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## G = C22×C102order 408 = 23·3·17

### Abelian group of type [2,2,102]

Aliases: C22×C102, SmallGroup(408,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C102
 Chief series C1 — C17 — C51 — C102 — C2×C102 — C22×C102
 Lower central C1 — C22×C102
 Upper central C1 — C22×C102

Generators and relations for C22×C102
G = < a,b,c | a2=b2=c102=1, ab=ba, ac=ca, bc=cb >

Subgroups: 64, all normal (8 characteristic)
C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C17, C22×C6, C34 [×7], C51, C2×C34 [×7], C102 [×7], C22×C34, C2×C102 [×7], C22×C102
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C17, C22×C6, C34 [×7], C51, C2×C34 [×7], C102 [×7], C22×C34, C2×C102 [×7], C22×C102

Smallest permutation representation of C22×C102
Regular action on 408 points
Generators in S408
(1 272)(2 273)(3 274)(4 275)(5 276)(6 277)(7 278)(8 279)(9 280)(10 281)(11 282)(12 283)(13 284)(14 285)(15 286)(16 287)(17 288)(18 289)(19 290)(20 291)(21 292)(22 293)(23 294)(24 295)(25 296)(26 297)(27 298)(28 299)(29 300)(30 301)(31 302)(32 303)(33 304)(34 305)(35 306)(36 205)(37 206)(38 207)(39 208)(40 209)(41 210)(42 211)(43 212)(44 213)(45 214)(46 215)(47 216)(48 217)(49 218)(50 219)(51 220)(52 221)(53 222)(54 223)(55 224)(56 225)(57 226)(58 227)(59 228)(60 229)(61 230)(62 231)(63 232)(64 233)(65 234)(66 235)(67 236)(68 237)(69 238)(70 239)(71 240)(72 241)(73 242)(74 243)(75 244)(76 245)(77 246)(78 247)(79 248)(80 249)(81 250)(82 251)(83 252)(84 253)(85 254)(86 255)(87 256)(88 257)(89 258)(90 259)(91 260)(92 261)(93 262)(94 263)(95 264)(96 265)(97 266)(98 267)(99 268)(100 269)(101 270)(102 271)(103 307)(104 308)(105 309)(106 310)(107 311)(108 312)(109 313)(110 314)(111 315)(112 316)(113 317)(114 318)(115 319)(116 320)(117 321)(118 322)(119 323)(120 324)(121 325)(122 326)(123 327)(124 328)(125 329)(126 330)(127 331)(128 332)(129 333)(130 334)(131 335)(132 336)(133 337)(134 338)(135 339)(136 340)(137 341)(138 342)(139 343)(140 344)(141 345)(142 346)(143 347)(144 348)(145 349)(146 350)(147 351)(148 352)(149 353)(150 354)(151 355)(152 356)(153 357)(154 358)(155 359)(156 360)(157 361)(158 362)(159 363)(160 364)(161 365)(162 366)(163 367)(164 368)(165 369)(166 370)(167 371)(168 372)(169 373)(170 374)(171 375)(172 376)(173 377)(174 378)(175 379)(176 380)(177 381)(178 382)(179 383)(180 384)(181 385)(182 386)(183 387)(184 388)(185 389)(186 390)(187 391)(188 392)(189 393)(190 394)(191 395)(192 396)(193 397)(194 398)(195 399)(196 400)(197 401)(198 402)(199 403)(200 404)(201 405)(202 406)(203 407)(204 408)
(1 180)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 192)(14 193)(15 194)(16 195)(17 196)(18 197)(19 198)(20 199)(21 200)(22 201)(23 202)(24 203)(25 204)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)(87 164)(88 165)(89 166)(90 167)(91 168)(92 169)(93 170)(94 171)(95 172)(96 173)(97 174)(98 175)(99 176)(100 177)(101 178)(102 179)(205 317)(206 318)(207 319)(208 320)(209 321)(210 322)(211 323)(212 324)(213 325)(214 326)(215 327)(216 328)(217 329)(218 330)(219 331)(220 332)(221 333)(222 334)(223 335)(224 336)(225 337)(226 338)(227 339)(228 340)(229 341)(230 342)(231 343)(232 344)(233 345)(234 346)(235 347)(236 348)(237 349)(238 350)(239 351)(240 352)(241 353)(242 354)(243 355)(244 356)(245 357)(246 358)(247 359)(248 360)(249 361)(250 362)(251 363)(252 364)(253 365)(254 366)(255 367)(256 368)(257 369)(258 370)(259 371)(260 372)(261 373)(262 374)(263 375)(264 376)(265 377)(266 378)(267 379)(268 380)(269 381)(270 382)(271 383)(272 384)(273 385)(274 386)(275 387)(276 388)(277 389)(278 390)(279 391)(280 392)(281 393)(282 394)(283 395)(284 396)(285 397)(286 398)(287 399)(288 400)(289 401)(290 402)(291 403)(292 404)(293 405)(294 406)(295 407)(296 408)(297 307)(298 308)(299 309)(300 310)(301 311)(302 312)(303 313)(304 314)(305 315)(306 316)
(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)

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

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

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

408 conjugacy classes

 class 1 2A ··· 2G 3A 3B 6A ··· 6N 17A ··· 17P 34A ··· 34DH 51A ··· 51AF 102A ··· 102HP order 1 2 ··· 2 3 3 6 ··· 6 17 ··· 17 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

408 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C17 C34 C51 C102 kernel C22×C102 C2×C102 C22×C34 C2×C34 C22×C6 C2×C6 C23 C22 # reps 1 7 2 14 16 112 32 224

Matrix representation of C22×C102 in GL3(𝔽103) generated by

 1 0 0 0 1 0 0 0 102
,
 1 0 0 0 102 0 0 0 1
,
 80 0 0 0 5 0 0 0 78
G:=sub<GL(3,GF(103))| [1,0,0,0,1,0,0,0,102],[1,0,0,0,102,0,0,0,1],[80,0,0,0,5,0,0,0,78] >;

C22×C102 in GAP, Magma, Sage, TeX

C_2^2\times C_{102}
% in TeX

G:=Group("C2^2xC102");
// GroupNames label

G:=SmallGroup(408,46);
// by ID

G=gap.SmallGroup(408,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^102=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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