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## G = C22×Dic31order 496 = 24·31

### Direct product of C22 and Dic31

Aliases: C22×Dic31, C62.9C23, C23.2D31, C22.11D62, (C2×C62)⋊3C4, C622(C2×C4), C312(C22×C4), (C22×C62).3C2, C2.2(C22×D31), (C2×C62).12C22, SmallGroup(496,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C22×Dic31
 Chief series C1 — C31 — C62 — Dic31 — C2×Dic31 — C22×Dic31
 Lower central C31 — C22×Dic31
 Upper central C1 — C23

Generators and relations for C22×Dic31
G = < a,b,c,d | a2=b2=c62=1, d2=c31, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 384 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C31, C62, C62, Dic31, C2×C62, C2×Dic31, C22×C62, C22×Dic31
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D31, Dic31, D62, C2×Dic31, C22×D31, C22×Dic31

Smallest permutation representation of C22×Dic31
Regular action on 496 points
Generators in S496
(1 184)(2 185)(3 186)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 181)(61 182)(62 183)(63 218)(64 219)(65 220)(66 221)(67 222)(68 223)(69 224)(70 225)(71 226)(72 227)(73 228)(74 229)(75 230)(76 231)(77 232)(78 233)(79 234)(80 235)(81 236)(82 237)(83 238)(84 239)(85 240)(86 241)(87 242)(88 243)(89 244)(90 245)(91 246)(92 247)(93 248)(94 187)(95 188)(96 189)(97 190)(98 191)(99 192)(100 193)(101 194)(102 195)(103 196)(104 197)(105 198)(106 199)(107 200)(108 201)(109 202)(110 203)(111 204)(112 205)(113 206)(114 207)(115 208)(116 209)(117 210)(118 211)(119 212)(120 213)(121 214)(122 215)(123 216)(124 217)(249 404)(250 405)(251 406)(252 407)(253 408)(254 409)(255 410)(256 411)(257 412)(258 413)(259 414)(260 415)(261 416)(262 417)(263 418)(264 419)(265 420)(266 421)(267 422)(268 423)(269 424)(270 425)(271 426)(272 427)(273 428)(274 429)(275 430)(276 431)(277 432)(278 433)(279 434)(280 373)(281 374)(282 375)(283 376)(284 377)(285 378)(286 379)(287 380)(288 381)(289 382)(290 383)(291 384)(292 385)(293 386)(294 387)(295 388)(296 389)(297 390)(298 391)(299 392)(300 393)(301 394)(302 395)(303 396)(304 397)(305 398)(306 399)(307 400)(308 401)(309 402)(310 403)(311 435)(312 436)(313 437)(314 438)(315 439)(316 440)(317 441)(318 442)(319 443)(320 444)(321 445)(322 446)(323 447)(324 448)(325 449)(326 450)(327 451)(328 452)(329 453)(330 454)(331 455)(332 456)(333 457)(334 458)(335 459)(336 460)(337 461)(338 462)(339 463)(340 464)(341 465)(342 466)(343 467)(344 468)(345 469)(346 470)(347 471)(348 472)(349 473)(350 474)(351 475)(352 476)(353 477)(354 478)(355 479)(356 480)(357 481)(358 482)(359 483)(360 484)(361 485)(362 486)(363 487)(364 488)(365 489)(366 490)(367 491)(368 492)(369 493)(370 494)(371 495)(372 496)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(125 234)(126 235)(127 236)(128 237)(129 238)(130 239)(131 240)(132 241)(133 242)(134 243)(135 244)(136 245)(137 246)(138 247)(139 248)(140 187)(141 188)(142 189)(143 190)(144 191)(145 192)(146 193)(147 194)(148 195)(149 196)(150 197)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)(163 210)(164 211)(165 212)(166 213)(167 214)(168 215)(169 216)(170 217)(171 218)(172 219)(173 220)(174 221)(175 222)(176 223)(177 224)(178 225)(179 226)(180 227)(181 228)(182 229)(183 230)(184 231)(185 232)(186 233)(249 311)(250 312)(251 313)(252 314)(253 315)(254 316)(255 317)(256 318)(257 319)(258 320)(259 321)(260 322)(261 323)(262 324)(263 325)(264 326)(265 327)(266 328)(267 329)(268 330)(269 331)(270 332)(271 333)(272 334)(273 335)(274 336)(275 337)(276 338)(277 339)(278 340)(279 341)(280 342)(281 343)(282 344)(283 345)(284 346)(285 347)(286 348)(287 349)(288 350)(289 351)(290 352)(291 353)(292 354)(293 355)(294 356)(295 357)(296 358)(297 359)(298 360)(299 361)(300 362)(301 363)(302 364)(303 365)(304 366)(305 367)(306 368)(307 369)(308 370)(309 371)(310 372)(373 466)(374 467)(375 468)(376 469)(377 470)(378 471)(379 472)(380 473)(381 474)(382 475)(383 476)(384 477)(385 478)(386 479)(387 480)(388 481)(389 482)(390 483)(391 484)(392 485)(393 486)(394 487)(395 488)(396 489)(397 490)(398 491)(399 492)(400 493)(401 494)(402 495)(403 496)(404 435)(405 436)(406 437)(407 438)(408 439)(409 440)(410 441)(411 442)(412 443)(413 444)(414 445)(415 446)(416 447)(417 448)(418 449)(419 450)(420 451)(421 452)(422 453)(423 454)(424 455)(425 456)(426 457)(427 458)(428 459)(429 460)(430 461)(431 462)(432 463)(433 464)(434 465)
(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)(373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434)(435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
(1 311 32 342)(2 372 33 341)(3 371 34 340)(4 370 35 339)(5 369 36 338)(6 368 37 337)(7 367 38 336)(8 366 39 335)(9 365 40 334)(10 364 41 333)(11 363 42 332)(12 362 43 331)(13 361 44 330)(14 360 45 329)(15 359 46 328)(16 358 47 327)(17 357 48 326)(18 356 49 325)(19 355 50 324)(20 354 51 323)(21 353 52 322)(22 352 53 321)(23 351 54 320)(24 350 55 319)(25 349 56 318)(26 348 57 317)(27 347 58 316)(28 346 59 315)(29 345 60 314)(30 344 61 313)(31 343 62 312)(63 262 94 293)(64 261 95 292)(65 260 96 291)(66 259 97 290)(67 258 98 289)(68 257 99 288)(69 256 100 287)(70 255 101 286)(71 254 102 285)(72 253 103 284)(73 252 104 283)(74 251 105 282)(75 250 106 281)(76 249 107 280)(77 310 108 279)(78 309 109 278)(79 308 110 277)(80 307 111 276)(81 306 112 275)(82 305 113 274)(83 304 114 273)(84 303 115 272)(85 302 116 271)(86 301 117 270)(87 300 118 269)(88 299 119 268)(89 298 120 267)(90 297 121 266)(91 296 122 265)(92 295 123 264)(93 294 124 263)(125 494 156 463)(126 493 157 462)(127 492 158 461)(128 491 159 460)(129 490 160 459)(130 489 161 458)(131 488 162 457)(132 487 163 456)(133 486 164 455)(134 485 165 454)(135 484 166 453)(136 483 167 452)(137 482 168 451)(138 481 169 450)(139 480 170 449)(140 479 171 448)(141 478 172 447)(142 477 173 446)(143 476 174 445)(144 475 175 444)(145 474 176 443)(146 473 177 442)(147 472 178 441)(148 471 179 440)(149 470 180 439)(150 469 181 438)(151 468 182 437)(152 467 183 436)(153 466 184 435)(154 465 185 496)(155 464 186 495)(187 386 218 417)(188 385 219 416)(189 384 220 415)(190 383 221 414)(191 382 222 413)(192 381 223 412)(193 380 224 411)(194 379 225 410)(195 378 226 409)(196 377 227 408)(197 376 228 407)(198 375 229 406)(199 374 230 405)(200 373 231 404)(201 434 232 403)(202 433 233 402)(203 432 234 401)(204 431 235 400)(205 430 236 399)(206 429 237 398)(207 428 238 397)(208 427 239 396)(209 426 240 395)(210 425 241 394)(211 424 242 393)(212 423 243 392)(213 422 244 391)(214 421 245 390)(215 420 246 389)(216 419 247 388)(217 418 248 387)

G:=sub<Sym(496)| (1,184)(2,185)(3,186)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,225)(71,226)(72,227)(73,228)(74,229)(75,230)(76,231)(77,232)(78,233)(79,234)(80,235)(81,236)(82,237)(83,238)(84,239)(85,240)(86,241)(87,242)(88,243)(89,244)(90,245)(91,246)(92,247)(93,248)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,194)(102,195)(103,196)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,209)(117,210)(118,211)(119,212)(120,213)(121,214)(122,215)(123,216)(124,217)(249,404)(250,405)(251,406)(252,407)(253,408)(254,409)(255,410)(256,411)(257,412)(258,413)(259,414)(260,415)(261,416)(262,417)(263,418)(264,419)(265,420)(266,421)(267,422)(268,423)(269,424)(270,425)(271,426)(272,427)(273,428)(274,429)(275,430)(276,431)(277,432)(278,433)(279,434)(280,373)(281,374)(282,375)(283,376)(284,377)(285,378)(286,379)(287,380)(288,381)(289,382)(290,383)(291,384)(292,385)(293,386)(294,387)(295,388)(296,389)(297,390)(298,391)(299,392)(300,393)(301,394)(302,395)(303,396)(304,397)(305,398)(306,399)(307,400)(308,401)(309,402)(310,403)(311,435)(312,436)(313,437)(314,438)(315,439)(316,440)(317,441)(318,442)(319,443)(320,444)(321,445)(322,446)(323,447)(324,448)(325,449)(326,450)(327,451)(328,452)(329,453)(330,454)(331,455)(332,456)(333,457)(334,458)(335,459)(336,460)(337,461)(338,462)(339,463)(340,464)(341,465)(342,466)(343,467)(344,468)(345,469)(346,470)(347,471)(348,472)(349,473)(350,474)(351,475)(352,476)(353,477)(354,478)(355,479)(356,480)(357,481)(358,482)(359,483)(360,484)(361,485)(362,486)(363,487)(364,488)(365,489)(366,490)(367,491)(368,492)(369,493)(370,494)(371,495)(372,496), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(125,234)(126,235)(127,236)(128,237)(129,238)(130,239)(131,240)(132,241)(133,242)(134,243)(135,244)(136,245)(137,246)(138,247)(139,248)(140,187)(141,188)(142,189)(143,190)(144,191)(145,192)(146,193)(147,194)(148,195)(149,196)(150,197)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215)(169,216)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223)(177,224)(178,225)(179,226)(180,227)(181,228)(182,229)(183,230)(184,231)(185,232)(186,233)(249,311)(250,312)(251,313)(252,314)(253,315)(254,316)(255,317)(256,318)(257,319)(258,320)(259,321)(260,322)(261,323)(262,324)(263,325)(264,326)(265,327)(266,328)(267,329)(268,330)(269,331)(270,332)(271,333)(272,334)(273,335)(274,336)(275,337)(276,338)(277,339)(278,340)(279,341)(280,342)(281,343)(282,344)(283,345)(284,346)(285,347)(286,348)(287,349)(288,350)(289,351)(290,352)(291,353)(292,354)(293,355)(294,356)(295,357)(296,358)(297,359)(298,360)(299,361)(300,362)(301,363)(302,364)(303,365)(304,366)(305,367)(306,368)(307,369)(308,370)(309,371)(310,372)(373,466)(374,467)(375,468)(376,469)(377,470)(378,471)(379,472)(380,473)(381,474)(382,475)(383,476)(384,477)(385,478)(386,479)(387,480)(388,481)(389,482)(390,483)(391,484)(392,485)(393,486)(394,487)(395,488)(396,489)(397,490)(398,491)(399,492)(400,493)(401,494)(402,495)(403,496)(404,435)(405,436)(406,437)(407,438)(408,439)(409,440)(410,441)(411,442)(412,443)(413,444)(414,445)(415,446)(416,447)(417,448)(418,449)(419,450)(420,451)(421,452)(422,453)(423,454)(424,455)(425,456)(426,457)(427,458)(428,459)(429,460)(430,461)(431,462)(432,463)(433,464)(434,465), (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)(373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,311,32,342)(2,372,33,341)(3,371,34,340)(4,370,35,339)(5,369,36,338)(6,368,37,337)(7,367,38,336)(8,366,39,335)(9,365,40,334)(10,364,41,333)(11,363,42,332)(12,362,43,331)(13,361,44,330)(14,360,45,329)(15,359,46,328)(16,358,47,327)(17,357,48,326)(18,356,49,325)(19,355,50,324)(20,354,51,323)(21,353,52,322)(22,352,53,321)(23,351,54,320)(24,350,55,319)(25,349,56,318)(26,348,57,317)(27,347,58,316)(28,346,59,315)(29,345,60,314)(30,344,61,313)(31,343,62,312)(63,262,94,293)(64,261,95,292)(65,260,96,291)(66,259,97,290)(67,258,98,289)(68,257,99,288)(69,256,100,287)(70,255,101,286)(71,254,102,285)(72,253,103,284)(73,252,104,283)(74,251,105,282)(75,250,106,281)(76,249,107,280)(77,310,108,279)(78,309,109,278)(79,308,110,277)(80,307,111,276)(81,306,112,275)(82,305,113,274)(83,304,114,273)(84,303,115,272)(85,302,116,271)(86,301,117,270)(87,300,118,269)(88,299,119,268)(89,298,120,267)(90,297,121,266)(91,296,122,265)(92,295,123,264)(93,294,124,263)(125,494,156,463)(126,493,157,462)(127,492,158,461)(128,491,159,460)(129,490,160,459)(130,489,161,458)(131,488,162,457)(132,487,163,456)(133,486,164,455)(134,485,165,454)(135,484,166,453)(136,483,167,452)(137,482,168,451)(138,481,169,450)(139,480,170,449)(140,479,171,448)(141,478,172,447)(142,477,173,446)(143,476,174,445)(144,475,175,444)(145,474,176,443)(146,473,177,442)(147,472,178,441)(148,471,179,440)(149,470,180,439)(150,469,181,438)(151,468,182,437)(152,467,183,436)(153,466,184,435)(154,465,185,496)(155,464,186,495)(187,386,218,417)(188,385,219,416)(189,384,220,415)(190,383,221,414)(191,382,222,413)(192,381,223,412)(193,380,224,411)(194,379,225,410)(195,378,226,409)(196,377,227,408)(197,376,228,407)(198,375,229,406)(199,374,230,405)(200,373,231,404)(201,434,232,403)(202,433,233,402)(203,432,234,401)(204,431,235,400)(205,430,236,399)(206,429,237,398)(207,428,238,397)(208,427,239,396)(209,426,240,395)(210,425,241,394)(211,424,242,393)(212,423,243,392)(213,422,244,391)(214,421,245,390)(215,420,246,389)(216,419,247,388)(217,418,248,387)>;

G:=Group( (1,184)(2,185)(3,186)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,225)(71,226)(72,227)(73,228)(74,229)(75,230)(76,231)(77,232)(78,233)(79,234)(80,235)(81,236)(82,237)(83,238)(84,239)(85,240)(86,241)(87,242)(88,243)(89,244)(90,245)(91,246)(92,247)(93,248)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,194)(102,195)(103,196)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,209)(117,210)(118,211)(119,212)(120,213)(121,214)(122,215)(123,216)(124,217)(249,404)(250,405)(251,406)(252,407)(253,408)(254,409)(255,410)(256,411)(257,412)(258,413)(259,414)(260,415)(261,416)(262,417)(263,418)(264,419)(265,420)(266,421)(267,422)(268,423)(269,424)(270,425)(271,426)(272,427)(273,428)(274,429)(275,430)(276,431)(277,432)(278,433)(279,434)(280,373)(281,374)(282,375)(283,376)(284,377)(285,378)(286,379)(287,380)(288,381)(289,382)(290,383)(291,384)(292,385)(293,386)(294,387)(295,388)(296,389)(297,390)(298,391)(299,392)(300,393)(301,394)(302,395)(303,396)(304,397)(305,398)(306,399)(307,400)(308,401)(309,402)(310,403)(311,435)(312,436)(313,437)(314,438)(315,439)(316,440)(317,441)(318,442)(319,443)(320,444)(321,445)(322,446)(323,447)(324,448)(325,449)(326,450)(327,451)(328,452)(329,453)(330,454)(331,455)(332,456)(333,457)(334,458)(335,459)(336,460)(337,461)(338,462)(339,463)(340,464)(341,465)(342,466)(343,467)(344,468)(345,469)(346,470)(347,471)(348,472)(349,473)(350,474)(351,475)(352,476)(353,477)(354,478)(355,479)(356,480)(357,481)(358,482)(359,483)(360,484)(361,485)(362,486)(363,487)(364,488)(365,489)(366,490)(367,491)(368,492)(369,493)(370,494)(371,495)(372,496), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(125,234)(126,235)(127,236)(128,237)(129,238)(130,239)(131,240)(132,241)(133,242)(134,243)(135,244)(136,245)(137,246)(138,247)(139,248)(140,187)(141,188)(142,189)(143,190)(144,191)(145,192)(146,193)(147,194)(148,195)(149,196)(150,197)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215)(169,216)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223)(177,224)(178,225)(179,226)(180,227)(181,228)(182,229)(183,230)(184,231)(185,232)(186,233)(249,311)(250,312)(251,313)(252,314)(253,315)(254,316)(255,317)(256,318)(257,319)(258,320)(259,321)(260,322)(261,323)(262,324)(263,325)(264,326)(265,327)(266,328)(267,329)(268,330)(269,331)(270,332)(271,333)(272,334)(273,335)(274,336)(275,337)(276,338)(277,339)(278,340)(279,341)(280,342)(281,343)(282,344)(283,345)(284,346)(285,347)(286,348)(287,349)(288,350)(289,351)(290,352)(291,353)(292,354)(293,355)(294,356)(295,357)(296,358)(297,359)(298,360)(299,361)(300,362)(301,363)(302,364)(303,365)(304,366)(305,367)(306,368)(307,369)(308,370)(309,371)(310,372)(373,466)(374,467)(375,468)(376,469)(377,470)(378,471)(379,472)(380,473)(381,474)(382,475)(383,476)(384,477)(385,478)(386,479)(387,480)(388,481)(389,482)(390,483)(391,484)(392,485)(393,486)(394,487)(395,488)(396,489)(397,490)(398,491)(399,492)(400,493)(401,494)(402,495)(403,496)(404,435)(405,436)(406,437)(407,438)(408,439)(409,440)(410,441)(411,442)(412,443)(413,444)(414,445)(415,446)(416,447)(417,448)(418,449)(419,450)(420,451)(421,452)(422,453)(423,454)(424,455)(425,456)(426,457)(427,458)(428,459)(429,460)(430,461)(431,462)(432,463)(433,464)(434,465), (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)(373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,311,32,342)(2,372,33,341)(3,371,34,340)(4,370,35,339)(5,369,36,338)(6,368,37,337)(7,367,38,336)(8,366,39,335)(9,365,40,334)(10,364,41,333)(11,363,42,332)(12,362,43,331)(13,361,44,330)(14,360,45,329)(15,359,46,328)(16,358,47,327)(17,357,48,326)(18,356,49,325)(19,355,50,324)(20,354,51,323)(21,353,52,322)(22,352,53,321)(23,351,54,320)(24,350,55,319)(25,349,56,318)(26,348,57,317)(27,347,58,316)(28,346,59,315)(29,345,60,314)(30,344,61,313)(31,343,62,312)(63,262,94,293)(64,261,95,292)(65,260,96,291)(66,259,97,290)(67,258,98,289)(68,257,99,288)(69,256,100,287)(70,255,101,286)(71,254,102,285)(72,253,103,284)(73,252,104,283)(74,251,105,282)(75,250,106,281)(76,249,107,280)(77,310,108,279)(78,309,109,278)(79,308,110,277)(80,307,111,276)(81,306,112,275)(82,305,113,274)(83,304,114,273)(84,303,115,272)(85,302,116,271)(86,301,117,270)(87,300,118,269)(88,299,119,268)(89,298,120,267)(90,297,121,266)(91,296,122,265)(92,295,123,264)(93,294,124,263)(125,494,156,463)(126,493,157,462)(127,492,158,461)(128,491,159,460)(129,490,160,459)(130,489,161,458)(131,488,162,457)(132,487,163,456)(133,486,164,455)(134,485,165,454)(135,484,166,453)(136,483,167,452)(137,482,168,451)(138,481,169,450)(139,480,170,449)(140,479,171,448)(141,478,172,447)(142,477,173,446)(143,476,174,445)(144,475,175,444)(145,474,176,443)(146,473,177,442)(147,472,178,441)(148,471,179,440)(149,470,180,439)(150,469,181,438)(151,468,182,437)(152,467,183,436)(153,466,184,435)(154,465,185,496)(155,464,186,495)(187,386,218,417)(188,385,219,416)(189,384,220,415)(190,383,221,414)(191,382,222,413)(192,381,223,412)(193,380,224,411)(194,379,225,410)(195,378,226,409)(196,377,227,408)(197,376,228,407)(198,375,229,406)(199,374,230,405)(200,373,231,404)(201,434,232,403)(202,433,233,402)(203,432,234,401)(204,431,235,400)(205,430,236,399)(206,429,237,398)(207,428,238,397)(208,427,239,396)(209,426,240,395)(210,425,241,394)(211,424,242,393)(212,423,243,392)(213,422,244,391)(214,421,245,390)(215,420,246,389)(216,419,247,388)(217,418,248,387) );

G=PermutationGroup([[(1,184),(2,185),(3,186),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,141),(21,142),(22,143),(23,144),(24,145),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,157),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,178),(58,179),(59,180),(60,181),(61,182),(62,183),(63,218),(64,219),(65,220),(66,221),(67,222),(68,223),(69,224),(70,225),(71,226),(72,227),(73,228),(74,229),(75,230),(76,231),(77,232),(78,233),(79,234),(80,235),(81,236),(82,237),(83,238),(84,239),(85,240),(86,241),(87,242),(88,243),(89,244),(90,245),(91,246),(92,247),(93,248),(94,187),(95,188),(96,189),(97,190),(98,191),(99,192),(100,193),(101,194),(102,195),(103,196),(104,197),(105,198),(106,199),(107,200),(108,201),(109,202),(110,203),(111,204),(112,205),(113,206),(114,207),(115,208),(116,209),(117,210),(118,211),(119,212),(120,213),(121,214),(122,215),(123,216),(124,217),(249,404),(250,405),(251,406),(252,407),(253,408),(254,409),(255,410),(256,411),(257,412),(258,413),(259,414),(260,415),(261,416),(262,417),(263,418),(264,419),(265,420),(266,421),(267,422),(268,423),(269,424),(270,425),(271,426),(272,427),(273,428),(274,429),(275,430),(276,431),(277,432),(278,433),(279,434),(280,373),(281,374),(282,375),(283,376),(284,377),(285,378),(286,379),(287,380),(288,381),(289,382),(290,383),(291,384),(292,385),(293,386),(294,387),(295,388),(296,389),(297,390),(298,391),(299,392),(300,393),(301,394),(302,395),(303,396),(304,397),(305,398),(306,399),(307,400),(308,401),(309,402),(310,403),(311,435),(312,436),(313,437),(314,438),(315,439),(316,440),(317,441),(318,442),(319,443),(320,444),(321,445),(322,446),(323,447),(324,448),(325,449),(326,450),(327,451),(328,452),(329,453),(330,454),(331,455),(332,456),(333,457),(334,458),(335,459),(336,460),(337,461),(338,462),(339,463),(340,464),(341,465),(342,466),(343,467),(344,468),(345,469),(346,470),(347,471),(348,472),(349,473),(350,474),(351,475),(352,476),(353,477),(354,478),(355,479),(356,480),(357,481),(358,482),(359,483),(360,484),(361,485),(362,486),(363,487),(364,488),(365,489),(366,490),(367,491),(368,492),(369,493),(370,494),(371,495),(372,496)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(125,234),(126,235),(127,236),(128,237),(129,238),(130,239),(131,240),(132,241),(133,242),(134,243),(135,244),(136,245),(137,246),(138,247),(139,248),(140,187),(141,188),(142,189),(143,190),(144,191),(145,192),(146,193),(147,194),(148,195),(149,196),(150,197),(151,198),(152,199),(153,200),(154,201),(155,202),(156,203),(157,204),(158,205),(159,206),(160,207),(161,208),(162,209),(163,210),(164,211),(165,212),(166,213),(167,214),(168,215),(169,216),(170,217),(171,218),(172,219),(173,220),(174,221),(175,222),(176,223),(177,224),(178,225),(179,226),(180,227),(181,228),(182,229),(183,230),(184,231),(185,232),(186,233),(249,311),(250,312),(251,313),(252,314),(253,315),(254,316),(255,317),(256,318),(257,319),(258,320),(259,321),(260,322),(261,323),(262,324),(263,325),(264,326),(265,327),(266,328),(267,329),(268,330),(269,331),(270,332),(271,333),(272,334),(273,335),(274,336),(275,337),(276,338),(277,339),(278,340),(279,341),(280,342),(281,343),(282,344),(283,345),(284,346),(285,347),(286,348),(287,349),(288,350),(289,351),(290,352),(291,353),(292,354),(293,355),(294,356),(295,357),(296,358),(297,359),(298,360),(299,361),(300,362),(301,363),(302,364),(303,365),(304,366),(305,367),(306,368),(307,369),(308,370),(309,371),(310,372),(373,466),(374,467),(375,468),(376,469),(377,470),(378,471),(379,472),(380,473),(381,474),(382,475),(383,476),(384,477),(385,478),(386,479),(387,480),(388,481),(389,482),(390,483),(391,484),(392,485),(393,486),(394,487),(395,488),(396,489),(397,490),(398,491),(399,492),(400,493),(401,494),(402,495),(403,496),(404,435),(405,436),(406,437),(407,438),(408,439),(409,440),(410,441),(411,442),(412,443),(413,444),(414,445),(415,446),(416,447),(417,448),(418,449),(419,450),(420,451),(421,452),(422,453),(423,454),(424,455),(425,456),(426,457),(427,458),(428,459),(429,460),(430,461),(431,462),(432,463),(433,464),(434,465)], [(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),(373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434),(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)], [(1,311,32,342),(2,372,33,341),(3,371,34,340),(4,370,35,339),(5,369,36,338),(6,368,37,337),(7,367,38,336),(8,366,39,335),(9,365,40,334),(10,364,41,333),(11,363,42,332),(12,362,43,331),(13,361,44,330),(14,360,45,329),(15,359,46,328),(16,358,47,327),(17,357,48,326),(18,356,49,325),(19,355,50,324),(20,354,51,323),(21,353,52,322),(22,352,53,321),(23,351,54,320),(24,350,55,319),(25,349,56,318),(26,348,57,317),(27,347,58,316),(28,346,59,315),(29,345,60,314),(30,344,61,313),(31,343,62,312),(63,262,94,293),(64,261,95,292),(65,260,96,291),(66,259,97,290),(67,258,98,289),(68,257,99,288),(69,256,100,287),(70,255,101,286),(71,254,102,285),(72,253,103,284),(73,252,104,283),(74,251,105,282),(75,250,106,281),(76,249,107,280),(77,310,108,279),(78,309,109,278),(79,308,110,277),(80,307,111,276),(81,306,112,275),(82,305,113,274),(83,304,114,273),(84,303,115,272),(85,302,116,271),(86,301,117,270),(87,300,118,269),(88,299,119,268),(89,298,120,267),(90,297,121,266),(91,296,122,265),(92,295,123,264),(93,294,124,263),(125,494,156,463),(126,493,157,462),(127,492,158,461),(128,491,159,460),(129,490,160,459),(130,489,161,458),(131,488,162,457),(132,487,163,456),(133,486,164,455),(134,485,165,454),(135,484,166,453),(136,483,167,452),(137,482,168,451),(138,481,169,450),(139,480,170,449),(140,479,171,448),(141,478,172,447),(142,477,173,446),(143,476,174,445),(144,475,175,444),(145,474,176,443),(146,473,177,442),(147,472,178,441),(148,471,179,440),(149,470,180,439),(150,469,181,438),(151,468,182,437),(152,467,183,436),(153,466,184,435),(154,465,185,496),(155,464,186,495),(187,386,218,417),(188,385,219,416),(189,384,220,415),(190,383,221,414),(191,382,222,413),(192,381,223,412),(193,380,224,411),(194,379,225,410),(195,378,226,409),(196,377,227,408),(197,376,228,407),(198,375,229,406),(199,374,230,405),(200,373,231,404),(201,434,232,403),(202,433,233,402),(203,432,234,401),(204,431,235,400),(205,430,236,399),(206,429,237,398),(207,428,238,397),(208,427,239,396),(209,426,240,395),(210,425,241,394),(211,424,242,393),(212,423,243,392),(213,422,244,391),(214,421,245,390),(215,420,246,389),(216,419,247,388),(217,418,248,387)]])

136 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 31A ··· 31O 62A ··· 62DA order 1 2 ··· 2 4 ··· 4 31 ··· 31 62 ··· 62 size 1 1 ··· 1 31 ··· 31 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D31 Dic31 D62 kernel C22×Dic31 C2×Dic31 C22×C62 C2×C62 C23 C22 C22 # reps 1 6 1 8 15 60 45

Matrix representation of C22×Dic31 in GL4(𝔽373) generated by

 1 0 0 0 0 1 0 0 0 0 372 0 0 0 0 372
,
 372 0 0 0 0 372 0 0 0 0 1 0 0 0 0 1
,
 372 0 0 0 0 1 0 0 0 0 0 1 0 0 372 194
,
 104 0 0 0 0 372 0 0 0 0 4 232 0 0 262 369
G:=sub<GL(4,GF(373))| [1,0,0,0,0,1,0,0,0,0,372,0,0,0,0,372],[372,0,0,0,0,372,0,0,0,0,1,0,0,0,0,1],[372,0,0,0,0,1,0,0,0,0,0,372,0,0,1,194],[104,0,0,0,0,372,0,0,0,0,4,262,0,0,232,369] >;

C22×Dic31 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{31}
% in TeX

G:=Group("C2^2xDic31");
// GroupNames label

G:=SmallGroup(496,35);
// by ID

G=gap.SmallGroup(496,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-31,40,12004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^62=1,d^2=c^31,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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