123 294 167 50 117)(8 124 295 168 51 118)(9 125 296 145 52 119)(10 126 297 146 53 120)(11 127 298 147 54 97)(12 128 299 148 55 98)(13 129 300 149 56 99)(14 130 301 150 57 100)(15 131 302 151 58 101)(16 132 303 152 59 102)(17 133 304 153 60 103)(18 134 305 154 61 104)(19 135 306 155 62 105)(20 136 307 156 63 106)(21 137 308 157 64 107)(22 138 309 158 65 108)(23 139 310 159 66 109)(24 140 311 160 67 110)(25 77 344 219 188 257)(26 78 345 220 189 258)(27 79 346 221 190 259)(28 80 347 222 191 260)(29 81 348 223 192 261)(30 82 349 224 169 262)(31 83 350 225 170 263)(32 84 351 226 171 264)(33 85 352 227 172 241)(34 86 353 228 173 242)(35 87 354 229 174 243)(36 88 355 230 175 244)(37 89 356 231 176 245)(38 90 357 232 177 246)(39 91 358 233 178 247)(40 92 359 234 179 248)(41 93 360 235 180 249)(42 94 337 236 181 250)(43 95 338 237 182 251)(44 96 339 238 183 252)(45 73 340 239 184 253)(46 74 341 240 185 254)(47 75 342 217 186 255)(48 76 343 218 187 256)(193 333 423 379 283 389)(194 334 424 380 284 390)(195 335 425 381 285 391)(196 336 426 382 286 392)(197 313 427 383 287 393)(198 314 428 384 288 394)(199 315 429 361 265 395)(200 316 430 362 266 396)(201 317 431 363 267 397)(202 318 432 364 268 398)(203 319 409 365 269 399)(204 320 410 366 270 400)(205 321 411 367 271 401)(206 322 412 368 272 402)(207 323 413 369 273 403)(208 324 414 370 274 404)(209 325 415 371 275 405)(210 326 416 372 276 406)(211 327 417 373 277 407)(212 328 418 374 278 408)(213 329 419 375 279 385)(214 330 420 376 280 386)(215 331 421 377 281 387)(216 332 422 378 282 388)
(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)

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

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

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

432 conjugacy classes

class 1 2A2B2C3A···3Z4A4B4C4D6A···6BZ8A···8H12A···12CZ24A···24GZ
order12223···344446···68···812···1224···24
size11111···111111···11···11···11···1

432 irreducible representations

dim111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C12C12C24
kernelC3×C6×C24C32×C24C3×C6×C12C6×C24C32×C12C3×C62C3×C24C6×C12C32×C6C3×C12C62C3×C6
# reps1212622522685252208

Matrix representation of C3×C6×C24 in GL3(𝔽73) generated by

6400
080
001
,
800
010
009
,
4600
0660
007
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,1],[8,0,0,0,1,0,0,0,9],[46,0,0,0,66,0,0,0,7] >;

C3×C6×C24 in GAP, Magma, Sage, TeX

C_3\times C_6\times C_{24}
% in TeX

G:=Group("C3xC6xC24");
// GroupNames label

G:=SmallGroup(432,515);
// by ID

G=gap.SmallGroup(432,515);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,-2,756,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^6=c^24=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽