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## G = C22×C6×C18order 432 = 24·33

### Abelian group of type [2,2,6,18]

Aliases: C22×C6×C18, SmallGroup(432,562)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6×C18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — C2×C6×C18 — C22×C6×C18
 Lower central C1 — C22×C6×C18
 Upper central C1 — C22×C6×C18

Generators and relations for C22×C6×C18
G = < a,b,c,d | a2=b2=c6=d18=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 670, all normal (8 characteristic)
C1, C2, C3, C3, C22, C6, C23, C9, C32, C2×C6, C24, C18, C3×C6, C22×C6, C3×C9, C2×C18, C62, C23×C6, C23×C6, C3×C18, C22×C18, C2×C62, C6×C18, C23×C18, C22×C62, C2×C6×C18, C22×C6×C18
Quotients: C1, C2, C3, C22, C6, C23, C9, C32, C2×C6, C24, C18, C3×C6, C22×C6, C3×C9, C2×C18, C62, C23×C6, C3×C18, C22×C18, C2×C62, C6×C18, C23×C18, C22×C62, C2×C6×C18, C22×C6×C18

Smallest permutation representation of C22×C6×C18
Regular action on 432 points
Generators in S432
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 55)(17 56)(18 57)(19 412)(20 413)(21 414)(22 397)(23 398)(24 399)(25 400)(26 401)(27 402)(28 403)(29 404)(30 405)(31 406)(32 407)(33 408)(34 409)(35 410)(36 411)(37 428)(38 429)(39 430)(40 431)(41 432)(42 415)(43 416)(44 417)(45 418)(46 419)(47 420)(48 421)(49 422)(50 423)(51 424)(52 425)(53 426)(54 427)(73 291)(74 292)(75 293)(76 294)(77 295)(78 296)(79 297)(80 298)(81 299)(82 300)(83 301)(84 302)(85 303)(86 304)(87 305)(88 306)(89 289)(90 290)(91 126)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(127 166)(128 167)(129 168)(130 169)(131 170)(132 171)(133 172)(134 173)(135 174)(136 175)(137 176)(138 177)(139 178)(140 179)(141 180)(142 163)(143 164)(144 165)(145 215)(146 216)(147 199)(148 200)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)(155 207)(156 208)(157 209)(158 210)(159 211)(160 212)(161 213)(162 214)(181 350)(182 351)(183 352)(184 353)(185 354)(186 355)(187 356)(188 357)(189 358)(190 359)(191 360)(192 343)(193 344)(194 345)(195 346)(196 347)(197 348)(198 349)(217 392)(218 393)(219 394)(220 395)(221 396)(222 379)(223 380)(224 381)(225 382)(226 383)(227 384)(228 385)(229 386)(230 387)(231 388)(232 389)(233 390)(234 391)(235 319)(236 320)(237 321)(238 322)(239 323)(240 324)(241 307)(242 308)(243 309)(244 310)(245 311)(246 312)(247 313)(248 314)(249 315)(250 316)(251 317)(252 318)(253 375)(254 376)(255 377)(256 378)(257 361)(258 362)(259 363)(260 364)(261 365)(262 366)(263 367)(264 368)(265 369)(266 370)(267 371)(268 372)(269 373)(270 374)(271 337)(272 338)(273 339)(274 340)(275 341)(276 342)(277 325)(278 326)(279 327)(280 328)(281 329)(282 330)(283 331)(284 332)(285 333)(286 334)(287 335)(288 336)
(1 409)(2 410)(3 411)(4 412)(5 413)(6 414)(7 397)(8 398)(9 399)(10 400)(11 401)(12 402)(13 403)(14 404)(15 405)(16 406)(17 407)(18 408)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 392)(38 393)(39 394)(40 395)(41 396)(42 379)(43 380)(44 381)(45 382)(46 383)(47 384)(48 385)(49 386)(50 387)(51 388)(52 389)(53 390)(54 391)(73 352)(74 353)(75 354)(76 355)(77 356)(78 357)(79 358)(80 359)(81 360)(82 343)(83 344)(84 345)(85 346)(86 347)(87 348)(88 349)(89 350)(90 351)(91 170)(92 171)(93 172)(94 173)(95 174)(96 175)(97 176)(98 177)(99 178)(100 179)(101 180)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 169)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(121 144)(122 127)(123 128)(124 129)(125 130)(126 131)(145 244)(146 245)(147 246)(148 247)(149 248)(150 249)(151 250)(152 251)(153 252)(154 235)(155 236)(156 237)(157 238)(158 239)(159 240)(160 241)(161 242)(162 243)(181 289)(182 290)(183 291)(184 292)(185 293)(186 294)(187 295)(188 296)(189 297)(190 298)(191 299)(192 300)(193 301)(194 302)(195 303)(196 304)(197 305)(198 306)(199 312)(200 313)(201 314)(202 315)(203 316)(204 317)(205 318)(206 319)(207 320)(208 321)(209 322)(210 323)(211 324)(212 307)(213 308)(214 309)(215 310)(216 311)(217 428)(218 429)(219 430)(220 431)(221 432)(222 415)(223 416)(224 417)(225 418)(226 419)(227 420)(228 421)(229 422)(230 423)(231 424)(232 425)(233 426)(234 427)(253 272)(254 273)(255 274)(256 275)(257 276)(258 277)(259 278)(260 279)(261 280)(262 281)(263 282)(264 283)(265 284)(266 285)(267 286)(268 287)(269 288)(270 271)(325 362)(326 363)(327 364)(328 365)(329 366)(330 367)(331 368)(332 369)(333 370)(334 371)(335 372)(336 373)(337 374)(338 375)(339 376)(340 377)(341 378)(342 361)
(1 76 241 273 233 97)(2 77 242 274 234 98)(3 78 243 275 217 99)(4 79 244 276 218 100)(5 80 245 277 219 101)(6 81 246 278 220 102)(7 82 247 279 221 103)(8 83 248 280 222 104)(9 84 249 281 223 105)(10 85 250 282 224 106)(11 86 251 283 225 107)(12 87 252 284 226 108)(13 88 235 285 227 91)(14 89 236 286 228 92)(15 90 237 287 229 93)(16 73 238 288 230 94)(17 74 239 271 231 95)(18 75 240 272 232 96)(19 189 215 361 38 140)(20 190 216 362 39 141)(21 191 199 363 40 142)(22 192 200 364 41 143)(23 193 201 365 42 144)(24 194 202 366 43 127)(25 195 203 367 44 128)(26 196 204 368 45 129)(27 197 205 369 46 130)(28 198 206 370 47 131)(29 181 207 371 48 132)(30 182 208 372 49 133)(31 183 209 373 50 134)(32 184 210 374 51 135)(33 185 211 375 52 136)(34 186 212 376 53 137)(35 187 213 377 54 138)(36 188 214 378 37 139)(55 291 322 336 387 111)(56 292 323 337 388 112)(57 293 324 338 389 113)(58 294 307 339 390 114)(59 295 308 340 391 115)(60 296 309 341 392 116)(61 297 310 342 393 117)(62 298 311 325 394 118)(63 299 312 326 395 119)(64 300 313 327 396 120)(65 301 314 328 379 121)(66 302 315 329 380 122)(67 303 316 330 381 123)(68 304 317 331 382 124)(69 305 318 332 383 125)(70 306 319 333 384 126)(71 289 320 334 385 109)(72 290 321 335 386 110)(145 257 429 179 412 358)(146 258 430 180 413 359)(147 259 431 163 414 360)(148 260 432 164 397 343)(149 261 415 165 398 344)(150 262 416 166 399 345)(151 263 417 167 400 346)(152 264 418 168 401 347)(153 265 419 169 402 348)(154 266 420 170 403 349)(155 267 421 171 404 350)(156 268 422 172 405 351)(157 269 423 173 406 352)(158 270 424 174 407 353)(159 253 425 175 408 354)(160 254 426 176 409 355)(161 255 427 177 410 356)(162 256 428 178 411 357)
(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)| (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,55)(17,56)(18,57)(19,412)(20,413)(21,414)(22,397)(23,398)(24,399)(25,400)(26,401)(27,402)(28,403)(29,404)(30,405)(31,406)(32,407)(33,408)(34,409)(35,410)(36,411)(37,428)(38,429)(39,430)(40,431)(41,432)(42,415)(43,416)(44,417)(45,418)(46,419)(47,420)(48,421)(49,422)(50,423)(51,424)(52,425)(53,426)(54,427)(73,291)(74,292)(75,293)(76,294)(77,295)(78,296)(79,297)(80,298)(81,299)(82,300)(83,301)(84,302)(85,303)(86,304)(87,305)(88,306)(89,289)(90,290)(91,126)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(127,166)(128,167)(129,168)(130,169)(131,170)(132,171)(133,172)(134,173)(135,174)(136,175)(137,176)(138,177)(139,178)(140,179)(141,180)(142,163)(143,164)(144,165)(145,215)(146,216)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208)(157,209)(158,210)(159,211)(160,212)(161,213)(162,214)(181,350)(182,351)(183,352)(184,353)(185,354)(186,355)(187,356)(188,357)(189,358)(190,359)(191,360)(192,343)(193,344)(194,345)(195,346)(196,347)(197,348)(198,349)(217,392)(218,393)(219,394)(220,395)(221,396)(222,379)(223,380)(224,381)(225,382)(226,383)(227,384)(228,385)(229,386)(230,387)(231,388)(232,389)(233,390)(234,391)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,307)(242,308)(243,309)(244,310)(245,311)(246,312)(247,313)(248,314)(249,315)(250,316)(251,317)(252,318)(253,375)(254,376)(255,377)(256,378)(257,361)(258,362)(259,363)(260,364)(261,365)(262,366)(263,367)(264,368)(265,369)(266,370)(267,371)(268,372)(269,373)(270,374)(271,337)(272,338)(273,339)(274,340)(275,341)(276,342)(277,325)(278,326)(279,327)(280,328)(281,329)(282,330)(283,331)(284,332)(285,333)(286,334)(287,335)(288,336), (1,409)(2,410)(3,411)(4,412)(5,413)(6,414)(7,397)(8,398)(9,399)(10,400)(11,401)(12,402)(13,403)(14,404)(15,405)(16,406)(17,407)(18,408)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,392)(38,393)(39,394)(40,395)(41,396)(42,379)(43,380)(44,381)(45,382)(46,383)(47,384)(48,385)(49,386)(50,387)(51,388)(52,389)(53,390)(54,391)(73,352)(74,353)(75,354)(76,355)(77,356)(78,357)(79,358)(80,359)(81,360)(82,343)(83,344)(84,345)(85,346)(86,347)(87,348)(88,349)(89,350)(90,351)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,169)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(121,144)(122,127)(123,128)(124,129)(125,130)(126,131)(145,244)(146,245)(147,246)(148,247)(149,248)(150,249)(151,250)(152,251)(153,252)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)(162,243)(181,289)(182,290)(183,291)(184,292)(185,293)(186,294)(187,295)(188,296)(189,297)(190,298)(191,299)(192,300)(193,301)(194,302)(195,303)(196,304)(197,305)(198,306)(199,312)(200,313)(201,314)(202,315)(203,316)(204,317)(205,318)(206,319)(207,320)(208,321)(209,322)(210,323)(211,324)(212,307)(213,308)(214,309)(215,310)(216,311)(217,428)(218,429)(219,430)(220,431)(221,432)(222,415)(223,416)(224,417)(225,418)(226,419)(227,420)(228,421)(229,422)(230,423)(231,424)(232,425)(233,426)(234,427)(253,272)(254,273)(255,274)(256,275)(257,276)(258,277)(259,278)(260,279)(261,280)(262,281)(263,282)(264,283)(265,284)(266,285)(267,286)(268,287)(269,288)(270,271)(325,362)(326,363)(327,364)(328,365)(329,366)(330,367)(331,368)(332,369)(333,370)(334,371)(335,372)(336,373)(337,374)(338,375)(339,376)(340,377)(341,378)(342,361), (1,76,241,273,233,97)(2,77,242,274,234,98)(3,78,243,275,217,99)(4,79,244,276,218,100)(5,80,245,277,219,101)(6,81,246,278,220,102)(7,82,247,279,221,103)(8,83,248,280,222,104)(9,84,249,281,223,105)(10,85,250,282,224,106)(11,86,251,283,225,107)(12,87,252,284,226,108)(13,88,235,285,227,91)(14,89,236,286,228,92)(15,90,237,287,229,93)(16,73,238,288,230,94)(17,74,239,271,231,95)(18,75,240,272,232,96)(19,189,215,361,38,140)(20,190,216,362,39,141)(21,191,199,363,40,142)(22,192,200,364,41,143)(23,193,201,365,42,144)(24,194,202,366,43,127)(25,195,203,367,44,128)(26,196,204,368,45,129)(27,197,205,369,46,130)(28,198,206,370,47,131)(29,181,207,371,48,132)(30,182,208,372,49,133)(31,183,209,373,50,134)(32,184,210,374,51,135)(33,185,211,375,52,136)(34,186,212,376,53,137)(35,187,213,377,54,138)(36,188,214,378,37,139)(55,291,322,336,387,111)(56,292,323,337,388,112)(57,293,324,338,389,113)(58,294,307,339,390,114)(59,295,308,340,391,115)(60,296,309,341,392,116)(61,297,310,342,393,117)(62,298,311,325,394,118)(63,299,312,326,395,119)(64,300,313,327,396,120)(65,301,314,328,379,121)(66,302,315,329,380,122)(67,303,316,330,381,123)(68,304,317,331,382,124)(69,305,318,332,383,125)(70,306,319,333,384,126)(71,289,320,334,385,109)(72,290,321,335,386,110)(145,257,429,179,412,358)(146,258,430,180,413,359)(147,259,431,163,414,360)(148,260,432,164,397,343)(149,261,415,165,398,344)(150,262,416,166,399,345)(151,263,417,167,400,346)(152,264,418,168,401,347)(153,265,419,169,402,348)(154,266,420,170,403,349)(155,267,421,171,404,350)(156,268,422,172,405,351)(157,269,423,173,406,352)(158,270,424,174,407,353)(159,253,425,175,408,354)(160,254,426,176,409,355)(161,255,427,177,410,356)(162,256,428,178,411,357), (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( (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,55)(17,56)(18,57)(19,412)(20,413)(21,414)(22,397)(23,398)(24,399)(25,400)(26,401)(27,402)(28,403)(29,404)(30,405)(31,406)(32,407)(33,408)(34,409)(35,410)(36,411)(37,428)(38,429)(39,430)(40,431)(41,432)(42,415)(43,416)(44,417)(45,418)(46,419)(47,420)(48,421)(49,422)(50,423)(51,424)(52,425)(53,426)(54,427)(73,291)(74,292)(75,293)(76,294)(77,295)(78,296)(79,297)(80,298)(81,299)(82,300)(83,301)(84,302)(85,303)(86,304)(87,305)(88,306)(89,289)(90,290)(91,126)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(127,166)(128,167)(129,168)(130,169)(131,170)(132,171)(133,172)(134,173)(135,174)(136,175)(137,176)(138,177)(139,178)(140,179)(141,180)(142,163)(143,164)(144,165)(145,215)(146,216)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208)(157,209)(158,210)(159,211)(160,212)(161,213)(162,214)(181,350)(182,351)(183,352)(184,353)(185,354)(186,355)(187,356)(188,357)(189,358)(190,359)(191,360)(192,343)(193,344)(194,345)(195,346)(196,347)(197,348)(198,349)(217,392)(218,393)(219,394)(220,395)(221,396)(222,379)(223,380)(224,381)(225,382)(226,383)(227,384)(228,385)(229,386)(230,387)(231,388)(232,389)(233,390)(234,391)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,307)(242,308)(243,309)(244,310)(245,311)(246,312)(247,313)(248,314)(249,315)(250,316)(251,317)(252,318)(253,375)(254,376)(255,377)(256,378)(257,361)(258,362)(259,363)(260,364)(261,365)(262,366)(263,367)(264,368)(265,369)(266,370)(267,371)(268,372)(269,373)(270,374)(271,337)(272,338)(273,339)(274,340)(275,341)(276,342)(277,325)(278,326)(279,327)(280,328)(281,329)(282,330)(283,331)(284,332)(285,333)(286,334)(287,335)(288,336), (1,409)(2,410)(3,411)(4,412)(5,413)(6,414)(7,397)(8,398)(9,399)(10,400)(11,401)(12,402)(13,403)(14,404)(15,405)(16,406)(17,407)(18,408)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,392)(38,393)(39,394)(40,395)(41,396)(42,379)(43,380)(44,381)(45,382)(46,383)(47,384)(48,385)(49,386)(50,387)(51,388)(52,389)(53,390)(54,391)(73,352)(74,353)(75,354)(76,355)(77,356)(78,357)(79,358)(80,359)(81,360)(82,343)(83,344)(84,345)(85,346)(86,347)(87,348)(88,349)(89,350)(90,351)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,169)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(121,144)(122,127)(123,128)(124,129)(125,130)(126,131)(145,244)(146,245)(147,246)(148,247)(149,248)(150,249)(151,250)(152,251)(153,252)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)(162,243)(181,289)(182,290)(183,291)(184,292)(185,293)(186,294)(187,295)(188,296)(189,297)(190,298)(191,299)(192,300)(193,301)(194,302)(195,303)(196,304)(197,305)(198,306)(199,312)(200,313)(201,314)(202,315)(203,316)(204,317)(205,318)(206,319)(207,320)(208,321)(209,322)(210,323)(211,324)(212,307)(213,308)(214,309)(215,310)(216,311)(217,428)(218,429)(219,430)(220,431)(221,432)(222,415)(223,416)(224,417)(225,418)(226,419)(227,420)(228,421)(229,422)(230,423)(231,424)(232,425)(233,426)(234,427)(253,272)(254,273)(255,274)(256,275)(257,276)(258,277)(259,278)(260,279)(261,280)(262,281)(263,282)(264,283)(265,284)(266,285)(267,286)(268,287)(269,288)(270,271)(325,362)(326,363)(327,364)(328,365)(329,366)(330,367)(331,368)(332,369)(333,370)(334,371)(335,372)(336,373)(337,374)(338,375)(339,376)(340,377)(341,378)(342,361), (1,76,241,273,233,97)(2,77,242,274,234,98)(3,78,243,275,217,99)(4,79,244,276,218,100)(5,80,245,277,219,101)(6,81,246,278,220,102)(7,82,247,279,221,103)(8,83,248,280,222,104)(9,84,249,281,223,105)(10,85,250,282,224,106)(11,86,251,283,225,107)(12,87,252,284,226,108)(13,88,235,285,227,91)(14,89,236,286,228,92)(15,90,237,287,229,93)(16,73,238,288,230,94)(17,74,239,271,231,95)(18,75,240,272,232,96)(19,189,215,361,38,140)(20,190,216,362,39,141)(21,191,199,363,40,142)(22,192,200,364,41,143)(23,193,201,365,42,144)(24,194,202,366,43,127)(25,195,203,367,44,128)(26,196,204,368,45,129)(27,197,205,369,46,130)(28,198,206,370,47,131)(29,181,207,371,48,132)(30,182,208,372,49,133)(31,183,209,373,50,134)(32,184,210,374,51,135)(33,185,211,375,52,136)(34,186,212,376,53,137)(35,187,213,377,54,138)(36,188,214,378,37,139)(55,291,322,336,387,111)(56,292,323,337,388,112)(57,293,324,338,389,113)(58,294,307,339,390,114)(59,295,308,340,391,115)(60,296,309,341,392,116)(61,297,310,342,393,117)(62,298,311,325,394,118)(63,299,312,326,395,119)(64,300,313,327,396,120)(65,301,314,328,379,121)(66,302,315,329,380,122)(67,303,316,330,381,123)(68,304,317,331,382,124)(69,305,318,332,383,125)(70,306,319,333,384,126)(71,289,320,334,385,109)(72,290,321,335,386,110)(145,257,429,179,412,358)(146,258,430,180,413,359)(147,259,431,163,414,360)(148,260,432,164,397,343)(149,261,415,165,398,344)(150,262,416,166,399,345)(151,263,417,167,400,346)(152,264,418,168,401,347)(153,265,419,169,402,348)(154,266,420,170,403,349)(155,267,421,171,404,350)(156,268,422,172,405,351)(157,269,423,173,406,352)(158,270,424,174,407,353)(159,253,425,175,408,354)(160,254,426,176,409,355)(161,255,427,177,410,356)(162,256,428,178,411,357), (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([[(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,55),(17,56),(18,57),(19,412),(20,413),(21,414),(22,397),(23,398),(24,399),(25,400),(26,401),(27,402),(28,403),(29,404),(30,405),(31,406),(32,407),(33,408),(34,409),(35,410),(36,411),(37,428),(38,429),(39,430),(40,431),(41,432),(42,415),(43,416),(44,417),(45,418),(46,419),(47,420),(48,421),(49,422),(50,423),(51,424),(52,425),(53,426),(54,427),(73,291),(74,292),(75,293),(76,294),(77,295),(78,296),(79,297),(80,298),(81,299),(82,300),(83,301),(84,302),(85,303),(86,304),(87,305),(88,306),(89,289),(90,290),(91,126),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,121),(105,122),(106,123),(107,124),(108,125),(127,166),(128,167),(129,168),(130,169),(131,170),(132,171),(133,172),(134,173),(135,174),(136,175),(137,176),(138,177),(139,178),(140,179),(141,180),(142,163),(143,164),(144,165),(145,215),(146,216),(147,199),(148,200),(149,201),(150,202),(151,203),(152,204),(153,205),(154,206),(155,207),(156,208),(157,209),(158,210),(159,211),(160,212),(161,213),(162,214),(181,350),(182,351),(183,352),(184,353),(185,354),(186,355),(187,356),(188,357),(189,358),(190,359),(191,360),(192,343),(193,344),(194,345),(195,346),(196,347),(197,348),(198,349),(217,392),(218,393),(219,394),(220,395),(221,396),(222,379),(223,380),(224,381),(225,382),(226,383),(227,384),(228,385),(229,386),(230,387),(231,388),(232,389),(233,390),(234,391),(235,319),(236,320),(237,321),(238,322),(239,323),(240,324),(241,307),(242,308),(243,309),(244,310),(245,311),(246,312),(247,313),(248,314),(249,315),(250,316),(251,317),(252,318),(253,375),(254,376),(255,377),(256,378),(257,361),(258,362),(259,363),(260,364),(261,365),(262,366),(263,367),(264,368),(265,369),(266,370),(267,371),(268,372),(269,373),(270,374),(271,337),(272,338),(273,339),(274,340),(275,341),(276,342),(277,325),(278,326),(279,327),(280,328),(281,329),(282,330),(283,331),(284,332),(285,333),(286,334),(287,335),(288,336)], [(1,409),(2,410),(3,411),(4,412),(5,413),(6,414),(7,397),(8,398),(9,399),(10,400),(11,401),(12,402),(13,403),(14,404),(15,405),(16,406),(17,407),(18,408),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,392),(38,393),(39,394),(40,395),(41,396),(42,379),(43,380),(44,381),(45,382),(46,383),(47,384),(48,385),(49,386),(50,387),(51,388),(52,389),(53,390),(54,391),(73,352),(74,353),(75,354),(76,355),(77,356),(78,357),(79,358),(80,359),(81,360),(82,343),(83,344),(84,345),(85,346),(86,347),(87,348),(88,349),(89,350),(90,351),(91,170),(92,171),(93,172),(94,173),(95,174),(96,175),(97,176),(98,177),(99,178),(100,179),(101,180),(102,163),(103,164),(104,165),(105,166),(106,167),(107,168),(108,169),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,141),(119,142),(120,143),(121,144),(122,127),(123,128),(124,129),(125,130),(126,131),(145,244),(146,245),(147,246),(148,247),(149,248),(150,249),(151,250),(152,251),(153,252),(154,235),(155,236),(156,237),(157,238),(158,239),(159,240),(160,241),(161,242),(162,243),(181,289),(182,290),(183,291),(184,292),(185,293),(186,294),(187,295),(188,296),(189,297),(190,298),(191,299),(192,300),(193,301),(194,302),(195,303),(196,304),(197,305),(198,306),(199,312),(200,313),(201,314),(202,315),(203,316),(204,317),(205,318),(206,319),(207,320),(208,321),(209,322),(210,323),(211,324),(212,307),(213,308),(214,309),(215,310),(216,311),(217,428),(218,429),(219,430),(220,431),(221,432),(222,415),(223,416),(224,417),(225,418),(226,419),(227,420),(228,421),(229,422),(230,423),(231,424),(232,425),(233,426),(234,427),(253,272),(254,273),(255,274),(256,275),(257,276),(258,277),(259,278),(260,279),(261,280),(262,281),(263,282),(264,283),(265,284),(266,285),(267,286),(268,287),(269,288),(270,271),(325,362),(326,363),(327,364),(328,365),(329,366),(330,367),(331,368),(332,369),(333,370),(334,371),(335,372),(336,373),(337,374),(338,375),(339,376),(340,377),(341,378),(342,361)], [(1,76,241,273,233,97),(2,77,242,274,234,98),(3,78,243,275,217,99),(4,79,244,276,218,100),(5,80,245,277,219,101),(6,81,246,278,220,102),(7,82,247,279,221,103),(8,83,248,280,222,104),(9,84,249,281,223,105),(10,85,250,282,224,106),(11,86,251,283,225,107),(12,87,252,284,226,108),(13,88,235,285,227,91),(14,89,236,286,228,92),(15,90,237,287,229,93),(16,73,238,288,230,94),(17,74,239,271,231,95),(18,75,240,272,232,96),(19,189,215,361,38,140),(20,190,216,362,39,141),(21,191,199,363,40,142),(22,192,200,364,41,143),(23,193,201,365,42,144),(24,194,202,366,43,127),(25,195,203,367,44,128),(26,196,204,368,45,129),(27,197,205,369,46,130),(28,198,206,370,47,131),(29,181,207,371,48,132),(30,182,208,372,49,133),(31,183,209,373,50,134),(32,184,210,374,51,135),(33,185,211,375,52,136),(34,186,212,376,53,137),(35,187,213,377,54,138),(36,188,214,378,37,139),(55,291,322,336,387,111),(56,292,323,337,388,112),(57,293,324,338,389,113),(58,294,307,339,390,114),(59,295,308,340,391,115),(60,296,309,341,392,116),(61,297,310,342,393,117),(62,298,311,325,394,118),(63,299,312,326,395,119),(64,300,313,327,396,120),(65,301,314,328,379,121),(66,302,315,329,380,122),(67,303,316,330,381,123),(68,304,317,331,382,124),(69,305,318,332,383,125),(70,306,319,333,384,126),(71,289,320,334,385,109),(72,290,321,335,386,110),(145,257,429,179,412,358),(146,258,430,180,413,359),(147,259,431,163,414,360),(148,260,432,164,397,343),(149,261,415,165,398,344),(150,262,416,166,399,345),(151,263,417,167,400,346),(152,264,418,168,401,347),(153,265,419,169,402,348),(154,266,420,170,403,349),(155,267,421,171,404,350),(156,268,422,172,405,351),(157,269,423,173,406,352),(158,270,424,174,407,353),(159,253,425,175,408,354),(160,254,426,176,409,355),(161,255,427,177,410,356),(162,256,428,178,411,357)], [(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 ··· 2O 3A ··· 3H 6A ··· 6DP 9A ··· 9R 18A ··· 18JJ order 1 2 ··· 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

432 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C3 C6 C6 C9 C18 kernel C22×C6×C18 C2×C6×C18 C23×C18 C22×C62 C22×C18 C2×C62 C23×C6 C22×C6 # reps 1 15 6 2 90 30 18 270

Matrix representation of C22×C6×C18 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 1
,
 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 18
,
 7 0 0 0 0 11 0 0 0 0 11 0 0 0 0 8
,
 9 0 0 0 0 2 0 0 0 0 17 0 0 0 0 1
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,1],[18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,18],[7,0,0,0,0,11,0,0,0,0,11,0,0,0,0,8],[9,0,0,0,0,2,0,0,0,0,17,0,0,0,0,1] >;

C22×C6×C18 in GAP, Magma, Sage, TeX

C_2^2\times C_6\times C_{18}
% in TeX

G:=Group("C2^2xC6xC18");
// GroupNames label

G:=SmallGroup(432,562);
// by ID

G=gap.SmallGroup(432,562);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,347]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^6=d^18=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