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## G = C22×C106order 424 = 23·53

### Abelian group of type [2,2,106]

Aliases: C22×C106, SmallGroup(424,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C106
 Chief series C1 — C53 — C106 — C2×C106 — C22×C106
 Lower central C1 — C22×C106
 Upper central C1 — C22×C106

Generators and relations for C22×C106
G = < a,b,c | a2=b2=c106=1, ab=ba, ac=ca, bc=cb >

Smallest permutation representation of C22×C106
Regular action on 424 points
Generators in S424
(1 309)(2 310)(3 311)(4 312)(5 313)(6 314)(7 315)(8 316)(9 317)(10 318)(11 213)(12 214)(13 215)(14 216)(15 217)(16 218)(17 219)(18 220)(19 221)(20 222)(21 223)(22 224)(23 225)(24 226)(25 227)(26 228)(27 229)(28 230)(29 231)(30 232)(31 233)(32 234)(33 235)(34 236)(35 237)(36 238)(37 239)(38 240)(39 241)(40 242)(41 243)(42 244)(43 245)(44 246)(45 247)(46 248)(47 249)(48 250)(49 251)(50 252)(51 253)(52 254)(53 255)(54 256)(55 257)(56 258)(57 259)(58 260)(59 261)(60 262)(61 263)(62 264)(63 265)(64 266)(65 267)(66 268)(67 269)(68 270)(69 271)(70 272)(71 273)(72 274)(73 275)(74 276)(75 277)(76 278)(77 279)(78 280)(79 281)(80 282)(81 283)(82 284)(83 285)(84 286)(85 287)(86 288)(87 289)(88 290)(89 291)(90 292)(91 293)(92 294)(93 295)(94 296)(95 297)(96 298)(97 299)(98 300)(99 301)(100 302)(101 303)(102 304)(103 305)(104 306)(105 307)(106 308)(107 416)(108 417)(109 418)(110 419)(111 420)(112 421)(113 422)(114 423)(115 424)(116 319)(117 320)(118 321)(119 322)(120 323)(121 324)(122 325)(123 326)(124 327)(125 328)(126 329)(127 330)(128 331)(129 332)(130 333)(131 334)(132 335)(133 336)(134 337)(135 338)(136 339)(137 340)(138 341)(139 342)(140 343)(141 344)(142 345)(143 346)(144 347)(145 348)(146 349)(147 350)(148 351)(149 352)(150 353)(151 354)(152 355)(153 356)(154 357)(155 358)(156 359)(157 360)(158 361)(159 362)(160 363)(161 364)(162 365)(163 366)(164 367)(165 368)(166 369)(167 370)(168 371)(169 372)(170 373)(171 374)(172 375)(173 376)(174 377)(175 378)(176 379)(177 380)(178 381)(179 382)(180 383)(181 384)(182 385)(183 386)(184 387)(185 388)(186 389)(187 390)(188 391)(189 392)(190 393)(191 394)(192 395)(193 396)(194 397)(195 398)(196 399)(197 400)(198 401)(199 402)(200 403)(201 404)(202 405)(203 406)(204 407)(205 408)(206 409)(207 410)(208 411)(209 412)(210 413)(211 414)(212 415)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 181)(7 182)(8 183)(9 184)(10 185)(11 186)(12 187)(13 188)(14 189)(15 190)(16 191)(17 192)(18 193)(19 194)(20 195)(21 196)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 204)(30 205)(31 206)(32 207)(33 208)(34 209)(35 210)(36 211)(37 212)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 161)(93 162)(94 163)(95 164)(96 165)(97 166)(98 167)(99 168)(100 169)(101 170)(102 171)(103 172)(104 173)(105 174)(106 175)(213 389)(214 390)(215 391)(216 392)(217 393)(218 394)(219 395)(220 396)(221 397)(222 398)(223 399)(224 400)(225 401)(226 402)(227 403)(228 404)(229 405)(230 406)(231 407)(232 408)(233 409)(234 410)(235 411)(236 412)(237 413)(238 414)(239 415)(240 416)(241 417)(242 418)(243 419)(244 420)(245 421)(246 422)(247 423)(248 424)(249 319)(250 320)(251 321)(252 322)(253 323)(254 324)(255 325)(256 326)(257 327)(258 328)(259 329)(260 330)(261 331)(262 332)(263 333)(264 334)(265 335)(266 336)(267 337)(268 338)(269 339)(270 340)(271 341)(272 342)(273 343)(274 344)(275 345)(276 346)(277 347)(278 348)(279 349)(280 350)(281 351)(282 352)(283 353)(284 354)(285 355)(286 356)(287 357)(288 358)(289 359)(290 360)(291 361)(292 362)(293 363)(294 364)(295 365)(296 366)(297 367)(298 368)(299 369)(300 370)(301 371)(302 372)(303 373)(304 374)(305 375)(306 376)(307 377)(308 378)(309 379)(310 380)(311 381)(312 382)(313 383)(314 384)(315 385)(316 386)(317 387)(318 388)
(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)

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

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

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

424 conjugacy classes

 class 1 2A ··· 2G 53A ··· 53AZ 106A ··· 106MZ order 1 2 ··· 2 53 ··· 53 106 ··· 106 size 1 1 ··· 1 1 ··· 1 1 ··· 1

424 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C53 C106 kernel C22×C106 C2×C106 C23 C22 # reps 1 7 52 364

Matrix representation of C22×C106 in GL3(𝔽107) generated by

 1 0 0 0 106 0 0 0 106
,
 1 0 0 0 1 0 0 0 106
,
 67 0 0 0 22 0 0 0 52
G:=sub<GL(3,GF(107))| [1,0,0,0,106,0,0,0,106],[1,0,0,0,1,0,0,0,106],[67,0,0,0,22,0,0,0,52] >;

C22×C106 in GAP, Magma, Sage, TeX

C_2^2\times C_{106}
% in TeX

G:=Group("C2^2xC106");
// GroupNames label

G:=SmallGroup(424,14);
// by ID

G=gap.SmallGroup(424,14);
# by ID

G:=PCGroup([4,-2,-2,-2,-53]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^106=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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