metabelian, supersoluble, monomial, A-group
Aliases: C72.8S3, C24.9D9, C12.7Dic9, C36.7Dic3, C9⋊(C3⋊C16), C3⋊(C9⋊C16), (C3×C9)⋊3C16, C6.3(C9⋊C8), C18.3(C3⋊C8), C8.2(C9⋊S3), (C3×C18).3C8, (C3×C36).6C4, (C3×C72).5C2, C24.8(C3⋊S3), (C3×C24).21S3, C3.(C24.S3), C2.(C36.S3), C32.3(C3⋊C16), C4.2(C9⋊Dic3), C12.4(C3⋊Dic3), C6.1(C32⋊4C8), (C3×C12).22Dic3, (C3×C6).8(C3⋊C8), SmallGroup(432,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C72.S3 |
Generators and relations for C72.S3
G = < a,b,c | a72=b3=1, c2=a9, ab=ba, cac-1=a17, cbc-1=b-1 >
27C16
Smallest permutation representation of C72.S3
►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)
(1 255 308)(2 256 309)(3 257 310)(4 258 311)(5 259 312)(6 260 313)(7 261 314)(8 262 315)(9 263 316)(10 264 317)(11 265 318)(12 266 319)(13 267 320)(14 268 321)(15 269 322)(16 270 323)(17 271 324)(18 272 325)(19 273 326)(20 274 327)(21 275 328)(22 276 329)(23 277 330)(24 278 331)(25 279 332)(26 280 333)(27 281 334)(28 282 335)(29 283 336)(30 284 337)(31 285 338)(32 286 339)(33 287 340)(34 288 341)(35 217 342)(36 218 343)(37 219 344)(38 220 345)(39 221 346)(40 222 347)(41 223 348)(42 224 349)(43 225 350)(44 226 351)(45 227 352)(46 228 353)(47 229 354)(48 230 355)(49 231 356)(50 232 357)(51 233 358)(52 234 359)(53 235 360)(54 236 289)(55 237 290)(56 238 291)(57 239 292)(58 240 293)(59 241 294)(60 242 295)(61 243 296)(62 244 297)(63 245 298)(64 246 299)(65 247 300)(66 248 301)(67 249 302)(68 250 303)(69 251 304)(70 252 305)(71 253 306)(72 254 307)(73 209 380)(74 210 381)(75 211 382)(76 212 383)(77 213 384)(78 214 385)(79 215 386)(80 216 387)(81 145 388)(82 146 389)(83 147 390)(84 148 391)(85 149 392)(86 150 393)(87 151 394)(88 152 395)(89 153 396)(90 154 397)(91 155 398)(92 156 399)(93 157 400)(94 158 401)(95 159 402)(96 160 403)(97 161 404)(98 162 405)(99 163 406)(100 164 407)(101 165 408)(102 166 409)(103 167 410)(104 168 411)(105 169 412)(106 170 413)(107 171 414)(108 172 415)(109 173 416)(110 174 417)(111 175 418)(112 176 419)(113 177 420)(114 178 421)(115 179 422)(116 180 423)(117 181 424)(118 182 425)(119 183 426)(120 184 427)(121 185 428)(122 186 429)(123 187 430)(124 188 431)(125 189 432)(126 190 361)(127 191 362)(128 192 363)(129 193 364)(130 194 365)(131 195 366)(132 196 367)(133 197 368)(134 198 369)(135 199 370)(136 200 371)(137 201 372)(138 202 373)(139 203 374)(140 204 375)(141 205 376)(142 206 377)(143 207 378)(144 208 379)
(1 212 10 149 19 158 28 167 37 176 46 185 55 194 64 203)(2 157 11 166 20 175 29 184 38 193 47 202 56 211 65 148)(3 174 12 183 21 192 30 201 39 210 48 147 57 156 66 165)(4 191 13 200 22 209 31 146 40 155 49 164 58 173 67 182)(5 208 14 145 23 154 32 163 41 172 50 181 59 190 68 199)(6 153 15 162 24 171 33 180 42 189 51 198 60 207 69 216)(7 170 16 179 25 188 34 197 43 206 52 215 61 152 70 161)(8 187 17 196 26 205 35 214 44 151 53 160 62 169 71 178)(9 204 18 213 27 150 36 159 45 168 54 177 63 186 72 195)(73 285 82 222 91 231 100 240 109 249 118 258 127 267 136 276)(74 230 83 239 92 248 101 257 110 266 119 275 128 284 137 221)(75 247 84 256 93 265 102 274 111 283 120 220 129 229 138 238)(76 264 85 273 94 282 103 219 112 228 121 237 130 246 139 255)(77 281 86 218 95 227 104 236 113 245 122 254 131 263 140 272)(78 226 87 235 96 244 105 253 114 262 123 271 132 280 141 217)(79 243 88 252 97 261 106 270 115 279 124 288 133 225 142 234)(80 260 89 269 98 278 107 287 116 224 125 233 134 242 143 251)(81 277 90 286 99 223 108 232 117 241 126 250 135 259 144 268)(289 420 298 429 307 366 316 375 325 384 334 393 343 402 352 411)(290 365 299 374 308 383 317 392 326 401 335 410 344 419 353 428)(291 382 300 391 309 400 318 409 327 418 336 427 345 364 354 373)(292 399 301 408 310 417 319 426 328 363 337 372 346 381 355 390)(293 416 302 425 311 362 320 371 329 380 338 389 347 398 356 407)(294 361 303 370 312 379 321 388 330 397 339 406 348 415 357 424)(295 378 304 387 313 396 322 405 331 414 340 423 349 432 358 369)(296 395 305 404 314 413 323 422 332 431 341 368 350 377 359 386)(297 412 306 421 315 430 324 367 333 376 342 385 351 394 360 403)
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), (1,255,308)(2,256,309)(3,257,310)(4,258,311)(5,259,312)(6,260,313)(7,261,314)(8,262,315)(9,263,316)(10,264,317)(11,265,318)(12,266,319)(13,267,320)(14,268,321)(15,269,322)(16,270,323)(17,271,324)(18,272,325)(19,273,326)(20,274,327)(21,275,328)(22,276,329)(23,277,330)(24,278,331)(25,279,332)(26,280,333)(27,281,334)(28,282,335)(29,283,336)(30,284,337)(31,285,338)(32,286,339)(33,287,340)(34,288,341)(35,217,342)(36,218,343)(37,219,344)(38,220,345)(39,221,346)(40,222,347)(41,223,348)(42,224,349)(43,225,350)(44,226,351)(45,227,352)(46,228,353)(47,229,354)(48,230,355)(49,231,356)(50,232,357)(51,233,358)(52,234,359)(53,235,360)(54,236,289)(55,237,290)(56,238,291)(57,239,292)(58,240,293)(59,241,294)(60,242,295)(61,243,296)(62,244,297)(63,245,298)(64,246,299)(65,247,300)(66,248,301)(67,249,302)(68,250,303)(69,251,304)(70,252,305)(71,253,306)(72,254,307)(73,209,380)(74,210,381)(75,211,382)(76,212,383)(77,213,384)(78,214,385)(79,215,386)(80,216,387)(81,145,388)(82,146,389)(83,147,390)(84,148,391)(85,149,392)(86,150,393)(87,151,394)(88,152,395)(89,153,396)(90,154,397)(91,155,398)(92,156,399)(93,157,400)(94,158,401)(95,159,402)(96,160,403)(97,161,404)(98,162,405)(99,163,406)(100,164,407)(101,165,408)(102,166,409)(103,167,410)(104,168,411)(105,169,412)(106,170,413)(107,171,414)(108,172,415)(109,173,416)(110,174,417)(111,175,418)(112,176,419)(113,177,420)(114,178,421)(115,179,422)(116,180,423)(117,181,424)(118,182,425)(119,183,426)(120,184,427)(121,185,428)(122,186,429)(123,187,430)(124,188,431)(125,189,432)(126,190,361)(127,191,362)(128,192,363)(129,193,364)(130,194,365)(131,195,366)(132,196,367)(133,197,368)(134,198,369)(135,199,370)(136,200,371)(137,201,372)(138,202,373)(139,203,374)(140,204,375)(141,205,376)(142,206,377)(143,207,378)(144,208,379), (1,212,10,149,19,158,28,167,37,176,46,185,55,194,64,203)(2,157,11,166,20,175,29,184,38,193,47,202,56,211,65,148)(3,174,12,183,21,192,30,201,39,210,48,147,57,156,66,165)(4,191,13,200,22,209,31,146,40,155,49,164,58,173,67,182)(5,208,14,145,23,154,32,163,41,172,50,181,59,190,68,199)(6,153,15,162,24,171,33,180,42,189,51,198,60,207,69,216)(7,170,16,179,25,188,34,197,43,206,52,215,61,152,70,161)(8,187,17,196,26,205,35,214,44,151,53,160,62,169,71,178)(9,204,18,213,27,150,36,159,45,168,54,177,63,186,72,195)(73,285,82,222,91,231,100,240,109,249,118,258,127,267,136,276)(74,230,83,239,92,248,101,257,110,266,119,275,128,284,137,221)(75,247,84,256,93,265,102,274,111,283,120,220,129,229,138,238)(76,264,85,273,94,282,103,219,112,228,121,237,130,246,139,255)(77,281,86,218,95,227,104,236,113,245,122,254,131,263,140,272)(78,226,87,235,96,244,105,253,114,262,123,271,132,280,141,217)(79,243,88,252,97,261,106,270,115,279,124,288,133,225,142,234)(80,260,89,269,98,278,107,287,116,224,125,233,134,242,143,251)(81,277,90,286,99,223,108,232,117,241,126,250,135,259,144,268)(289,420,298,429,307,366,316,375,325,384,334,393,343,402,352,411)(290,365,299,374,308,383,317,392,326,401,335,410,344,419,353,428)(291,382,300,391,309,400,318,409,327,418,336,427,345,364,354,373)(292,399,301,408,310,417,319,426,328,363,337,372,346,381,355,390)(293,416,302,425,311,362,320,371,329,380,338,389,347,398,356,407)(294,361,303,370,312,379,321,388,330,397,339,406,348,415,357,424)(295,378,304,387,313,396,322,405,331,414,340,423,349,432,358,369)(296,395,305,404,314,413,323,422,332,431,341,368,350,377,359,386)(297,412,306,421,315,430,324,367,333,376,342,385,351,394,360,403)>;
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), (1,255,308)(2,256,309)(3,257,310)(4,258,311)(5,259,312)(6,260,313)(7,261,314)(8,262,315)(9,263,316)(10,264,317)(11,265,318)(12,266,319)(13,267,320)(14,268,321)(15,269,322)(16,270,323)(17,271,324)(18,272,325)(19,273,326)(20,274,327)(21,275,328)(22,276,329)(23,277,330)(24,278,331)(25,279,332)(26,280,333)(27,281,334)(28,282,335)(29,283,336)(30,284,337)(31,285,338)(32,286,339)(33,287,340)(34,288,341)(35,217,342)(36,218,343)(37,219,344)(38,220,345)(39,221,346)(40,222,347)(41,223,348)(42,224,349)(43,225,350)(44,226,351)(45,227,352)(46,228,353)(47,229,354)(48,230,355)(49,231,356)(50,232,357)(51,233,358)(52,234,359)(53,235,360)(54,236,289)(55,237,290)(56,238,291)(57,239,292)(58,240,293)(59,241,294)(60,242,295)(61,243,296)(62,244,297)(63,245,298)(64,246,299)(65,247,300)(66,248,301)(67,249,302)(68,250,303)(69,251,304)(70,252,305)(71,253,306)(72,254,307)(73,209,380)(74,210,381)(75,211,382)(76,212,383)(77,213,384)(78,214,385)(79,215,386)(80,216,387)(81,145,388)(82,146,389)(83,147,390)(84,148,391)(85,149,392)(86,150,393)(87,151,394)(88,152,395)(89,153,396)(90,154,397)(91,155,398)(92,156,399)(93,157,400)(94,158,401)(95,159,402)(96,160,403)(97,161,404)(98,162,405)(99,163,406)(100,164,407)(101,165,408)(102,166,409)(103,167,410)(104,168,411)(105,169,412)(106,170,413)(107,171,414)(108,172,415)(109,173,416)(110,174,417)(111,175,418)(112,176,419)(113,177,420)(114,178,421)(115,179,422)(116,180,423)(117,181,424)(118,182,425)(119,183,426)(120,184,427)(121,185,428)(122,186,429)(123,187,430)(124,188,431)(125,189,432)(126,190,361)(127,191,362)(128,192,363)(129,193,364)(130,194,365)(131,195,366)(132,196,367)(133,197,368)(134,198,369)(135,199,370)(136,200,371)(137,201,372)(138,202,373)(139,203,374)(140,204,375)(141,205,376)(142,206,377)(143,207,378)(144,208,379), (1,212,10,149,19,158,28,167,37,176,46,185,55,194,64,203)(2,157,11,166,20,175,29,184,38,193,47,202,56,211,65,148)(3,174,12,183,21,192,30,201,39,210,48,147,57,156,66,165)(4,191,13,200,22,209,31,146,40,155,49,164,58,173,67,182)(5,208,14,145,23,154,32,163,41,172,50,181,59,190,68,199)(6,153,15,162,24,171,33,180,42,189,51,198,60,207,69,216)(7,170,16,179,25,188,34,197,43,206,52,215,61,152,70,161)(8,187,17,196,26,205,35,214,44,151,53,160,62,169,71,178)(9,204,18,213,27,150,36,159,45,168,54,177,63,186,72,195)(73,285,82,222,91,231,100,240,109,249,118,258,127,267,136,276)(74,230,83,239,92,248,101,257,110,266,119,275,128,284,137,221)(75,247,84,256,93,265,102,274,111,283,120,220,129,229,138,238)(76,264,85,273,94,282,103,219,112,228,121,237,130,246,139,255)(77,281,86,218,95,227,104,236,113,245,122,254,131,263,140,272)(78,226,87,235,96,244,105,253,114,262,123,271,132,280,141,217)(79,243,88,252,97,261,106,270,115,279,124,288,133,225,142,234)(80,260,89,269,98,278,107,287,116,224,125,233,134,242,143,251)(81,277,90,286,99,223,108,232,117,241,126,250,135,259,144,268)(289,420,298,429,307,366,316,375,325,384,334,393,343,402,352,411)(290,365,299,374,308,383,317,392,326,401,335,410,344,419,353,428)(291,382,300,391,309,400,318,409,327,418,336,427,345,364,354,373)(292,399,301,408,310,417,319,426,328,363,337,372,346,381,355,390)(293,416,302,425,311,362,320,371,329,380,338,389,347,398,356,407)(294,361,303,370,312,379,321,388,330,397,339,406,348,415,357,424)(295,378,304,387,313,396,322,405,331,414,340,423,349,432,358,369)(296,395,305,404,314,413,323,422,332,431,341,368,350,377,359,386)(297,412,306,421,315,430,324,367,333,376,342,385,351,394,360,403) );
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)], [(1,255,308),(2,256,309),(3,257,310),(4,258,311),(5,259,312),(6,260,313),(7,261,314),(8,262,315),(9,263,316),(10,264,317),(11,265,318),(12,266,319),(13,267,320),(14,268,321),(15,269,322),(16,270,323),(17,271,324),(18,272,325),(19,273,326),(20,274,327),(21,275,328),(22,276,329),(23,277,330),(24,278,331),(25,279,332),(26,280,333),(27,281,334),(28,282,335),(29,283,336),(30,284,337),(31,285,338),(32,286,339),(33,287,340),(34,288,341),(35,217,342),(36,218,343),(37,219,344),(38,220,345),(39,221,346),(40,222,347),(41,223,348),(42,224,349),(43,225,350),(44,226,351),(45,227,352),(46,228,353),(47,229,354),(48,230,355),(49,231,356),(50,232,357),(51,233,358),(52,234,359),(53,235,360),(54,236,289),(55,237,290),(56,238,291),(57,239,292),(58,240,293),(59,241,294),(60,242,295),(61,243,296),(62,244,297),(63,245,298),(64,246,299),(65,247,300),(66,248,301),(67,249,302),(68,250,303),(69,251,304),(70,252,305),(71,253,306),(72,254,307),(73,209,380),(74,210,381),(75,211,382),(76,212,383),(77,213,384),(78,214,385),(79,215,386),(80,216,387),(81,145,388),(82,146,389),(83,147,390),(84,148,391),(85,149,392),(86,150,393),(87,151,394),(88,152,395),(89,153,396),(90,154,397),(91,155,398),(92,156,399),(93,157,400),(94,158,401),(95,159,402),(96,160,403),(97,161,404),(98,162,405),(99,163,406),(100,164,407),(101,165,408),(102,166,409),(103,167,410),(104,168,411),(105,169,412),(106,170,413),(107,171,414),(108,172,415),(109,173,416),(110,174,417),(111,175,418),(112,176,419),(113,177,420),(114,178,421),(115,179,422),(116,180,423),(117,181,424),(118,182,425),(119,183,426),(120,184,427),(121,185,428),(122,186,429),(123,187,430),(124,188,431),(125,189,432),(126,190,361),(127,191,362),(128,192,363),(129,193,364),(130,194,365),(131,195,366),(132,196,367),(133,197,368),(134,198,369),(135,199,370),(136,200,371),(137,201,372),(138,202,373),(139,203,374),(140,204,375),(141,205,376),(142,206,377),(143,207,378),(144,208,379)], [(1,212,10,149,19,158,28,167,37,176,46,185,55,194,64,203),(2,157,11,166,20,175,29,184,38,193,47,202,56,211,65,148),(3,174,12,183,21,192,30,201,39,210,48,147,57,156,66,165),(4,191,13,200,22,209,31,146,40,155,49,164,58,173,67,182),(5,208,14,145,23,154,32,163,41,172,50,181,59,190,68,199),(6,153,15,162,24,171,33,180,42,189,51,198,60,207,69,216),(7,170,16,179,25,188,34,197,43,206,52,215,61,152,70,161),(8,187,17,196,26,205,35,214,44,151,53,160,62,169,71,178),(9,204,18,213,27,150,36,159,45,168,54,177,63,186,72,195),(73,285,82,222,91,231,100,240,109,249,118,258,127,267,136,276),(74,230,83,239,92,248,101,257,110,266,119,275,128,284,137,221),(75,247,84,256,93,265,102,274,111,283,120,220,129,229,138,238),(76,264,85,273,94,282,103,219,112,228,121,237,130,246,139,255),(77,281,86,218,95,227,104,236,113,245,122,254,131,263,140,272),(78,226,87,235,96,244,105,253,114,262,123,271,132,280,141,217),(79,243,88,252,97,261,106,270,115,279,124,288,133,225,142,234),(80,260,89,269,98,278,107,287,116,224,125,233,134,242,143,251),(81,277,90,286,99,223,108,232,117,241,126,250,135,259,144,268),(289,420,298,429,307,366,316,375,325,384,334,393,343,402,352,411),(290,365,299,374,308,383,317,392,326,401,335,410,344,419,353,428),(291,382,300,391,309,400,318,409,327,418,336,427,345,364,354,373),(292,399,301,408,310,417,319,426,328,363,337,372,346,381,355,390),(293,416,302,425,311,362,320,371,329,380,338,389,347,398,356,407),(294,361,303,370,312,379,321,388,330,397,339,406,348,415,357,424),(295,378,304,387,313,396,322,405,331,414,340,423,349,432,358,369),(296,395,305,404,314,413,323,422,332,431,341,368,350,377,359,386),(297,412,306,421,315,430,324,367,333,376,342,385,351,394,360,403)]])
120 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | ··· | 12H | 16A | ··· | 16H | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 27 | ··· | 27 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | S3 | Dic3 | Dic3 | D9 | C3⋊C8 | C3⋊C8 | Dic9 | C3⋊C16 | C3⋊C16 | C9⋊C8 | C9⋊C16 |
| kernel | C72.S3 | C3×C72 | C3×C36 | C3×C18 | C3×C9 | C72 | C3×C24 | C36 | C3×C12 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 8 | 3 | 1 | 3 | 1 | 9 | 6 | 2 | 9 | 12 | 4 | 18 | 36 |
Matrix representation of C72.S3 ►in GL5(𝔽433)
| 79 | 0 | 0 | 0 | 0 |
| 0 | 51 | 186 | 0 | 0 |
| 0 | 247 | 298 | 0 | 0 |
| 0 | 0 | 0 | 179 | 104 |
| 0 | 0 | 0 | 254 | 75 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 432 | 432 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 431 | 430 |
| 0 | 0 | 0 | 1 | 1 |
| 265 | 0 | 0 | 0 | 0 |
| 0 | 284 | 183 | 0 | 0 |
| 0 | 332 | 149 | 0 | 0 |
| 0 | 0 | 0 | 367 | 294 |
| 0 | 0 | 0 | 164 | 66 |
G:=sub<GL(5,GF(433))| [79,0,0,0,0,0,51,247,0,0,0,186,298,0,0,0,0,0,179,254,0,0,0,104,75],[1,0,0,0,0,0,432,1,0,0,0,432,0,0,0,0,0,0,431,1,0,0,0,430,1],[265,0,0,0,0,0,284,332,0,0,0,183,149,0,0,0,0,0,367,164,0,0,0,294,66] >;
C72.S3 in GAP, Magma, Sage, TeX
C_{72}.S_3 % in TeX
G:=Group("C72.S3"); // GroupNames label
G:=SmallGroup(432,32);
// by ID
G=gap.SmallGroup(432,32);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,14,36,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^72=b^3=1,c^2=a^9,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations
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