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G = C22×C108order 432 = 24·33

Abelian group of type [2,2,108]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C108, SmallGroup(432,53)

Series: Derived Chief Lower central Upper central

C1 — C22×C108
C1C3C9C18C54C108C2×C108 — C22×C108
C1 — C22×C108
C1 — C22×C108

Generators and relations for C22×C108
 G = < a,b,c | a2=b2=c108=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (16 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C6, C6 [×6], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C18, C18 [×6], C2×C12 [×6], C22×C6, C27, C36 [×4], C2×C18 [×7], C22×C12, C54, C54 [×6], C2×C36 [×6], C22×C18, C108 [×4], C2×C54 [×7], C22×C36, C2×C108 [×6], C22×C54, C22×C108
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C18 [×7], C2×C12 [×6], C22×C6, C27, C36 [×4], C2×C18 [×7], C22×C12, C54 [×7], C2×C36 [×6], C22×C18, C108 [×4], C2×C54 [×7], C22×C36, C2×C108 [×6], C22×C54, C22×C108

Smallest permutation representation of C22×C108
Regular action on 432 points
Generators in S432
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 129)(17 130)(18 131)(19 132)(20 133)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(41 154)(42 155)(43 156)(44 157)(45 158)(46 159)(47 160)(48 161)(49 162)(50 163)(51 164)(52 165)(53 166)(54 167)(55 168)(56 169)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 181)(69 182)(70 183)(71 184)(72 185)(73 186)(74 187)(75 188)(76 189)(77 190)(78 191)(79 192)(80 193)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 202)(90 203)(91 204)(92 205)(93 206)(94 207)(95 208)(96 209)(97 210)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 109)(105 110)(106 111)(107 112)(108 113)(217 432)(218 325)(219 326)(220 327)(221 328)(222 329)(223 330)(224 331)(225 332)(226 333)(227 334)(228 335)(229 336)(230 337)(231 338)(232 339)(233 340)(234 341)(235 342)(236 343)(237 344)(238 345)(239 346)(240 347)(241 348)(242 349)(243 350)(244 351)(245 352)(246 353)(247 354)(248 355)(249 356)(250 357)(251 358)(252 359)(253 360)(254 361)(255 362)(256 363)(257 364)(258 365)(259 366)(260 367)(261 368)(262 369)(263 370)(264 371)(265 372)(266 373)(267 374)(268 375)(269 376)(270 377)(271 378)(272 379)(273 380)(274 381)(275 382)(276 383)(277 384)(278 385)(279 386)(280 387)(281 388)(282 389)(283 390)(284 391)(285 392)(286 393)(287 394)(288 395)(289 396)(290 397)(291 398)(292 399)(293 400)(294 401)(295 402)(296 403)(297 404)(298 405)(299 406)(300 407)(301 408)(302 409)(303 410)(304 411)(305 412)(306 413)(307 414)(308 415)(309 416)(310 417)(311 418)(312 419)(313 420)(314 421)(315 422)(316 423)(317 424)(318 425)(319 426)(320 427)(321 428)(322 429)(323 430)(324 431)
(1 291)(2 292)(3 293)(4 294)(5 295)(6 296)(7 297)(8 298)(9 299)(10 300)(11 301)(12 302)(13 303)(14 304)(15 305)(16 306)(17 307)(18 308)(19 309)(20 310)(21 311)(22 312)(23 313)(24 314)(25 315)(26 316)(27 317)(28 318)(29 319)(30 320)(31 321)(32 322)(33 323)(34 324)(35 217)(36 218)(37 219)(38 220)(39 221)(40 222)(41 223)(42 224)(43 225)(44 226)(45 227)(46 228)(47 229)(48 230)(49 231)(50 232)(51 233)(52 234)(53 235)(54 236)(55 237)(56 238)(57 239)(58 240)(59 241)(60 242)(61 243)(62 244)(63 245)(64 246)(65 247)(66 248)(67 249)(68 250)(69 251)(70 252)(71 253)(72 254)(73 255)(74 256)(75 257)(76 258)(77 259)(78 260)(79 261)(80 262)(81 263)(82 264)(83 265)(84 266)(85 267)(86 268)(87 269)(88 270)(89 271)(90 272)(91 273)(92 274)(93 275)(94 276)(95 277)(96 278)(97 279)(98 280)(99 281)(100 282)(101 283)(102 284)(103 285)(104 286)(105 287)(106 288)(107 289)(108 290)(109 393)(110 394)(111 395)(112 396)(113 397)(114 398)(115 399)(116 400)(117 401)(118 402)(119 403)(120 404)(121 405)(122 406)(123 407)(124 408)(125 409)(126 410)(127 411)(128 412)(129 413)(130 414)(131 415)(132 416)(133 417)(134 418)(135 419)(136 420)(137 421)(138 422)(139 423)(140 424)(141 425)(142 426)(143 427)(144 428)(145 429)(146 430)(147 431)(148 432)(149 325)(150 326)(151 327)(152 328)(153 329)(154 330)(155 331)(156 332)(157 333)(158 334)(159 335)(160 336)(161 337)(162 338)(163 339)(164 340)(165 341)(166 342)(167 343)(168 344)(169 345)(170 346)(171 347)(172 348)(173 349)(174 350)(175 351)(176 352)(177 353)(178 354)(179 355)(180 356)(181 357)(182 358)(183 359)(184 360)(185 361)(186 362)(187 363)(188 364)(189 365)(190 366)(191 367)(192 368)(193 369)(194 370)(195 371)(196 372)(197 373)(198 374)(199 375)(200 376)(201 377)(202 378)(203 379)(204 380)(205 381)(206 382)(207 383)(208 384)(209 385)(210 386)(211 387)(212 388)(213 389)(214 390)(215 391)(216 392)
(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)| (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,161)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,171)(59,172)(60,173)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,109)(105,110)(106,111)(107,112)(108,113)(217,432)(218,325)(219,326)(220,327)(221,328)(222,329)(223,330)(224,331)(225,332)(226,333)(227,334)(228,335)(229,336)(230,337)(231,338)(232,339)(233,340)(234,341)(235,342)(236,343)(237,344)(238,345)(239,346)(240,347)(241,348)(242,349)(243,350)(244,351)(245,352)(246,353)(247,354)(248,355)(249,356)(250,357)(251,358)(252,359)(253,360)(254,361)(255,362)(256,363)(257,364)(258,365)(259,366)(260,367)(261,368)(262,369)(263,370)(264,371)(265,372)(266,373)(267,374)(268,375)(269,376)(270,377)(271,378)(272,379)(273,380)(274,381)(275,382)(276,383)(277,384)(278,385)(279,386)(280,387)(281,388)(282,389)(283,390)(284,391)(285,392)(286,393)(287,394)(288,395)(289,396)(290,397)(291,398)(292,399)(293,400)(294,401)(295,402)(296,403)(297,404)(298,405)(299,406)(300,407)(301,408)(302,409)(303,410)(304,411)(305,412)(306,413)(307,414)(308,415)(309,416)(310,417)(311,418)(312,419)(313,420)(314,421)(315,422)(316,423)(317,424)(318,425)(319,426)(320,427)(321,428)(322,429)(323,430)(324,431), (1,291)(2,292)(3,293)(4,294)(5,295)(6,296)(7,297)(8,298)(9,299)(10,300)(11,301)(12,302)(13,303)(14,304)(15,305)(16,306)(17,307)(18,308)(19,309)(20,310)(21,311)(22,312)(23,313)(24,314)(25,315)(26,316)(27,317)(28,318)(29,319)(30,320)(31,321)(32,322)(33,323)(34,324)(35,217)(36,218)(37,219)(38,220)(39,221)(40,222)(41,223)(42,224)(43,225)(44,226)(45,227)(46,228)(47,229)(48,230)(49,231)(50,232)(51,233)(52,234)(53,235)(54,236)(55,237)(56,238)(57,239)(58,240)(59,241)(60,242)(61,243)(62,244)(63,245)(64,246)(65,247)(66,248)(67,249)(68,250)(69,251)(70,252)(71,253)(72,254)(73,255)(74,256)(75,257)(76,258)(77,259)(78,260)(79,261)(80,262)(81,263)(82,264)(83,265)(84,266)(85,267)(86,268)(87,269)(88,270)(89,271)(90,272)(91,273)(92,274)(93,275)(94,276)(95,277)(96,278)(97,279)(98,280)(99,281)(100,282)(101,283)(102,284)(103,285)(104,286)(105,287)(106,288)(107,289)(108,290)(109,393)(110,394)(111,395)(112,396)(113,397)(114,398)(115,399)(116,400)(117,401)(118,402)(119,403)(120,404)(121,405)(122,406)(123,407)(124,408)(125,409)(126,410)(127,411)(128,412)(129,413)(130,414)(131,415)(132,416)(133,417)(134,418)(135,419)(136,420)(137,421)(138,422)(139,423)(140,424)(141,425)(142,426)(143,427)(144,428)(145,429)(146,430)(147,431)(148,432)(149,325)(150,326)(151,327)(152,328)(153,329)(154,330)(155,331)(156,332)(157,333)(158,334)(159,335)(160,336)(161,337)(162,338)(163,339)(164,340)(165,341)(166,342)(167,343)(168,344)(169,345)(170,346)(171,347)(172,348)(173,349)(174,350)(175,351)(176,352)(177,353)(178,354)(179,355)(180,356)(181,357)(182,358)(183,359)(184,360)(185,361)(186,362)(187,363)(188,364)(189,365)(190,366)(191,367)(192,368)(193,369)(194,370)(195,371)(196,372)(197,373)(198,374)(199,375)(200,376)(201,377)(202,378)(203,379)(204,380)(205,381)(206,382)(207,383)(208,384)(209,385)(210,386)(211,387)(212,388)(213,389)(214,390)(215,391)(216,392), (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( (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,161)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,171)(59,172)(60,173)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,109)(105,110)(106,111)(107,112)(108,113)(217,432)(218,325)(219,326)(220,327)(221,328)(222,329)(223,330)(224,331)(225,332)(226,333)(227,334)(228,335)(229,336)(230,337)(231,338)(232,339)(233,340)(234,341)(235,342)(236,343)(237,344)(238,345)(239,346)(240,347)(241,348)(242,349)(243,350)(244,351)(245,352)(246,353)(247,354)(248,355)(249,356)(250,357)(251,358)(252,359)(253,360)(254,361)(255,362)(256,363)(257,364)(258,365)(259,366)(260,367)(261,368)(262,369)(263,370)(264,371)(265,372)(266,373)(267,374)(268,375)(269,376)(270,377)(271,378)(272,379)(273,380)(274,381)(275,382)(276,383)(277,384)(278,385)(279,386)(280,387)(281,388)(282,389)(283,390)(284,391)(285,392)(286,393)(287,394)(288,395)(289,396)(290,397)(291,398)(292,399)(293,400)(294,401)(295,402)(296,403)(297,404)(298,405)(299,406)(300,407)(301,408)(302,409)(303,410)(304,411)(305,412)(306,413)(307,414)(308,415)(309,416)(310,417)(311,418)(312,419)(313,420)(314,421)(315,422)(316,423)(317,424)(318,425)(319,426)(320,427)(321,428)(322,429)(323,430)(324,431), (1,291)(2,292)(3,293)(4,294)(5,295)(6,296)(7,297)(8,298)(9,299)(10,300)(11,301)(12,302)(13,303)(14,304)(15,305)(16,306)(17,307)(18,308)(19,309)(20,310)(21,311)(22,312)(23,313)(24,314)(25,315)(26,316)(27,317)(28,318)(29,319)(30,320)(31,321)(32,322)(33,323)(34,324)(35,217)(36,218)(37,219)(38,220)(39,221)(40,222)(41,223)(42,224)(43,225)(44,226)(45,227)(46,228)(47,229)(48,230)(49,231)(50,232)(51,233)(52,234)(53,235)(54,236)(55,237)(56,238)(57,239)(58,240)(59,241)(60,242)(61,243)(62,244)(63,245)(64,246)(65,247)(66,248)(67,249)(68,250)(69,251)(70,252)(71,253)(72,254)(73,255)(74,256)(75,257)(76,258)(77,259)(78,260)(79,261)(80,262)(81,263)(82,264)(83,265)(84,266)(85,267)(86,268)(87,269)(88,270)(89,271)(90,272)(91,273)(92,274)(93,275)(94,276)(95,277)(96,278)(97,279)(98,280)(99,281)(100,282)(101,283)(102,284)(103,285)(104,286)(105,287)(106,288)(107,289)(108,290)(109,393)(110,394)(111,395)(112,396)(113,397)(114,398)(115,399)(116,400)(117,401)(118,402)(119,403)(120,404)(121,405)(122,406)(123,407)(124,408)(125,409)(126,410)(127,411)(128,412)(129,413)(130,414)(131,415)(132,416)(133,417)(134,418)(135,419)(136,420)(137,421)(138,422)(139,423)(140,424)(141,425)(142,426)(143,427)(144,428)(145,429)(146,430)(147,431)(148,432)(149,325)(150,326)(151,327)(152,328)(153,329)(154,330)(155,331)(156,332)(157,333)(158,334)(159,335)(160,336)(161,337)(162,338)(163,339)(164,340)(165,341)(166,342)(167,343)(168,344)(169,345)(170,346)(171,347)(172,348)(173,349)(174,350)(175,351)(176,352)(177,353)(178,354)(179,355)(180,356)(181,357)(182,358)(183,359)(184,360)(185,361)(186,362)(187,363)(188,364)(189,365)(190,366)(191,367)(192,368)(193,369)(194,370)(195,371)(196,372)(197,373)(198,374)(199,375)(200,376)(201,377)(202,378)(203,379)(204,380)(205,381)(206,382)(207,383)(208,384)(209,385)(210,386)(211,387)(212,388)(213,389)(214,390)(215,391)(216,392), (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([(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,129),(17,130),(18,131),(19,132),(20,133),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,148),(36,149),(37,150),(38,151),(39,152),(40,153),(41,154),(42,155),(43,156),(44,157),(45,158),(46,159),(47,160),(48,161),(49,162),(50,163),(51,164),(52,165),(53,166),(54,167),(55,168),(56,169),(57,170),(58,171),(59,172),(60,173),(61,174),(62,175),(63,176),(64,177),(65,178),(66,179),(67,180),(68,181),(69,182),(70,183),(71,184),(72,185),(73,186),(74,187),(75,188),(76,189),(77,190),(78,191),(79,192),(80,193),(81,194),(82,195),(83,196),(84,197),(85,198),(86,199),(87,200),(88,201),(89,202),(90,203),(91,204),(92,205),(93,206),(94,207),(95,208),(96,209),(97,210),(98,211),(99,212),(100,213),(101,214),(102,215),(103,216),(104,109),(105,110),(106,111),(107,112),(108,113),(217,432),(218,325),(219,326),(220,327),(221,328),(222,329),(223,330),(224,331),(225,332),(226,333),(227,334),(228,335),(229,336),(230,337),(231,338),(232,339),(233,340),(234,341),(235,342),(236,343),(237,344),(238,345),(239,346),(240,347),(241,348),(242,349),(243,350),(244,351),(245,352),(246,353),(247,354),(248,355),(249,356),(250,357),(251,358),(252,359),(253,360),(254,361),(255,362),(256,363),(257,364),(258,365),(259,366),(260,367),(261,368),(262,369),(263,370),(264,371),(265,372),(266,373),(267,374),(268,375),(269,376),(270,377),(271,378),(272,379),(273,380),(274,381),(275,382),(276,383),(277,384),(278,385),(279,386),(280,387),(281,388),(282,389),(283,390),(284,391),(285,392),(286,393),(287,394),(288,395),(289,396),(290,397),(291,398),(292,399),(293,400),(294,401),(295,402),(296,403),(297,404),(298,405),(299,406),(300,407),(301,408),(302,409),(303,410),(304,411),(305,412),(306,413),(307,414),(308,415),(309,416),(310,417),(311,418),(312,419),(313,420),(314,421),(315,422),(316,423),(317,424),(318,425),(319,426),(320,427),(321,428),(322,429),(323,430),(324,431)], [(1,291),(2,292),(3,293),(4,294),(5,295),(6,296),(7,297),(8,298),(9,299),(10,300),(11,301),(12,302),(13,303),(14,304),(15,305),(16,306),(17,307),(18,308),(19,309),(20,310),(21,311),(22,312),(23,313),(24,314),(25,315),(26,316),(27,317),(28,318),(29,319),(30,320),(31,321),(32,322),(33,323),(34,324),(35,217),(36,218),(37,219),(38,220),(39,221),(40,222),(41,223),(42,224),(43,225),(44,226),(45,227),(46,228),(47,229),(48,230),(49,231),(50,232),(51,233),(52,234),(53,235),(54,236),(55,237),(56,238),(57,239),(58,240),(59,241),(60,242),(61,243),(62,244),(63,245),(64,246),(65,247),(66,248),(67,249),(68,250),(69,251),(70,252),(71,253),(72,254),(73,255),(74,256),(75,257),(76,258),(77,259),(78,260),(79,261),(80,262),(81,263),(82,264),(83,265),(84,266),(85,267),(86,268),(87,269),(88,270),(89,271),(90,272),(91,273),(92,274),(93,275),(94,276),(95,277),(96,278),(97,279),(98,280),(99,281),(100,282),(101,283),(102,284),(103,285),(104,286),(105,287),(106,288),(107,289),(108,290),(109,393),(110,394),(111,395),(112,396),(113,397),(114,398),(115,399),(116,400),(117,401),(118,402),(119,403),(120,404),(121,405),(122,406),(123,407),(124,408),(125,409),(126,410),(127,411),(128,412),(129,413),(130,414),(131,415),(132,416),(133,417),(134,418),(135,419),(136,420),(137,421),(138,422),(139,423),(140,424),(141,425),(142,426),(143,427),(144,428),(145,429),(146,430),(147,431),(148,432),(149,325),(150,326),(151,327),(152,328),(153,329),(154,330),(155,331),(156,332),(157,333),(158,334),(159,335),(160,336),(161,337),(162,338),(163,339),(164,340),(165,341),(166,342),(167,343),(168,344),(169,345),(170,346),(171,347),(172,348),(173,349),(174,350),(175,351),(176,352),(177,353),(178,354),(179,355),(180,356),(181,357),(182,358),(183,359),(184,360),(185,361),(186,362),(187,363),(188,364),(189,365),(190,366),(191,367),(192,368),(193,369),(194,370),(195,371),(196,372),(197,373),(198,374),(199,375),(200,376),(201,377),(202,378),(203,379),(204,380),(205,381),(206,382),(207,383),(208,384),(209,385),(210,386),(211,387),(212,388),(213,389),(214,390),(215,391),(216,392)], [(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···2G3A3B4A···4H6A···6N9A···9F12A···12P18A···18AP27A···27R36A···36AV54A···54DV108A···108EN
order12···2334···46···69···912···1218···1827···2736···3654···54108···108
size11···1111···11···11···11···11···11···11···11···11···1

432 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C6C6C9C12C18C18C27C36C54C54C108
kernelC22×C108C2×C108C22×C54C22×C36C2×C54C2×C36C22×C18C22×C12C2×C18C2×C12C22×C6C22×C4C2×C6C2×C4C23C22
# reps16128122616366184810818144

Matrix representation of C22×C108 in GL3(𝔽109) generated by

10800
010
00108
,
10800
010
001
,
1500
0170
0054
G:=sub<GL(3,GF(109))| [108,0,0,0,1,0,0,0,108],[108,0,0,0,1,0,0,0,1],[15,0,0,0,17,0,0,0,54] >;

C22×C108 in GAP, Magma, Sage, TeX

C_2^2\times C_{108}
% in TeX

G:=Group("C2^2xC108");
// GroupNames label

G:=SmallGroup(432,53);
// by ID

G=gap.SmallGroup(432,53);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,192,166]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^108=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽