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## G = C22×C108order 432 = 24·33

### Abelian group of type [2,2,108]

Aliases: C22×C108, SmallGroup(432,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C108
 Chief series C1 — C3 — C9 — C18 — C54 — C108 — C2×C108 — C22×C108
 Lower central C1 — C22×C108
 Upper central 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, C3, C4, C22, C6, C6, C2×C4, C23, C9, C12, C2×C6, C22×C4, C18, C18, C2×C12, C22×C6, C27, C36, C2×C18, C22×C12, C54, C54, C2×C36, C22×C18, C108, C2×C54, C22×C36, C2×C108, C22×C54, C22×C108
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C12, C2×C6, C22×C4, C18, C2×C12, C22×C6, C27, C36, C2×C18, C22×C12, C54, C2×C36, C22×C18, C108, C2×C54, C22×C36, C2×C108, C22×C54, C22×C108

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

432 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C27 C36 C54 C54 C108 kernel C22×C108 C2×C108 C22×C54 C22×C36 C2×C54 C2×C36 C22×C18 C22×C12 C2×C18 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 6 1 2 8 12 2 6 16 36 6 18 48 108 18 144

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

 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
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

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