metabelian, supersoluble, monomial
Aliases: C36.19D6, C12.19D18, (C3×C9)⋊9Q16, (C3×Q8).9D9, (Q8×C9).9S3, C3⋊3(C9⋊Q16), C9⋊3(C3⋊Q16), (C3×C12).87D6, (C3×C18).49D4, Q8.3(C9⋊S3), C6.26(C9⋊D4), C3.(C32⋊7Q16), C18.26(C3⋊D4), C36.S3.2C2, C12.D9.4C2, (C3×C36).22C22, (Q8×C32).20S3, C6.18(C32⋊7D4), C32.5(C3⋊Q16), C2.6(C6.D18), C4.3(C2×C9⋊S3), (Q8×C3×C9).2C2, C12.3(C2×C3⋊S3), (C3×Q8).9(C3⋊S3), (C3×C6).101(C3⋊D4), SmallGroup(432,194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.19D6
G = < a,b,c | a36=1, b6=a18, c2=a27, bab-1=a19, cac-1=a17, cbc-1=a27b5 >
Subgroups: 440 in 90 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C9, C32, Dic3, C12, C12, C12, Q16, C18, C3×C6, C3⋊C8, Dic6, C3×Q8, C3×Q8, C3×C9, Dic9, C36, C36, C3⋊Dic3, C3×C12, C3×C12, C3⋊Q16, C3×C18, C9⋊C8, Dic18, Q8×C9, C32⋊4C8, C32⋊4Q8, Q8×C32, C9⋊Dic3, C3×C36, C3×C36, C9⋊Q16, C32⋊7Q16, C36.S3, C12.D9, Q8×C3×C9, C36.19D6
Quotients: C1, C2, C22, S3, D4, D6, Q16, D9, C3⋊S3, C3⋊D4, D18, C2×C3⋊S3, C3⋊Q16, C9⋊S3, C9⋊D4, C32⋊7D4, C2×C9⋊S3, C9⋊Q16, C32⋊7Q16, C6.D18, C36.19D6
►Regular action on 432 points
Generators in S432
(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)
(1 386 326 204 258 301 19 368 344 186 276 319)(2 369 327 187 259 320 20 387 345 205 277 302)(3 388 328 206 260 303 21 370 346 188 278 321)(4 371 329 189 261 322 22 389 347 207 279 304)(5 390 330 208 262 305 23 372 348 190 280 323)(6 373 331 191 263 324 24 391 349 209 281 306)(7 392 332 210 264 307 25 374 350 192 282 289)(8 375 333 193 265 290 26 393 351 211 283 308)(9 394 334 212 266 309 27 376 352 194 284 291)(10 377 335 195 267 292 28 395 353 213 285 310)(11 396 336 214 268 311 29 378 354 196 286 293)(12 379 337 197 269 294 30 361 355 215 287 312)(13 362 338 216 270 313 31 380 356 198 288 295)(14 381 339 199 271 296 32 363 357 181 253 314)(15 364 340 182 272 315 33 382 358 200 254 297)(16 383 341 201 273 298 34 365 359 183 255 316)(17 366 342 184 274 317 35 384 360 202 256 299)(18 385 343 203 275 300 36 367 325 185 257 318)(37 407 135 102 150 227 55 425 117 84 168 245)(38 426 136 85 151 246 56 408 118 103 169 228)(39 409 137 104 152 229 57 427 119 86 170 247)(40 428 138 87 153 248 58 410 120 105 171 230)(41 411 139 106 154 231 59 429 121 88 172 249)(42 430 140 89 155 250 60 412 122 107 173 232)(43 413 141 108 156 233 61 431 123 90 174 251)(44 432 142 91 157 252 62 414 124 73 175 234)(45 415 143 74 158 235 63 397 125 92 176 217)(46 398 144 93 159 218 64 416 126 75 177 236)(47 417 109 76 160 237 65 399 127 94 178 219)(48 400 110 95 161 220 66 418 128 77 179 238)(49 419 111 78 162 239 67 401 129 96 180 221)(50 402 112 97 163 222 68 420 130 79 145 240)(51 421 113 80 164 241 69 403 131 98 146 223)(52 404 114 99 165 224 70 422 132 81 147 242)(53 423 115 82 166 243 71 405 133 100 148 225)(54 406 116 101 167 226 72 424 134 83 149 244)
(1 129 28 120 19 111 10 138)(2 110 29 137 20 128 11 119)(3 127 30 118 21 109 12 136)(4 144 31 135 22 126 13 117)(5 125 32 116 23 143 14 134)(6 142 33 133 24 124 15 115)(7 123 34 114 25 141 16 132)(8 140 35 131 26 122 17 113)(9 121 36 112 27 139 18 130)(37 347 64 338 55 329 46 356)(38 328 65 355 56 346 47 337)(39 345 66 336 57 327 48 354)(40 326 67 353 58 344 49 335)(41 343 68 334 59 325 50 352)(42 360 69 351 60 342 51 333)(43 341 70 332 61 359 52 350)(44 358 71 349 62 340 53 331)(45 339 72 330 63 357 54 348)(73 306 100 297 91 324 82 315)(74 323 101 314 92 305 83 296)(75 304 102 295 93 322 84 313)(76 321 103 312 94 303 85 294)(77 302 104 293 95 320 86 311)(78 319 105 310 96 301 87 292)(79 300 106 291 97 318 88 309)(80 317 107 308 98 299 89 290)(81 298 108 289 99 316 90 307)(145 284 172 275 163 266 154 257)(146 265 173 256 164 283 155 274)(147 282 174 273 165 264 156 255)(148 263 175 254 166 281 157 272)(149 280 176 271 167 262 158 253)(150 261 177 288 168 279 159 270)(151 278 178 269 169 260 160 287)(152 259 179 286 170 277 161 268)(153 276 180 267 171 258 162 285)(181 217 208 244 199 235 190 226)(182 234 209 225 200 252 191 243)(183 251 210 242 201 233 192 224)(184 232 211 223 202 250 193 241)(185 249 212 240 203 231 194 222)(186 230 213 221 204 248 195 239)(187 247 214 238 205 229 196 220)(188 228 215 219 206 246 197 237)(189 245 216 236 207 227 198 218)(361 417 388 408 379 399 370 426)(362 398 389 425 380 416 371 407)(363 415 390 406 381 397 372 424)(364 432 391 423 382 414 373 405)(365 413 392 404 383 431 374 422)(366 430 393 421 384 412 375 403)(367 411 394 402 385 429 376 420)(368 428 395 419 386 410 377 401)(369 409 396 400 387 427 378 418)
G:=sub<Sym(432)| (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), (1,386,326,204,258,301,19,368,344,186,276,319)(2,369,327,187,259,320,20,387,345,205,277,302)(3,388,328,206,260,303,21,370,346,188,278,321)(4,371,329,189,261,322,22,389,347,207,279,304)(5,390,330,208,262,305,23,372,348,190,280,323)(6,373,331,191,263,324,24,391,349,209,281,306)(7,392,332,210,264,307,25,374,350,192,282,289)(8,375,333,193,265,290,26,393,351,211,283,308)(9,394,334,212,266,309,27,376,352,194,284,291)(10,377,335,195,267,292,28,395,353,213,285,310)(11,396,336,214,268,311,29,378,354,196,286,293)(12,379,337,197,269,294,30,361,355,215,287,312)(13,362,338,216,270,313,31,380,356,198,288,295)(14,381,339,199,271,296,32,363,357,181,253,314)(15,364,340,182,272,315,33,382,358,200,254,297)(16,383,341,201,273,298,34,365,359,183,255,316)(17,366,342,184,274,317,35,384,360,202,256,299)(18,385,343,203,275,300,36,367,325,185,257,318)(37,407,135,102,150,227,55,425,117,84,168,245)(38,426,136,85,151,246,56,408,118,103,169,228)(39,409,137,104,152,229,57,427,119,86,170,247)(40,428,138,87,153,248,58,410,120,105,171,230)(41,411,139,106,154,231,59,429,121,88,172,249)(42,430,140,89,155,250,60,412,122,107,173,232)(43,413,141,108,156,233,61,431,123,90,174,251)(44,432,142,91,157,252,62,414,124,73,175,234)(45,415,143,74,158,235,63,397,125,92,176,217)(46,398,144,93,159,218,64,416,126,75,177,236)(47,417,109,76,160,237,65,399,127,94,178,219)(48,400,110,95,161,220,66,418,128,77,179,238)(49,419,111,78,162,239,67,401,129,96,180,221)(50,402,112,97,163,222,68,420,130,79,145,240)(51,421,113,80,164,241,69,403,131,98,146,223)(52,404,114,99,165,224,70,422,132,81,147,242)(53,423,115,82,166,243,71,405,133,100,148,225)(54,406,116,101,167,226,72,424,134,83,149,244), (1,129,28,120,19,111,10,138)(2,110,29,137,20,128,11,119)(3,127,30,118,21,109,12,136)(4,144,31,135,22,126,13,117)(5,125,32,116,23,143,14,134)(6,142,33,133,24,124,15,115)(7,123,34,114,25,141,16,132)(8,140,35,131,26,122,17,113)(9,121,36,112,27,139,18,130)(37,347,64,338,55,329,46,356)(38,328,65,355,56,346,47,337)(39,345,66,336,57,327,48,354)(40,326,67,353,58,344,49,335)(41,343,68,334,59,325,50,352)(42,360,69,351,60,342,51,333)(43,341,70,332,61,359,52,350)(44,358,71,349,62,340,53,331)(45,339,72,330,63,357,54,348)(73,306,100,297,91,324,82,315)(74,323,101,314,92,305,83,296)(75,304,102,295,93,322,84,313)(76,321,103,312,94,303,85,294)(77,302,104,293,95,320,86,311)(78,319,105,310,96,301,87,292)(79,300,106,291,97,318,88,309)(80,317,107,308,98,299,89,290)(81,298,108,289,99,316,90,307)(145,284,172,275,163,266,154,257)(146,265,173,256,164,283,155,274)(147,282,174,273,165,264,156,255)(148,263,175,254,166,281,157,272)(149,280,176,271,167,262,158,253)(150,261,177,288,168,279,159,270)(151,278,178,269,169,260,160,287)(152,259,179,286,170,277,161,268)(153,276,180,267,171,258,162,285)(181,217,208,244,199,235,190,226)(182,234,209,225,200,252,191,243)(183,251,210,242,201,233,192,224)(184,232,211,223,202,250,193,241)(185,249,212,240,203,231,194,222)(186,230,213,221,204,248,195,239)(187,247,214,238,205,229,196,220)(188,228,215,219,206,246,197,237)(189,245,216,236,207,227,198,218)(361,417,388,408,379,399,370,426)(362,398,389,425,380,416,371,407)(363,415,390,406,381,397,372,424)(364,432,391,423,382,414,373,405)(365,413,392,404,383,431,374,422)(366,430,393,421,384,412,375,403)(367,411,394,402,385,429,376,420)(368,428,395,419,386,410,377,401)(369,409,396,400,387,427,378,418)>;
G:=Group( (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), (1,386,326,204,258,301,19,368,344,186,276,319)(2,369,327,187,259,320,20,387,345,205,277,302)(3,388,328,206,260,303,21,370,346,188,278,321)(4,371,329,189,261,322,22,389,347,207,279,304)(5,390,330,208,262,305,23,372,348,190,280,323)(6,373,331,191,263,324,24,391,349,209,281,306)(7,392,332,210,264,307,25,374,350,192,282,289)(8,375,333,193,265,290,26,393,351,211,283,308)(9,394,334,212,266,309,27,376,352,194,284,291)(10,377,335,195,267,292,28,395,353,213,285,310)(11,396,336,214,268,311,29,378,354,196,286,293)(12,379,337,197,269,294,30,361,355,215,287,312)(13,362,338,216,270,313,31,380,356,198,288,295)(14,381,339,199,271,296,32,363,357,181,253,314)(15,364,340,182,272,315,33,382,358,200,254,297)(16,383,341,201,273,298,34,365,359,183,255,316)(17,366,342,184,274,317,35,384,360,202,256,299)(18,385,343,203,275,300,36,367,325,185,257,318)(37,407,135,102,150,227,55,425,117,84,168,245)(38,426,136,85,151,246,56,408,118,103,169,228)(39,409,137,104,152,229,57,427,119,86,170,247)(40,428,138,87,153,248,58,410,120,105,171,230)(41,411,139,106,154,231,59,429,121,88,172,249)(42,430,140,89,155,250,60,412,122,107,173,232)(43,413,141,108,156,233,61,431,123,90,174,251)(44,432,142,91,157,252,62,414,124,73,175,234)(45,415,143,74,158,235,63,397,125,92,176,217)(46,398,144,93,159,218,64,416,126,75,177,236)(47,417,109,76,160,237,65,399,127,94,178,219)(48,400,110,95,161,220,66,418,128,77,179,238)(49,419,111,78,162,239,67,401,129,96,180,221)(50,402,112,97,163,222,68,420,130,79,145,240)(51,421,113,80,164,241,69,403,131,98,146,223)(52,404,114,99,165,224,70,422,132,81,147,242)(53,423,115,82,166,243,71,405,133,100,148,225)(54,406,116,101,167,226,72,424,134,83,149,244), (1,129,28,120,19,111,10,138)(2,110,29,137,20,128,11,119)(3,127,30,118,21,109,12,136)(4,144,31,135,22,126,13,117)(5,125,32,116,23,143,14,134)(6,142,33,133,24,124,15,115)(7,123,34,114,25,141,16,132)(8,140,35,131,26,122,17,113)(9,121,36,112,27,139,18,130)(37,347,64,338,55,329,46,356)(38,328,65,355,56,346,47,337)(39,345,66,336,57,327,48,354)(40,326,67,353,58,344,49,335)(41,343,68,334,59,325,50,352)(42,360,69,351,60,342,51,333)(43,341,70,332,61,359,52,350)(44,358,71,349,62,340,53,331)(45,339,72,330,63,357,54,348)(73,306,100,297,91,324,82,315)(74,323,101,314,92,305,83,296)(75,304,102,295,93,322,84,313)(76,321,103,312,94,303,85,294)(77,302,104,293,95,320,86,311)(78,319,105,310,96,301,87,292)(79,300,106,291,97,318,88,309)(80,317,107,308,98,299,89,290)(81,298,108,289,99,316,90,307)(145,284,172,275,163,266,154,257)(146,265,173,256,164,283,155,274)(147,282,174,273,165,264,156,255)(148,263,175,254,166,281,157,272)(149,280,176,271,167,262,158,253)(150,261,177,288,168,279,159,270)(151,278,178,269,169,260,160,287)(152,259,179,286,170,277,161,268)(153,276,180,267,171,258,162,285)(181,217,208,244,199,235,190,226)(182,234,209,225,200,252,191,243)(183,251,210,242,201,233,192,224)(184,232,211,223,202,250,193,241)(185,249,212,240,203,231,194,222)(186,230,213,221,204,248,195,239)(187,247,214,238,205,229,196,220)(188,228,215,219,206,246,197,237)(189,245,216,236,207,227,198,218)(361,417,388,408,379,399,370,426)(362,398,389,425,380,416,371,407)(363,415,390,406,381,397,372,424)(364,432,391,423,382,414,373,405)(365,413,392,404,383,431,374,422)(366,430,393,421,384,412,375,403)(367,411,394,402,385,429,376,420)(368,428,395,419,386,410,377,401)(369,409,396,400,387,427,378,418) );
G=PermutationGroup([[(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)], [(1,386,326,204,258,301,19,368,344,186,276,319),(2,369,327,187,259,320,20,387,345,205,277,302),(3,388,328,206,260,303,21,370,346,188,278,321),(4,371,329,189,261,322,22,389,347,207,279,304),(5,390,330,208,262,305,23,372,348,190,280,323),(6,373,331,191,263,324,24,391,349,209,281,306),(7,392,332,210,264,307,25,374,350,192,282,289),(8,375,333,193,265,290,26,393,351,211,283,308),(9,394,334,212,266,309,27,376,352,194,284,291),(10,377,335,195,267,292,28,395,353,213,285,310),(11,396,336,214,268,311,29,378,354,196,286,293),(12,379,337,197,269,294,30,361,355,215,287,312),(13,362,338,216,270,313,31,380,356,198,288,295),(14,381,339,199,271,296,32,363,357,181,253,314),(15,364,340,182,272,315,33,382,358,200,254,297),(16,383,341,201,273,298,34,365,359,183,255,316),(17,366,342,184,274,317,35,384,360,202,256,299),(18,385,343,203,275,300,36,367,325,185,257,318),(37,407,135,102,150,227,55,425,117,84,168,245),(38,426,136,85,151,246,56,408,118,103,169,228),(39,409,137,104,152,229,57,427,119,86,170,247),(40,428,138,87,153,248,58,410,120,105,171,230),(41,411,139,106,154,231,59,429,121,88,172,249),(42,430,140,89,155,250,60,412,122,107,173,232),(43,413,141,108,156,233,61,431,123,90,174,251),(44,432,142,91,157,252,62,414,124,73,175,234),(45,415,143,74,158,235,63,397,125,92,176,217),(46,398,144,93,159,218,64,416,126,75,177,236),(47,417,109,76,160,237,65,399,127,94,178,219),(48,400,110,95,161,220,66,418,128,77,179,238),(49,419,111,78,162,239,67,401,129,96,180,221),(50,402,112,97,163,222,68,420,130,79,145,240),(51,421,113,80,164,241,69,403,131,98,146,223),(52,404,114,99,165,224,70,422,132,81,147,242),(53,423,115,82,166,243,71,405,133,100,148,225),(54,406,116,101,167,226,72,424,134,83,149,244)], [(1,129,28,120,19,111,10,138),(2,110,29,137,20,128,11,119),(3,127,30,118,21,109,12,136),(4,144,31,135,22,126,13,117),(5,125,32,116,23,143,14,134),(6,142,33,133,24,124,15,115),(7,123,34,114,25,141,16,132),(8,140,35,131,26,122,17,113),(9,121,36,112,27,139,18,130),(37,347,64,338,55,329,46,356),(38,328,65,355,56,346,47,337),(39,345,66,336,57,327,48,354),(40,326,67,353,58,344,49,335),(41,343,68,334,59,325,50,352),(42,360,69,351,60,342,51,333),(43,341,70,332,61,359,52,350),(44,358,71,349,62,340,53,331),(45,339,72,330,63,357,54,348),(73,306,100,297,91,324,82,315),(74,323,101,314,92,305,83,296),(75,304,102,295,93,322,84,313),(76,321,103,312,94,303,85,294),(77,302,104,293,95,320,86,311),(78,319,105,310,96,301,87,292),(79,300,106,291,97,318,88,309),(80,317,107,308,98,299,89,290),(81,298,108,289,99,316,90,307),(145,284,172,275,163,266,154,257),(146,265,173,256,164,283,155,274),(147,282,174,273,165,264,156,255),(148,263,175,254,166,281,157,272),(149,280,176,271,167,262,158,253),(150,261,177,288,168,279,159,270),(151,278,178,269,169,260,160,287),(152,259,179,286,170,277,161,268),(153,276,180,267,171,258,162,285),(181,217,208,244,199,235,190,226),(182,234,209,225,200,252,191,243),(183,251,210,242,201,233,192,224),(184,232,211,223,202,250,193,241),(185,249,212,240,203,231,194,222),(186,230,213,221,204,248,195,239),(187,247,214,238,205,229,196,220),(188,228,215,219,206,246,197,237),(189,245,216,236,207,227,198,218),(361,417,388,408,379,399,370,426),(362,398,389,425,380,416,371,407),(363,415,390,406,381,397,372,424),(364,432,391,423,382,414,373,405),(365,413,392,404,383,431,374,422),(366,430,393,421,384,412,375,403),(367,411,394,402,385,429,376,420),(368,428,395,419,386,410,377,401),(369,409,396,400,387,427,378,418)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 9A | ··· | 9I | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AA |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 108 | 2 | 2 | 2 | 2 | 54 | 54 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | Q16 | D9 | C3⋊D4 | C3⋊D4 | D18 | C9⋊D4 | C3⋊Q16 | C3⋊Q16 | C9⋊Q16 |
| kernel | C36.19D6 | C36.S3 | C12.D9 | Q8×C3×C9 | Q8×C9 | Q8×C32 | C3×C18 | C36 | C3×C12 | C3×C9 | C3×Q8 | C18 | C3×C6 | C12 | C6 | C9 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 1 | 2 | 9 | 6 | 2 | 9 | 18 | 3 | 1 | 9 |
Matrix representation of C36.19D6 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 23 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 13 | 30 | 0 | 0 | 0 | 0 |
| 43 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 41 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 61 |
| 0 | 0 | 0 | 0 | 67 | 0 |
| 11 | 25 | 0 | 0 | 0 | 0 |
| 36 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 57 | 0 | 0 |
| 0 | 0 | 6 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 16 | 41 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,1,0,0,0,0,0,0,4,23,0,0,0,0,0,55,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[13,43,0,0,0,0,30,43,0,0,0,0,0,0,9,41,0,0,0,0,0,65,0,0,0,0,0,0,0,67,0,0,0,0,61,0],[11,36,0,0,0,0,25,62,0,0,0,0,0,0,30,6,0,0,0,0,57,43,0,0,0,0,0,0,0,16,0,0,0,0,41,41] >;
C36.19D6 in GAP, Magma, Sage, TeX
C_{36}._{19}D_6 % in TeX
G:=Group("C36.19D6"); // GroupNames label
G:=SmallGroup(432,194);
// by ID
G=gap.SmallGroup(432,194);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^6=a^18,c^2=a^27,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^27*b^5>;
// generators/relations