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## G = C2×C36.S3order 432 = 24·33

### Direct product of C2 and C36.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C36.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C36.S3 — C2×C36.S3
 Lower central C3×C9 — C2×C36.S3
 Upper central C1 — C2×C4

Generators and relations for C2×C36.S3
G = < a,b,c,d | a2=b36=1, c3=b12, d2=b9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b17, dcd-1=b24c2 >

Subgroups: 308 in 110 conjugacy classes, 83 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C2×C8, C18, C3×C6, C3×C6, C3⋊C8, C2×C12, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C2×C3⋊C8, C3×C18, C3×C18, C9⋊C8, C2×C36, C324C8, C6×C12, C3×C36, C6×C18, C2×C9⋊C8, C2×C324C8, C36.S3, C6×C36, C2×C36.S3
Quotients:

Smallest permutation representation of C2×C36.S3
Regular action on 432 points
Generators in S432
(1 338)(2 339)(3 340)(4 341)(5 342)(6 343)(7 344)(8 345)(9 346)(10 347)(11 348)(12 349)(13 350)(14 351)(15 352)(16 353)(17 354)(18 355)(19 356)(20 357)(21 358)(22 359)(23 360)(24 325)(25 326)(26 327)(27 328)(28 329)(29 330)(30 331)(31 332)(32 333)(33 334)(34 335)(35 336)(36 337)(37 276)(38 277)(39 278)(40 279)(41 280)(42 281)(43 282)(44 283)(45 284)(46 285)(47 286)(48 287)(49 288)(50 253)(51 254)(52 255)(53 256)(54 257)(55 258)(56 259)(57 260)(58 261)(59 262)(60 263)(61 264)(62 265)(63 266)(64 267)(65 268)(66 269)(67 270)(68 271)(69 272)(70 273)(71 274)(72 275)(73 217)(74 218)(75 219)(76 220)(77 221)(78 222)(79 223)(80 224)(81 225)(82 226)(83 227)(84 228)(85 229)(86 230)(87 231)(88 232)(89 233)(90 234)(91 235)(92 236)(93 237)(94 238)(95 239)(96 240)(97 241)(98 242)(99 243)(100 244)(101 245)(102 246)(103 247)(104 248)(105 249)(106 250)(107 251)(108 252)(109 422)(110 423)(111 424)(112 425)(113 426)(114 427)(115 428)(116 429)(117 430)(118 431)(119 432)(120 397)(121 398)(122 399)(123 400)(124 401)(125 402)(126 403)(127 404)(128 405)(129 406)(130 407)(131 408)(132 409)(133 410)(134 411)(135 412)(136 413)(137 414)(138 415)(139 416)(140 417)(141 418)(142 419)(143 420)(144 421)(145 317)(146 318)(147 319)(148 320)(149 321)(150 322)(151 323)(152 324)(153 289)(154 290)(155 291)(156 292)(157 293)(158 294)(159 295)(160 296)(161 297)(162 298)(163 299)(164 300)(165 301)(166 302)(167 303)(168 304)(169 305)(170 306)(171 307)(172 308)(173 309)(174 310)(175 311)(176 312)(177 313)(178 314)(179 315)(180 316)(181 376)(182 377)(183 378)(184 379)(185 380)(186 381)(187 382)(188 383)(189 384)(190 385)(191 386)(192 387)(193 388)(194 389)(195 390)(196 391)(197 392)(198 393)(199 394)(200 395)(201 396)(202 361)(203 362)(204 363)(205 364)(206 365)(207 366)(208 367)(209 368)(210 369)(211 370)(212 371)(213 372)(214 373)(215 374)(216 375)
(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)
(1 369 416 13 381 428 25 393 404)(2 370 417 14 382 429 26 394 405)(3 371 418 15 383 430 27 395 406)(4 372 419 16 384 431 28 396 407)(5 373 420 17 385 432 29 361 408)(6 374 421 18 386 397 30 362 409)(7 375 422 19 387 398 31 363 410)(8 376 423 20 388 399 32 364 411)(9 377 424 21 389 400 33 365 412)(10 378 425 22 390 401 34 366 413)(11 379 426 23 391 402 35 367 414)(12 380 427 24 392 403 36 368 415)(37 317 96 49 293 108 61 305 84)(38 318 97 50 294 73 62 306 85)(39 319 98 51 295 74 63 307 86)(40 320 99 52 296 75 64 308 87)(41 321 100 53 297 76 65 309 88)(42 322 101 54 298 77 66 310 89)(43 323 102 55 299 78 67 311 90)(44 324 103 56 300 79 68 312 91)(45 289 104 57 301 80 69 313 92)(46 290 105 58 302 81 70 314 93)(47 291 106 59 303 82 71 315 94)(48 292 107 60 304 83 72 316 95)(109 356 192 121 332 204 133 344 216)(110 357 193 122 333 205 134 345 181)(111 358 194 123 334 206 135 346 182)(112 359 195 124 335 207 136 347 183)(113 360 196 125 336 208 137 348 184)(114 325 197 126 337 209 138 349 185)(115 326 198 127 338 210 139 350 186)(116 327 199 128 339 211 140 351 187)(117 328 200 129 340 212 141 352 188)(118 329 201 130 341 213 142 353 189)(119 330 202 131 342 214 143 354 190)(120 331 203 132 343 215 144 355 191)(145 240 288 157 252 264 169 228 276)(146 241 253 158 217 265 170 229 277)(147 242 254 159 218 266 171 230 278)(148 243 255 160 219 267 172 231 279)(149 244 256 161 220 268 173 232 280)(150 245 257 162 221 269 174 233 281)(151 246 258 163 222 270 175 234 282)(152 247 259 164 223 271 176 235 283)(153 248 260 165 224 272 177 236 284)(154 249 261 166 225 273 178 237 285)(155 250 262 167 226 274 179 238 286)(156 251 263 168 227 275 180 239 287)
(1 150 10 159 19 168 28 177)(2 167 11 176 20 149 29 158)(3 148 12 157 21 166 30 175)(4 165 13 174 22 147 31 156)(5 146 14 155 23 164 32 173)(6 163 15 172 24 145 33 154)(7 180 16 153 25 162 34 171)(8 161 17 170 26 179 35 152)(9 178 18 151 27 160 36 169)(37 206 46 215 55 188 64 197)(38 187 47 196 56 205 65 214)(39 204 48 213 57 186 66 195)(40 185 49 194 58 203 67 212)(41 202 50 211 59 184 68 193)(42 183 51 192 60 201 69 210)(43 200 52 209 61 182 70 191)(44 181 53 190 62 199 71 208)(45 198 54 207 63 216 72 189)(73 128 82 137 91 110 100 119)(74 109 83 118 92 127 101 136)(75 126 84 135 93 144 102 117)(76 143 85 116 94 125 103 134)(77 124 86 133 95 142 104 115)(78 141 87 114 96 123 105 132)(79 122 88 131 97 140 106 113)(80 139 89 112 98 121 107 130)(81 120 90 129 99 138 108 111)(217 405 226 414 235 423 244 432)(218 422 227 431 236 404 245 413)(219 403 228 412 237 421 246 430)(220 420 229 429 238 402 247 411)(221 401 230 410 239 419 248 428)(222 418 231 427 240 400 249 409)(223 399 232 408 241 417 250 426)(224 416 233 425 242 398 251 407)(225 397 234 406 243 415 252 424)(253 370 262 379 271 388 280 361)(254 387 263 396 272 369 281 378)(255 368 264 377 273 386 282 395)(256 385 265 394 274 367 283 376)(257 366 266 375 275 384 284 393)(258 383 267 392 276 365 285 374)(259 364 268 373 277 382 286 391)(260 381 269 390 278 363 287 372)(261 362 270 371 279 380 288 389)(289 326 298 335 307 344 316 353)(290 343 299 352 308 325 317 334)(291 360 300 333 309 342 318 351)(292 341 301 350 310 359 319 332)(293 358 302 331 311 340 320 349)(294 339 303 348 312 357 321 330)(295 356 304 329 313 338 322 347)(296 337 305 346 314 355 323 328)(297 354 306 327 315 336 324 345)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A ··· 8H 9A ··· 9I 12A ··· 12P 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 ··· 8 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 27 ··· 27 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - - + - + - + - image C1 C2 C2 C4 C4 C8 S3 S3 Dic3 D6 Dic3 Dic3 D6 Dic3 D9 C3⋊C8 C3⋊C8 Dic9 D18 Dic9 C9⋊C8 kernel C2×C36.S3 C36.S3 C6×C36 C3×C36 C6×C18 C3×C18 C2×C36 C6×C12 C36 C36 C2×C18 C3×C12 C3×C12 C62 C2×C12 C18 C3×C6 C12 C12 C2×C6 C6 # reps 1 2 1 2 2 8 3 1 3 3 3 1 1 1 9 12 4 9 9 9 36

Matrix representation of C2×C36.S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 34 65 0 0 0 0 8 26 0 0 0 0 0 0 26 65 0 0 0 0 8 34 0 0 0 0 0 0 70 42 0 0 0 0 31 28
,
 45 31 0 0 0 0 42 3 0 0 0 0 0 0 3 31 0 0 0 0 42 45 0 0 0 0 0 0 3 31 0 0 0 0 42 45
,
 34 62 0 0 0 0 23 39 0 0 0 0 0 0 71 29 0 0 0 0 31 2 0 0 0 0 0 0 71 53 0 0 0 0 55 2

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,8,0,0,0,0,65,26,0,0,0,0,0,0,26,8,0,0,0,0,65,34,0,0,0,0,0,0,70,31,0,0,0,0,42,28],[45,42,0,0,0,0,31,3,0,0,0,0,0,0,3,42,0,0,0,0,31,45,0,0,0,0,0,0,3,42,0,0,0,0,31,45],[34,23,0,0,0,0,62,39,0,0,0,0,0,0,71,31,0,0,0,0,29,2,0,0,0,0,0,0,71,55,0,0,0,0,53,2] >;

C2×C36.S3 in GAP, Magma, Sage, TeX

C_2\times C_{36}.S_3
% in TeX

G:=Group("C2xC36.S3");
// GroupNames label

G:=SmallGroup(432,178);
// by ID

G=gap.SmallGroup(432,178);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,58,6164,662,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^36=1,c^3=b^12,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^17,d*c*d^-1=b^24*c^2>;
// generators/relations

׿
×
𝔽