metabelian, supersoluble, monomial
Aliases: C72.3S3, C24.3D9, C6.7D36, C3⋊1Dic36, C9⋊1Dic12, C18.7D12, C36.56D6, C12.56D18, C32.4Dic12, C8.(C9⋊S3), (C3×C9)⋊6Q16, (C3×C72).3C2, C24.1(C3⋊S3), (C3×C24).11S3, (C3×C6).54D12, (C3×C18).29D4, C3.(C32⋊5Q16), (C3×C12).191D6, C6.1(C12⋊S3), C2.3(C36⋊S3), C12.D9.2C2, (C3×C36).58C22, C4.8(C2×C9⋊S3), C12.59(C2×C3⋊S3), SmallGroup(432,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.D9
G = < a,b,c | a24=b9=1, c2=a12, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 572 in 90 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C32, Dic3, C12, C12, Q16, C18, C3×C6, C24, C24, Dic6, C3×C9, Dic9, C36, C3⋊Dic3, C3×C12, Dic12, C3×C18, C72, Dic18, C3×C24, C32⋊4Q8, C9⋊Dic3, C3×C36, Dic36, C32⋊5Q16, C3×C72, C12.D9, C24.D9
Quotients: C1, C2, C22, S3, D4, D6, Q16, D9, C3⋊S3, D12, D18, C2×C3⋊S3, Dic12, C9⋊S3, D36, C12⋊S3, C2×C9⋊S3, Dic36, C32⋊5Q16, C36⋊S3, C24.D9
►Regular action on 432 points
Generators in S432
(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 73 373 165 119 239 432 282 170)(2 74 374 166 120 240 409 283 171)(3 75 375 167 97 217 410 284 172)(4 76 376 168 98 218 411 285 173)(5 77 377 145 99 219 412 286 174)(6 78 378 146 100 220 413 287 175)(7 79 379 147 101 221 414 288 176)(8 80 380 148 102 222 415 265 177)(9 81 381 149 103 223 416 266 178)(10 82 382 150 104 224 417 267 179)(11 83 383 151 105 225 418 268 180)(12 84 384 152 106 226 419 269 181)(13 85 361 153 107 227 420 270 182)(14 86 362 154 108 228 421 271 183)(15 87 363 155 109 229 422 272 184)(16 88 364 156 110 230 423 273 185)(17 89 365 157 111 231 424 274 186)(18 90 366 158 112 232 425 275 187)(19 91 367 159 113 233 426 276 188)(20 92 368 160 114 234 427 277 189)(21 93 369 161 115 235 428 278 190)(22 94 370 162 116 236 429 279 191)(23 95 371 163 117 237 430 280 192)(24 96 372 164 118 238 431 281 169)(25 311 350 202 324 57 392 144 254)(26 312 351 203 325 58 393 121 255)(27 289 352 204 326 59 394 122 256)(28 290 353 205 327 60 395 123 257)(29 291 354 206 328 61 396 124 258)(30 292 355 207 329 62 397 125 259)(31 293 356 208 330 63 398 126 260)(32 294 357 209 331 64 399 127 261)(33 295 358 210 332 65 400 128 262)(34 296 359 211 333 66 401 129 263)(35 297 360 212 334 67 402 130 264)(36 298 337 213 335 68 403 131 241)(37 299 338 214 336 69 404 132 242)(38 300 339 215 313 70 405 133 243)(39 301 340 216 314 71 406 134 244)(40 302 341 193 315 72 407 135 245)(41 303 342 194 316 49 408 136 246)(42 304 343 195 317 50 385 137 247)(43 305 344 196 318 51 386 138 248)(44 306 345 197 319 52 387 139 249)(45 307 346 198 320 53 388 140 250)(46 308 347 199 321 54 389 141 251)(47 309 348 200 322 55 390 142 252)(48 310 349 201 323 56 391 143 253)
(1 289 13 301)(2 312 14 300)(3 311 15 299)(4 310 16 298)(5 309 17 297)(6 308 18 296)(7 307 19 295)(8 306 20 294)(9 305 21 293)(10 304 22 292)(11 303 23 291)(12 302 24 290)(25 87 37 75)(26 86 38 74)(27 85 39 73)(28 84 40 96)(29 83 41 95)(30 82 42 94)(31 81 43 93)(32 80 44 92)(33 79 45 91)(34 78 46 90)(35 77 47 89)(36 76 48 88)(49 237 61 225)(50 236 62 224)(51 235 63 223)(52 234 64 222)(53 233 65 221)(54 232 66 220)(55 231 67 219)(56 230 68 218)(57 229 69 217)(58 228 70 240)(59 227 71 239)(60 226 72 238)(97 392 109 404)(98 391 110 403)(99 390 111 402)(100 389 112 401)(101 388 113 400)(102 387 114 399)(103 386 115 398)(104 385 116 397)(105 408 117 396)(106 407 118 395)(107 406 119 394)(108 405 120 393)(121 154 133 166)(122 153 134 165)(123 152 135 164)(124 151 136 163)(125 150 137 162)(126 149 138 161)(127 148 139 160)(128 147 140 159)(129 146 141 158)(130 145 142 157)(131 168 143 156)(132 167 144 155)(169 353 181 341)(170 352 182 340)(171 351 183 339)(172 350 184 338)(173 349 185 337)(174 348 186 360)(175 347 187 359)(176 346 188 358)(177 345 189 357)(178 344 190 356)(179 343 191 355)(180 342 192 354)(193 281 205 269)(194 280 206 268)(195 279 207 267)(196 278 208 266)(197 277 209 265)(198 276 210 288)(199 275 211 287)(200 274 212 286)(201 273 213 285)(202 272 214 284)(203 271 215 283)(204 270 216 282)(241 376 253 364)(242 375 254 363)(243 374 255 362)(244 373 256 361)(245 372 257 384)(246 371 258 383)(247 370 259 382)(248 369 260 381)(249 368 261 380)(250 367 262 379)(251 366 263 378)(252 365 264 377)(313 409 325 421)(314 432 326 420)(315 431 327 419)(316 430 328 418)(317 429 329 417)(318 428 330 416)(319 427 331 415)(320 426 332 414)(321 425 333 413)(322 424 334 412)(323 423 335 411)(324 422 336 410)
G:=sub<Sym(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), (1,73,373,165,119,239,432,282,170)(2,74,374,166,120,240,409,283,171)(3,75,375,167,97,217,410,284,172)(4,76,376,168,98,218,411,285,173)(5,77,377,145,99,219,412,286,174)(6,78,378,146,100,220,413,287,175)(7,79,379,147,101,221,414,288,176)(8,80,380,148,102,222,415,265,177)(9,81,381,149,103,223,416,266,178)(10,82,382,150,104,224,417,267,179)(11,83,383,151,105,225,418,268,180)(12,84,384,152,106,226,419,269,181)(13,85,361,153,107,227,420,270,182)(14,86,362,154,108,228,421,271,183)(15,87,363,155,109,229,422,272,184)(16,88,364,156,110,230,423,273,185)(17,89,365,157,111,231,424,274,186)(18,90,366,158,112,232,425,275,187)(19,91,367,159,113,233,426,276,188)(20,92,368,160,114,234,427,277,189)(21,93,369,161,115,235,428,278,190)(22,94,370,162,116,236,429,279,191)(23,95,371,163,117,237,430,280,192)(24,96,372,164,118,238,431,281,169)(25,311,350,202,324,57,392,144,254)(26,312,351,203,325,58,393,121,255)(27,289,352,204,326,59,394,122,256)(28,290,353,205,327,60,395,123,257)(29,291,354,206,328,61,396,124,258)(30,292,355,207,329,62,397,125,259)(31,293,356,208,330,63,398,126,260)(32,294,357,209,331,64,399,127,261)(33,295,358,210,332,65,400,128,262)(34,296,359,211,333,66,401,129,263)(35,297,360,212,334,67,402,130,264)(36,298,337,213,335,68,403,131,241)(37,299,338,214,336,69,404,132,242)(38,300,339,215,313,70,405,133,243)(39,301,340,216,314,71,406,134,244)(40,302,341,193,315,72,407,135,245)(41,303,342,194,316,49,408,136,246)(42,304,343,195,317,50,385,137,247)(43,305,344,196,318,51,386,138,248)(44,306,345,197,319,52,387,139,249)(45,307,346,198,320,53,388,140,250)(46,308,347,199,321,54,389,141,251)(47,309,348,200,322,55,390,142,252)(48,310,349,201,323,56,391,143,253), (1,289,13,301)(2,312,14,300)(3,311,15,299)(4,310,16,298)(5,309,17,297)(6,308,18,296)(7,307,19,295)(8,306,20,294)(9,305,21,293)(10,304,22,292)(11,303,23,291)(12,302,24,290)(25,87,37,75)(26,86,38,74)(27,85,39,73)(28,84,40,96)(29,83,41,95)(30,82,42,94)(31,81,43,93)(32,80,44,92)(33,79,45,91)(34,78,46,90)(35,77,47,89)(36,76,48,88)(49,237,61,225)(50,236,62,224)(51,235,63,223)(52,234,64,222)(53,233,65,221)(54,232,66,220)(55,231,67,219)(56,230,68,218)(57,229,69,217)(58,228,70,240)(59,227,71,239)(60,226,72,238)(97,392,109,404)(98,391,110,403)(99,390,111,402)(100,389,112,401)(101,388,113,400)(102,387,114,399)(103,386,115,398)(104,385,116,397)(105,408,117,396)(106,407,118,395)(107,406,119,394)(108,405,120,393)(121,154,133,166)(122,153,134,165)(123,152,135,164)(124,151,136,163)(125,150,137,162)(126,149,138,161)(127,148,139,160)(128,147,140,159)(129,146,141,158)(130,145,142,157)(131,168,143,156)(132,167,144,155)(169,353,181,341)(170,352,182,340)(171,351,183,339)(172,350,184,338)(173,349,185,337)(174,348,186,360)(175,347,187,359)(176,346,188,358)(177,345,189,357)(178,344,190,356)(179,343,191,355)(180,342,192,354)(193,281,205,269)(194,280,206,268)(195,279,207,267)(196,278,208,266)(197,277,209,265)(198,276,210,288)(199,275,211,287)(200,274,212,286)(201,273,213,285)(202,272,214,284)(203,271,215,283)(204,270,216,282)(241,376,253,364)(242,375,254,363)(243,374,255,362)(244,373,256,361)(245,372,257,384)(246,371,258,383)(247,370,259,382)(248,369,260,381)(249,368,261,380)(250,367,262,379)(251,366,263,378)(252,365,264,377)(313,409,325,421)(314,432,326,420)(315,431,327,419)(316,430,328,418)(317,429,329,417)(318,428,330,416)(319,427,331,415)(320,426,332,414)(321,425,333,413)(322,424,334,412)(323,423,335,411)(324,422,336,410)>;
G:=Group( (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), (1,73,373,165,119,239,432,282,170)(2,74,374,166,120,240,409,283,171)(3,75,375,167,97,217,410,284,172)(4,76,376,168,98,218,411,285,173)(5,77,377,145,99,219,412,286,174)(6,78,378,146,100,220,413,287,175)(7,79,379,147,101,221,414,288,176)(8,80,380,148,102,222,415,265,177)(9,81,381,149,103,223,416,266,178)(10,82,382,150,104,224,417,267,179)(11,83,383,151,105,225,418,268,180)(12,84,384,152,106,226,419,269,181)(13,85,361,153,107,227,420,270,182)(14,86,362,154,108,228,421,271,183)(15,87,363,155,109,229,422,272,184)(16,88,364,156,110,230,423,273,185)(17,89,365,157,111,231,424,274,186)(18,90,366,158,112,232,425,275,187)(19,91,367,159,113,233,426,276,188)(20,92,368,160,114,234,427,277,189)(21,93,369,161,115,235,428,278,190)(22,94,370,162,116,236,429,279,191)(23,95,371,163,117,237,430,280,192)(24,96,372,164,118,238,431,281,169)(25,311,350,202,324,57,392,144,254)(26,312,351,203,325,58,393,121,255)(27,289,352,204,326,59,394,122,256)(28,290,353,205,327,60,395,123,257)(29,291,354,206,328,61,396,124,258)(30,292,355,207,329,62,397,125,259)(31,293,356,208,330,63,398,126,260)(32,294,357,209,331,64,399,127,261)(33,295,358,210,332,65,400,128,262)(34,296,359,211,333,66,401,129,263)(35,297,360,212,334,67,402,130,264)(36,298,337,213,335,68,403,131,241)(37,299,338,214,336,69,404,132,242)(38,300,339,215,313,70,405,133,243)(39,301,340,216,314,71,406,134,244)(40,302,341,193,315,72,407,135,245)(41,303,342,194,316,49,408,136,246)(42,304,343,195,317,50,385,137,247)(43,305,344,196,318,51,386,138,248)(44,306,345,197,319,52,387,139,249)(45,307,346,198,320,53,388,140,250)(46,308,347,199,321,54,389,141,251)(47,309,348,200,322,55,390,142,252)(48,310,349,201,323,56,391,143,253), (1,289,13,301)(2,312,14,300)(3,311,15,299)(4,310,16,298)(5,309,17,297)(6,308,18,296)(7,307,19,295)(8,306,20,294)(9,305,21,293)(10,304,22,292)(11,303,23,291)(12,302,24,290)(25,87,37,75)(26,86,38,74)(27,85,39,73)(28,84,40,96)(29,83,41,95)(30,82,42,94)(31,81,43,93)(32,80,44,92)(33,79,45,91)(34,78,46,90)(35,77,47,89)(36,76,48,88)(49,237,61,225)(50,236,62,224)(51,235,63,223)(52,234,64,222)(53,233,65,221)(54,232,66,220)(55,231,67,219)(56,230,68,218)(57,229,69,217)(58,228,70,240)(59,227,71,239)(60,226,72,238)(97,392,109,404)(98,391,110,403)(99,390,111,402)(100,389,112,401)(101,388,113,400)(102,387,114,399)(103,386,115,398)(104,385,116,397)(105,408,117,396)(106,407,118,395)(107,406,119,394)(108,405,120,393)(121,154,133,166)(122,153,134,165)(123,152,135,164)(124,151,136,163)(125,150,137,162)(126,149,138,161)(127,148,139,160)(128,147,140,159)(129,146,141,158)(130,145,142,157)(131,168,143,156)(132,167,144,155)(169,353,181,341)(170,352,182,340)(171,351,183,339)(172,350,184,338)(173,349,185,337)(174,348,186,360)(175,347,187,359)(176,346,188,358)(177,345,189,357)(178,344,190,356)(179,343,191,355)(180,342,192,354)(193,281,205,269)(194,280,206,268)(195,279,207,267)(196,278,208,266)(197,277,209,265)(198,276,210,288)(199,275,211,287)(200,274,212,286)(201,273,213,285)(202,272,214,284)(203,271,215,283)(204,270,216,282)(241,376,253,364)(242,375,254,363)(243,374,255,362)(244,373,256,361)(245,372,257,384)(246,371,258,383)(247,370,259,382)(248,369,260,381)(249,368,261,380)(250,367,262,379)(251,366,263,378)(252,365,264,377)(313,409,325,421)(314,432,326,420)(315,431,327,419)(316,430,328,418)(317,429,329,417)(318,428,330,416)(319,427,331,415)(320,426,332,414)(321,425,333,413)(322,424,334,412)(323,423,335,411)(324,422,336,410) );
G=PermutationGroup([[(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)], [(1,73,373,165,119,239,432,282,170),(2,74,374,166,120,240,409,283,171),(3,75,375,167,97,217,410,284,172),(4,76,376,168,98,218,411,285,173),(5,77,377,145,99,219,412,286,174),(6,78,378,146,100,220,413,287,175),(7,79,379,147,101,221,414,288,176),(8,80,380,148,102,222,415,265,177),(9,81,381,149,103,223,416,266,178),(10,82,382,150,104,224,417,267,179),(11,83,383,151,105,225,418,268,180),(12,84,384,152,106,226,419,269,181),(13,85,361,153,107,227,420,270,182),(14,86,362,154,108,228,421,271,183),(15,87,363,155,109,229,422,272,184),(16,88,364,156,110,230,423,273,185),(17,89,365,157,111,231,424,274,186),(18,90,366,158,112,232,425,275,187),(19,91,367,159,113,233,426,276,188),(20,92,368,160,114,234,427,277,189),(21,93,369,161,115,235,428,278,190),(22,94,370,162,116,236,429,279,191),(23,95,371,163,117,237,430,280,192),(24,96,372,164,118,238,431,281,169),(25,311,350,202,324,57,392,144,254),(26,312,351,203,325,58,393,121,255),(27,289,352,204,326,59,394,122,256),(28,290,353,205,327,60,395,123,257),(29,291,354,206,328,61,396,124,258),(30,292,355,207,329,62,397,125,259),(31,293,356,208,330,63,398,126,260),(32,294,357,209,331,64,399,127,261),(33,295,358,210,332,65,400,128,262),(34,296,359,211,333,66,401,129,263),(35,297,360,212,334,67,402,130,264),(36,298,337,213,335,68,403,131,241),(37,299,338,214,336,69,404,132,242),(38,300,339,215,313,70,405,133,243),(39,301,340,216,314,71,406,134,244),(40,302,341,193,315,72,407,135,245),(41,303,342,194,316,49,408,136,246),(42,304,343,195,317,50,385,137,247),(43,305,344,196,318,51,386,138,248),(44,306,345,197,319,52,387,139,249),(45,307,346,198,320,53,388,140,250),(46,308,347,199,321,54,389,141,251),(47,309,348,200,322,55,390,142,252),(48,310,349,201,323,56,391,143,253)], [(1,289,13,301),(2,312,14,300),(3,311,15,299),(4,310,16,298),(5,309,17,297),(6,308,18,296),(7,307,19,295),(8,306,20,294),(9,305,21,293),(10,304,22,292),(11,303,23,291),(12,302,24,290),(25,87,37,75),(26,86,38,74),(27,85,39,73),(28,84,40,96),(29,83,41,95),(30,82,42,94),(31,81,43,93),(32,80,44,92),(33,79,45,91),(34,78,46,90),(35,77,47,89),(36,76,48,88),(49,237,61,225),(50,236,62,224),(51,235,63,223),(52,234,64,222),(53,233,65,221),(54,232,66,220),(55,231,67,219),(56,230,68,218),(57,229,69,217),(58,228,70,240),(59,227,71,239),(60,226,72,238),(97,392,109,404),(98,391,110,403),(99,390,111,402),(100,389,112,401),(101,388,113,400),(102,387,114,399),(103,386,115,398),(104,385,116,397),(105,408,117,396),(106,407,118,395),(107,406,119,394),(108,405,120,393),(121,154,133,166),(122,153,134,165),(123,152,135,164),(124,151,136,163),(125,150,137,162),(126,149,138,161),(127,148,139,160),(128,147,140,159),(129,146,141,158),(130,145,142,157),(131,168,143,156),(132,167,144,155),(169,353,181,341),(170,352,182,340),(171,351,183,339),(172,350,184,338),(173,349,185,337),(174,348,186,360),(175,347,187,359),(176,346,188,358),(177,345,189,357),(178,344,190,356),(179,343,191,355),(180,342,192,354),(193,281,205,269),(194,280,206,268),(195,279,207,267),(196,278,208,266),(197,277,209,265),(198,276,210,288),(199,275,211,287),(200,274,212,286),(201,273,213,285),(202,272,214,284),(203,271,215,283),(204,270,216,282),(241,376,253,364),(242,375,254,363),(243,374,255,362),(244,373,256,361),(245,372,257,384),(246,371,258,383),(247,370,259,382),(248,369,260,381),(249,368,261,380),(250,367,262,379),(251,366,263,378),(252,365,264,377),(313,409,325,421),(314,432,326,420),(315,431,327,419),(316,430,328,418),(317,429,329,417),(318,428,330,416),(319,427,331,415),(320,426,332,414),(321,425,333,413),(322,424,334,412),(323,423,335,411),(324,422,336,410)]])
111 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 108 | 108 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | Q16 | D9 | D12 | D12 | D18 | Dic12 | Dic12 | D36 | Dic36 |
| kernel | C24.D9 | C3×C72 | C12.D9 | C72 | C3×C24 | C3×C18 | C36 | C3×C12 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 2 | 9 | 6 | 2 | 9 | 12 | 4 | 18 | 36 |
Matrix representation of C24.D9 ►in GL4(𝔽73) generated by
| 50 | 55 | 0 | 0 |
| 18 | 68 | 0 | 0 |
| 0 | 0 | 50 | 18 |
| 0 | 0 | 55 | 68 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 70 | 45 |
| 0 | 0 | 28 | 42 |
| 27 | 27 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [50,18,0,0,55,68,0,0,0,0,50,55,0,0,18,68],[72,1,0,0,72,0,0,0,0,0,70,28,0,0,45,42],[27,0,0,0,27,46,0,0,0,0,46,0,0,0,27,27] >;
C24.D9 in GAP, Magma, Sage, TeX
C_{24}.D_9 % in TeX
G:=Group("C24.D9"); // GroupNames label
G:=SmallGroup(432,168);
// by ID
G=gap.SmallGroup(432,168);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^24=b^9=1,c^2=a^12,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations