metabelian, supersoluble, monomial
Aliases: C6.7Dic18, C18.7Dic6, C62.124D6, (C2×C36).4S3, (C6×C36).5C2, C9⋊Dic3⋊3C4, C6.14(C4×D9), (C2×C12).4D9, (C3×C18).8Q8, C18.15(C4×S3), (C6×C12).25S3, (C2×C6).36D18, (C3×C18).45D4, (C2×C18).36D6, C9⋊3(Dic3⋊C4), C3⋊3(Dic9⋊C4), C6.22(C9⋊D4), (C3×C6).24Dic6, C18.22(C3⋊D4), (C6×C18).38C22, C3.(C6.Dic6), C6.4(C32⋊4Q8), C2.1(C12.D9), C6.14(C32⋊7D4), C2.1(C6.D18), C32.5(Dic3⋊C4), (C3×C9)⋊8(C4⋊C4), C6.9(C4×C3⋊S3), C2.4(C4×C9⋊S3), (C2×C4).1(C9⋊S3), (C3×C6).71(C4×S3), C22.4(C2×C9⋊S3), (C2×C12).1(C3⋊S3), (C3×C18).25(C2×C4), (C2×C9⋊Dic3).5C2, (C3×C6).97(C3⋊D4), (C2×C6).30(C2×C3⋊S3), SmallGroup(432,181)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.Dic18
G = < a,b,c | a18=b12=1, c2=a9b6, ab=ba, cac-1=a-1, cbc-1=a9b-1 >
Subgroups: 612 in 130 conjugacy classes, 65 normal (27 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, Dic9, C36, C2×C18, C3⋊Dic3, C3×C12, C62, Dic3⋊C4, C3×C18, C2×Dic9, C2×C36, C2×C3⋊Dic3, C6×C12, C9⋊Dic3, C9⋊Dic3, C3×C36, C6×C18, Dic9⋊C4, C6.Dic6, C2×C9⋊Dic3, C6×C36, C6.Dic18
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D9, C3⋊S3, Dic6, C4×S3, C3⋊D4, D18, C2×C3⋊S3, Dic3⋊C4, C9⋊S3, Dic18, C4×D9, C9⋊D4, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C2×C9⋊S3, Dic9⋊C4, C6.Dic6, C12.D9, C4×C9⋊S3, C6.D18, C6.Dic18
►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)
(1 382 66 39 404 94 136 347 362 259 229 79)(2 383 67 40 405 95 137 348 363 260 230 80)(3 384 68 41 406 96 138 349 364 261 231 81)(4 385 69 42 407 97 139 350 365 262 232 82)(5 386 70 43 408 98 140 351 366 263 233 83)(6 387 71 44 409 99 141 352 367 264 234 84)(7 388 72 45 410 100 142 353 368 265 217 85)(8 389 55 46 411 101 143 354 369 266 218 86)(9 390 56 47 412 102 144 355 370 267 219 87)(10 391 57 48 413 103 127 356 371 268 220 88)(11 392 58 49 414 104 128 357 372 269 221 89)(12 393 59 50 397 105 129 358 373 270 222 90)(13 394 60 51 398 106 130 359 374 253 223 73)(14 395 61 52 399 107 131 360 375 254 224 74)(15 396 62 53 400 108 132 343 376 255 225 75)(16 379 63 54 401 91 133 344 377 256 226 76)(17 380 64 37 402 92 134 345 378 257 227 77)(18 381 65 38 403 93 135 346 361 258 228 78)(19 171 296 148 123 205 338 417 184 315 277 246)(20 172 297 149 124 206 339 418 185 316 278 247)(21 173 298 150 125 207 340 419 186 317 279 248)(22 174 299 151 126 208 341 420 187 318 280 249)(23 175 300 152 109 209 342 421 188 319 281 250)(24 176 301 153 110 210 325 422 189 320 282 251)(25 177 302 154 111 211 326 423 190 321 283 252)(26 178 303 155 112 212 327 424 191 322 284 235)(27 179 304 156 113 213 328 425 192 323 285 236)(28 180 305 157 114 214 329 426 193 324 286 237)(29 163 306 158 115 215 330 427 194 307 287 238)(30 164 289 159 116 216 331 428 195 308 288 239)(31 165 290 160 117 199 332 429 196 309 271 240)(32 166 291 161 118 200 333 430 197 310 272 241)(33 167 292 162 119 201 334 431 198 311 273 242)(34 168 293 145 120 202 335 432 181 312 274 243)(35 169 294 146 121 203 336 415 182 313 275 244)(36 170 295 147 122 204 337 416 183 314 276 245)
(1 178 127 415)(2 177 128 432)(3 176 129 431)(4 175 130 430)(5 174 131 429)(6 173 132 428)(7 172 133 427)(8 171 134 426)(9 170 135 425)(10 169 136 424)(11 168 137 423)(12 167 138 422)(13 166 139 421)(14 165 140 420)(15 164 141 419)(16 163 142 418)(17 180 143 417)(18 179 144 416)(19 354 329 380)(20 353 330 379)(21 352 331 396)(22 351 332 395)(23 350 333 394)(24 349 334 393)(25 348 335 392)(26 347 336 391)(27 346 337 390)(28 345 338 389)(29 344 339 388)(30 343 340 387)(31 360 341 386)(32 359 342 385)(33 358 325 384)(34 357 326 383)(35 356 327 382)(36 355 328 381)(37 277 266 114)(38 276 267 113)(39 275 268 112)(40 274 269 111)(41 273 270 110)(42 272 253 109)(43 271 254 126)(44 288 255 125)(45 287 256 124)(46 286 257 123)(47 285 258 122)(48 284 259 121)(49 283 260 120)(50 282 261 119)(51 281 262 118)(52 280 263 117)(53 279 264 116)(54 278 265 115)(55 246 378 214)(56 245 361 213)(57 244 362 212)(58 243 363 211)(59 242 364 210)(60 241 365 209)(61 240 366 208)(62 239 367 207)(63 238 368 206)(64 237 369 205)(65 236 370 204)(66 235 371 203)(67 252 372 202)(68 251 373 201)(69 250 374 200)(70 249 375 199)(71 248 376 216)(72 247 377 215)(73 300 97 197)(74 299 98 196)(75 298 99 195)(76 297 100 194)(77 296 101 193)(78 295 102 192)(79 294 103 191)(80 293 104 190)(81 292 105 189)(82 291 106 188)(83 290 107 187)(84 289 108 186)(85 306 91 185)(86 305 92 184)(87 304 93 183)(88 303 94 182)(89 302 95 181)(90 301 96 198)(145 405 321 221)(146 404 322 220)(147 403 323 219)(148 402 324 218)(149 401 307 217)(150 400 308 234)(151 399 309 233)(152 398 310 232)(153 397 311 231)(154 414 312 230)(155 413 313 229)(156 412 314 228)(157 411 315 227)(158 410 316 226)(159 409 317 225)(160 408 318 224)(161 407 319 223)(162 406 320 222)
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), (1,382,66,39,404,94,136,347,362,259,229,79)(2,383,67,40,405,95,137,348,363,260,230,80)(3,384,68,41,406,96,138,349,364,261,231,81)(4,385,69,42,407,97,139,350,365,262,232,82)(5,386,70,43,408,98,140,351,366,263,233,83)(6,387,71,44,409,99,141,352,367,264,234,84)(7,388,72,45,410,100,142,353,368,265,217,85)(8,389,55,46,411,101,143,354,369,266,218,86)(9,390,56,47,412,102,144,355,370,267,219,87)(10,391,57,48,413,103,127,356,371,268,220,88)(11,392,58,49,414,104,128,357,372,269,221,89)(12,393,59,50,397,105,129,358,373,270,222,90)(13,394,60,51,398,106,130,359,374,253,223,73)(14,395,61,52,399,107,131,360,375,254,224,74)(15,396,62,53,400,108,132,343,376,255,225,75)(16,379,63,54,401,91,133,344,377,256,226,76)(17,380,64,37,402,92,134,345,378,257,227,77)(18,381,65,38,403,93,135,346,361,258,228,78)(19,171,296,148,123,205,338,417,184,315,277,246)(20,172,297,149,124,206,339,418,185,316,278,247)(21,173,298,150,125,207,340,419,186,317,279,248)(22,174,299,151,126,208,341,420,187,318,280,249)(23,175,300,152,109,209,342,421,188,319,281,250)(24,176,301,153,110,210,325,422,189,320,282,251)(25,177,302,154,111,211,326,423,190,321,283,252)(26,178,303,155,112,212,327,424,191,322,284,235)(27,179,304,156,113,213,328,425,192,323,285,236)(28,180,305,157,114,214,329,426,193,324,286,237)(29,163,306,158,115,215,330,427,194,307,287,238)(30,164,289,159,116,216,331,428,195,308,288,239)(31,165,290,160,117,199,332,429,196,309,271,240)(32,166,291,161,118,200,333,430,197,310,272,241)(33,167,292,162,119,201,334,431,198,311,273,242)(34,168,293,145,120,202,335,432,181,312,274,243)(35,169,294,146,121,203,336,415,182,313,275,244)(36,170,295,147,122,204,337,416,183,314,276,245), (1,178,127,415)(2,177,128,432)(3,176,129,431)(4,175,130,430)(5,174,131,429)(6,173,132,428)(7,172,133,427)(8,171,134,426)(9,170,135,425)(10,169,136,424)(11,168,137,423)(12,167,138,422)(13,166,139,421)(14,165,140,420)(15,164,141,419)(16,163,142,418)(17,180,143,417)(18,179,144,416)(19,354,329,380)(20,353,330,379)(21,352,331,396)(22,351,332,395)(23,350,333,394)(24,349,334,393)(25,348,335,392)(26,347,336,391)(27,346,337,390)(28,345,338,389)(29,344,339,388)(30,343,340,387)(31,360,341,386)(32,359,342,385)(33,358,325,384)(34,357,326,383)(35,356,327,382)(36,355,328,381)(37,277,266,114)(38,276,267,113)(39,275,268,112)(40,274,269,111)(41,273,270,110)(42,272,253,109)(43,271,254,126)(44,288,255,125)(45,287,256,124)(46,286,257,123)(47,285,258,122)(48,284,259,121)(49,283,260,120)(50,282,261,119)(51,281,262,118)(52,280,263,117)(53,279,264,116)(54,278,265,115)(55,246,378,214)(56,245,361,213)(57,244,362,212)(58,243,363,211)(59,242,364,210)(60,241,365,209)(61,240,366,208)(62,239,367,207)(63,238,368,206)(64,237,369,205)(65,236,370,204)(66,235,371,203)(67,252,372,202)(68,251,373,201)(69,250,374,200)(70,249,375,199)(71,248,376,216)(72,247,377,215)(73,300,97,197)(74,299,98,196)(75,298,99,195)(76,297,100,194)(77,296,101,193)(78,295,102,192)(79,294,103,191)(80,293,104,190)(81,292,105,189)(82,291,106,188)(83,290,107,187)(84,289,108,186)(85,306,91,185)(86,305,92,184)(87,304,93,183)(88,303,94,182)(89,302,95,181)(90,301,96,198)(145,405,321,221)(146,404,322,220)(147,403,323,219)(148,402,324,218)(149,401,307,217)(150,400,308,234)(151,399,309,233)(152,398,310,232)(153,397,311,231)(154,414,312,230)(155,413,313,229)(156,412,314,228)(157,411,315,227)(158,410,316,226)(159,409,317,225)(160,408,318,224)(161,407,319,223)(162,406,320,222)>;
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), (1,382,66,39,404,94,136,347,362,259,229,79)(2,383,67,40,405,95,137,348,363,260,230,80)(3,384,68,41,406,96,138,349,364,261,231,81)(4,385,69,42,407,97,139,350,365,262,232,82)(5,386,70,43,408,98,140,351,366,263,233,83)(6,387,71,44,409,99,141,352,367,264,234,84)(7,388,72,45,410,100,142,353,368,265,217,85)(8,389,55,46,411,101,143,354,369,266,218,86)(9,390,56,47,412,102,144,355,370,267,219,87)(10,391,57,48,413,103,127,356,371,268,220,88)(11,392,58,49,414,104,128,357,372,269,221,89)(12,393,59,50,397,105,129,358,373,270,222,90)(13,394,60,51,398,106,130,359,374,253,223,73)(14,395,61,52,399,107,131,360,375,254,224,74)(15,396,62,53,400,108,132,343,376,255,225,75)(16,379,63,54,401,91,133,344,377,256,226,76)(17,380,64,37,402,92,134,345,378,257,227,77)(18,381,65,38,403,93,135,346,361,258,228,78)(19,171,296,148,123,205,338,417,184,315,277,246)(20,172,297,149,124,206,339,418,185,316,278,247)(21,173,298,150,125,207,340,419,186,317,279,248)(22,174,299,151,126,208,341,420,187,318,280,249)(23,175,300,152,109,209,342,421,188,319,281,250)(24,176,301,153,110,210,325,422,189,320,282,251)(25,177,302,154,111,211,326,423,190,321,283,252)(26,178,303,155,112,212,327,424,191,322,284,235)(27,179,304,156,113,213,328,425,192,323,285,236)(28,180,305,157,114,214,329,426,193,324,286,237)(29,163,306,158,115,215,330,427,194,307,287,238)(30,164,289,159,116,216,331,428,195,308,288,239)(31,165,290,160,117,199,332,429,196,309,271,240)(32,166,291,161,118,200,333,430,197,310,272,241)(33,167,292,162,119,201,334,431,198,311,273,242)(34,168,293,145,120,202,335,432,181,312,274,243)(35,169,294,146,121,203,336,415,182,313,275,244)(36,170,295,147,122,204,337,416,183,314,276,245), (1,178,127,415)(2,177,128,432)(3,176,129,431)(4,175,130,430)(5,174,131,429)(6,173,132,428)(7,172,133,427)(8,171,134,426)(9,170,135,425)(10,169,136,424)(11,168,137,423)(12,167,138,422)(13,166,139,421)(14,165,140,420)(15,164,141,419)(16,163,142,418)(17,180,143,417)(18,179,144,416)(19,354,329,380)(20,353,330,379)(21,352,331,396)(22,351,332,395)(23,350,333,394)(24,349,334,393)(25,348,335,392)(26,347,336,391)(27,346,337,390)(28,345,338,389)(29,344,339,388)(30,343,340,387)(31,360,341,386)(32,359,342,385)(33,358,325,384)(34,357,326,383)(35,356,327,382)(36,355,328,381)(37,277,266,114)(38,276,267,113)(39,275,268,112)(40,274,269,111)(41,273,270,110)(42,272,253,109)(43,271,254,126)(44,288,255,125)(45,287,256,124)(46,286,257,123)(47,285,258,122)(48,284,259,121)(49,283,260,120)(50,282,261,119)(51,281,262,118)(52,280,263,117)(53,279,264,116)(54,278,265,115)(55,246,378,214)(56,245,361,213)(57,244,362,212)(58,243,363,211)(59,242,364,210)(60,241,365,209)(61,240,366,208)(62,239,367,207)(63,238,368,206)(64,237,369,205)(65,236,370,204)(66,235,371,203)(67,252,372,202)(68,251,373,201)(69,250,374,200)(70,249,375,199)(71,248,376,216)(72,247,377,215)(73,300,97,197)(74,299,98,196)(75,298,99,195)(76,297,100,194)(77,296,101,193)(78,295,102,192)(79,294,103,191)(80,293,104,190)(81,292,105,189)(82,291,106,188)(83,290,107,187)(84,289,108,186)(85,306,91,185)(86,305,92,184)(87,304,93,183)(88,303,94,182)(89,302,95,181)(90,301,96,198)(145,405,321,221)(146,404,322,220)(147,403,323,219)(148,402,324,218)(149,401,307,217)(150,400,308,234)(151,399,309,233)(152,398,310,232)(153,397,311,231)(154,414,312,230)(155,413,313,229)(156,412,314,228)(157,411,315,227)(158,410,316,226)(159,409,317,225)(160,408,318,224)(161,407,319,223)(162,406,320,222) );
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)], [(1,382,66,39,404,94,136,347,362,259,229,79),(2,383,67,40,405,95,137,348,363,260,230,80),(3,384,68,41,406,96,138,349,364,261,231,81),(4,385,69,42,407,97,139,350,365,262,232,82),(5,386,70,43,408,98,140,351,366,263,233,83),(6,387,71,44,409,99,141,352,367,264,234,84),(7,388,72,45,410,100,142,353,368,265,217,85),(8,389,55,46,411,101,143,354,369,266,218,86),(9,390,56,47,412,102,144,355,370,267,219,87),(10,391,57,48,413,103,127,356,371,268,220,88),(11,392,58,49,414,104,128,357,372,269,221,89),(12,393,59,50,397,105,129,358,373,270,222,90),(13,394,60,51,398,106,130,359,374,253,223,73),(14,395,61,52,399,107,131,360,375,254,224,74),(15,396,62,53,400,108,132,343,376,255,225,75),(16,379,63,54,401,91,133,344,377,256,226,76),(17,380,64,37,402,92,134,345,378,257,227,77),(18,381,65,38,403,93,135,346,361,258,228,78),(19,171,296,148,123,205,338,417,184,315,277,246),(20,172,297,149,124,206,339,418,185,316,278,247),(21,173,298,150,125,207,340,419,186,317,279,248),(22,174,299,151,126,208,341,420,187,318,280,249),(23,175,300,152,109,209,342,421,188,319,281,250),(24,176,301,153,110,210,325,422,189,320,282,251),(25,177,302,154,111,211,326,423,190,321,283,252),(26,178,303,155,112,212,327,424,191,322,284,235),(27,179,304,156,113,213,328,425,192,323,285,236),(28,180,305,157,114,214,329,426,193,324,286,237),(29,163,306,158,115,215,330,427,194,307,287,238),(30,164,289,159,116,216,331,428,195,308,288,239),(31,165,290,160,117,199,332,429,196,309,271,240),(32,166,291,161,118,200,333,430,197,310,272,241),(33,167,292,162,119,201,334,431,198,311,273,242),(34,168,293,145,120,202,335,432,181,312,274,243),(35,169,294,146,121,203,336,415,182,313,275,244),(36,170,295,147,122,204,337,416,183,314,276,245)], [(1,178,127,415),(2,177,128,432),(3,176,129,431),(4,175,130,430),(5,174,131,429),(6,173,132,428),(7,172,133,427),(8,171,134,426),(9,170,135,425),(10,169,136,424),(11,168,137,423),(12,167,138,422),(13,166,139,421),(14,165,140,420),(15,164,141,419),(16,163,142,418),(17,180,143,417),(18,179,144,416),(19,354,329,380),(20,353,330,379),(21,352,331,396),(22,351,332,395),(23,350,333,394),(24,349,334,393),(25,348,335,392),(26,347,336,391),(27,346,337,390),(28,345,338,389),(29,344,339,388),(30,343,340,387),(31,360,341,386),(32,359,342,385),(33,358,325,384),(34,357,326,383),(35,356,327,382),(36,355,328,381),(37,277,266,114),(38,276,267,113),(39,275,268,112),(40,274,269,111),(41,273,270,110),(42,272,253,109),(43,271,254,126),(44,288,255,125),(45,287,256,124),(46,286,257,123),(47,285,258,122),(48,284,259,121),(49,283,260,120),(50,282,261,119),(51,281,262,118),(52,280,263,117),(53,279,264,116),(54,278,265,115),(55,246,378,214),(56,245,361,213),(57,244,362,212),(58,243,363,211),(59,242,364,210),(60,241,365,209),(61,240,366,208),(62,239,367,207),(63,238,368,206),(64,237,369,205),(65,236,370,204),(66,235,371,203),(67,252,372,202),(68,251,373,201),(69,250,374,200),(70,249,375,199),(71,248,376,216),(72,247,377,215),(73,300,97,197),(74,299,98,196),(75,298,99,195),(76,297,100,194),(77,296,101,193),(78,295,102,192),(79,294,103,191),(80,293,104,190),(81,292,105,189),(82,291,106,188),(83,290,107,187),(84,289,108,186),(85,306,91,185),(86,305,92,184),(87,304,93,183),(88,303,94,182),(89,302,95,181),(90,301,96,198),(145,405,321,221),(146,404,322,220),(147,403,323,219),(148,402,324,218),(149,401,307,217),(150,400,308,234),(151,399,309,233),(152,398,310,232),(153,397,311,231),(154,414,312,230),(155,413,313,229),(156,412,314,228),(157,411,315,227),(158,410,316,226),(159,409,317,225),(160,408,318,224),(161,407,319,223),(162,406,320,222)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 9A | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | S3 | S3 | D4 | Q8 | D6 | D6 | D9 | Dic6 | C4×S3 | C3⋊D4 | Dic6 | C4×S3 | C3⋊D4 | D18 | Dic18 | C4×D9 | C9⋊D4 |
| kernel | C6.Dic18 | C2×C9⋊Dic3 | C6×C36 | C9⋊Dic3 | C2×C36 | C6×C12 | C3×C18 | C3×C18 | C2×C18 | C62 | C2×C12 | C18 | C18 | C18 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 4 | 3 | 1 | 1 | 1 | 3 | 1 | 9 | 6 | 6 | 6 | 2 | 2 | 2 | 9 | 18 | 18 | 18 |
Matrix representation of C6.Dic18 ►in GL4(𝔽37) generated by
| 26 | 20 | 0 | 0 |
| 17 | 6 | 0 | 0 |
| 0 | 0 | 11 | 17 |
| 0 | 0 | 20 | 31 |
| 6 | 6 | 0 | 0 |
| 31 | 0 | 0 | 0 |
| 0 | 0 | 10 | 5 |
| 0 | 0 | 32 | 5 |
| 33 | 8 | 0 | 0 |
| 12 | 4 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 31 |
G:=sub<GL(4,GF(37))| [26,17,0,0,20,6,0,0,0,0,11,20,0,0,17,31],[6,31,0,0,6,0,0,0,0,0,10,32,0,0,5,5],[33,12,0,0,8,4,0,0,0,0,6,0,0,0,6,31] >;
C6.Dic18 in GAP, Magma, Sage, TeX
C_6.{\rm Dic}_{18} % in TeX
G:=Group("C6.Dic18"); // GroupNames label
G:=SmallGroup(432,181);
// by ID
G=gap.SmallGroup(432,181);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^18=b^12=1,c^2=a^9*b^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations