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## G = C3×C6×C24order 432 = 24·33

### Abelian group of type [3,6,24]

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6×C24
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C32×C12 — C32×C24 — C3×C6×C24
 Lower central C1 — C3×C6×C24
 Upper central 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, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, C33, C3×C12, C62, C2×C24, C32×C6, C32×C6, C3×C24, C6×C12, C32×C12, C3×C62, C6×C24, C32×C24, C3×C6×C12, C3×C6×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, C33, C3×C12, C62, C2×C24, C32×C6, C3×C24, C6×C12, C32×C12, C3×C62, C6×C24, C32×C24, C3×C6×C12, C3×C6×C24

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

432 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 kernel C3×C6×C24 C32×C24 C3×C6×C12 C6×C24 C32×C12 C3×C62 C3×C24 C6×C12 C32×C6 C3×C12 C62 C3×C6 # reps 1 2 1 26 2 2 52 26 8 52 52 208

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

 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
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

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