direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×Dic17, C34⋊2C12, C102⋊4C4, C6.16D34, C102.16C22, (C2×C34).C6, C17⋊3(C2×C12), C51⋊11(C2×C4), C34.4(C2×C6), (C2×C6).2D17, C2.2(C6×D17), C22.(C3×D17), (C2×C102).2C2, SmallGroup(408,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C6×Dic17 |
Generators and relations for C6×Dic17
G = < a,b,c | a6=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×Dic17
►Regular action on 408 points
Generators in S408
(1 164 97 115 43 175)(2 165 98 116 44 176)(3 166 99 117 45 177)(4 167 100 118 46 178)(5 168 101 119 47 179)(6 169 102 120 48 180)(7 170 69 121 49 181)(8 137 70 122 50 182)(9 138 71 123 51 183)(10 139 72 124 52 184)(11 140 73 125 53 185)(12 141 74 126 54 186)(13 142 75 127 55 187)(14 143 76 128 56 188)(15 144 77 129 57 189)(16 145 78 130 58 190)(17 146 79 131 59 191)(18 147 80 132 60 192)(19 148 81 133 61 193)(20 149 82 134 62 194)(21 150 83 135 63 195)(22 151 84 136 64 196)(23 152 85 103 65 197)(24 153 86 104 66 198)(25 154 87 105 67 199)(26 155 88 106 68 200)(27 156 89 107 35 201)(28 157 90 108 36 202)(29 158 91 109 37 203)(30 159 92 110 38 204)(31 160 93 111 39 171)(32 161 94 112 40 172)(33 162 95 113 41 173)(34 163 96 114 42 174)(205 358 273 307 256 392)(206 359 274 308 257 393)(207 360 275 309 258 394)(208 361 276 310 259 395)(209 362 277 311 260 396)(210 363 278 312 261 397)(211 364 279 313 262 398)(212 365 280 314 263 399)(213 366 281 315 264 400)(214 367 282 316 265 401)(215 368 283 317 266 402)(216 369 284 318 267 403)(217 370 285 319 268 404)(218 371 286 320 269 405)(219 372 287 321 270 406)(220 373 288 322 271 407)(221 374 289 323 272 408)(222 341 290 324 239 375)(223 342 291 325 240 376)(224 343 292 326 241 377)(225 344 293 327 242 378)(226 345 294 328 243 379)(227 346 295 329 244 380)(228 347 296 330 245 381)(229 348 297 331 246 382)(230 349 298 332 247 383)(231 350 299 333 248 384)(232 351 300 334 249 385)(233 352 301 335 250 386)(234 353 302 336 251 387)(235 354 303 337 252 388)(236 355 304 338 253 389)(237 356 305 339 254 390)(238 357 306 340 255 391)
(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)
(1 205 18 222)(2 238 19 221)(3 237 20 220)(4 236 21 219)(5 235 22 218)(6 234 23 217)(7 233 24 216)(8 232 25 215)(9 231 26 214)(10 230 27 213)(11 229 28 212)(12 228 29 211)(13 227 30 210)(14 226 31 209)(15 225 32 208)(16 224 33 207)(17 223 34 206)(35 264 52 247)(36 263 53 246)(37 262 54 245)(38 261 55 244)(39 260 56 243)(40 259 57 242)(41 258 58 241)(42 257 59 240)(43 256 60 239)(44 255 61 272)(45 254 62 271)(46 253 63 270)(47 252 64 269)(48 251 65 268)(49 250 66 267)(50 249 67 266)(51 248 68 265)(69 301 86 284)(70 300 87 283)(71 299 88 282)(72 298 89 281)(73 297 90 280)(74 296 91 279)(75 295 92 278)(76 294 93 277)(77 293 94 276)(78 292 95 275)(79 291 96 274)(80 290 97 273)(81 289 98 306)(82 288 99 305)(83 287 100 304)(84 286 101 303)(85 285 102 302)(103 319 120 336)(104 318 121 335)(105 317 122 334)(106 316 123 333)(107 315 124 332)(108 314 125 331)(109 313 126 330)(110 312 127 329)(111 311 128 328)(112 310 129 327)(113 309 130 326)(114 308 131 325)(115 307 132 324)(116 340 133 323)(117 339 134 322)(118 338 135 321)(119 337 136 320)(137 351 154 368)(138 350 155 367)(139 349 156 366)(140 348 157 365)(141 347 158 364)(142 346 159 363)(143 345 160 362)(144 344 161 361)(145 343 162 360)(146 342 163 359)(147 341 164 358)(148 374 165 357)(149 373 166 356)(150 372 167 355)(151 371 168 354)(152 370 169 353)(153 369 170 352)(171 396 188 379)(172 395 189 378)(173 394 190 377)(174 393 191 376)(175 392 192 375)(176 391 193 408)(177 390 194 407)(178 389 195 406)(179 388 196 405)(180 387 197 404)(181 386 198 403)(182 385 199 402)(183 384 200 401)(184 383 201 400)(185 382 202 399)(186 381 203 398)(187 380 204 397)
G:=sub<Sym(408)| (1,164,97,115,43,175)(2,165,98,116,44,176)(3,166,99,117,45,177)(4,167,100,118,46,178)(5,168,101,119,47,179)(6,169,102,120,48,180)(7,170,69,121,49,181)(8,137,70,122,50,182)(9,138,71,123,51,183)(10,139,72,124,52,184)(11,140,73,125,53,185)(12,141,74,126,54,186)(13,142,75,127,55,187)(14,143,76,128,56,188)(15,144,77,129,57,189)(16,145,78,130,58,190)(17,146,79,131,59,191)(18,147,80,132,60,192)(19,148,81,133,61,193)(20,149,82,134,62,194)(21,150,83,135,63,195)(22,151,84,136,64,196)(23,152,85,103,65,197)(24,153,86,104,66,198)(25,154,87,105,67,199)(26,155,88,106,68,200)(27,156,89,107,35,201)(28,157,90,108,36,202)(29,158,91,109,37,203)(30,159,92,110,38,204)(31,160,93,111,39,171)(32,161,94,112,40,172)(33,162,95,113,41,173)(34,163,96,114,42,174)(205,358,273,307,256,392)(206,359,274,308,257,393)(207,360,275,309,258,394)(208,361,276,310,259,395)(209,362,277,311,260,396)(210,363,278,312,261,397)(211,364,279,313,262,398)(212,365,280,314,263,399)(213,366,281,315,264,400)(214,367,282,316,265,401)(215,368,283,317,266,402)(216,369,284,318,267,403)(217,370,285,319,268,404)(218,371,286,320,269,405)(219,372,287,321,270,406)(220,373,288,322,271,407)(221,374,289,323,272,408)(222,341,290,324,239,375)(223,342,291,325,240,376)(224,343,292,326,241,377)(225,344,293,327,242,378)(226,345,294,328,243,379)(227,346,295,329,244,380)(228,347,296,330,245,381)(229,348,297,331,246,382)(230,349,298,332,247,383)(231,350,299,333,248,384)(232,351,300,334,249,385)(233,352,301,335,250,386)(234,353,302,336,251,387)(235,354,303,337,252,388)(236,355,304,338,253,389)(237,356,305,339,254,390)(238,357,306,340,255,391), (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), (1,205,18,222)(2,238,19,221)(3,237,20,220)(4,236,21,219)(5,235,22,218)(6,234,23,217)(7,233,24,216)(8,232,25,215)(9,231,26,214)(10,230,27,213)(11,229,28,212)(12,228,29,211)(13,227,30,210)(14,226,31,209)(15,225,32,208)(16,224,33,207)(17,223,34,206)(35,264,52,247)(36,263,53,246)(37,262,54,245)(38,261,55,244)(39,260,56,243)(40,259,57,242)(41,258,58,241)(42,257,59,240)(43,256,60,239)(44,255,61,272)(45,254,62,271)(46,253,63,270)(47,252,64,269)(48,251,65,268)(49,250,66,267)(50,249,67,266)(51,248,68,265)(69,301,86,284)(70,300,87,283)(71,299,88,282)(72,298,89,281)(73,297,90,280)(74,296,91,279)(75,295,92,278)(76,294,93,277)(77,293,94,276)(78,292,95,275)(79,291,96,274)(80,290,97,273)(81,289,98,306)(82,288,99,305)(83,287,100,304)(84,286,101,303)(85,285,102,302)(103,319,120,336)(104,318,121,335)(105,317,122,334)(106,316,123,333)(107,315,124,332)(108,314,125,331)(109,313,126,330)(110,312,127,329)(111,311,128,328)(112,310,129,327)(113,309,130,326)(114,308,131,325)(115,307,132,324)(116,340,133,323)(117,339,134,322)(118,338,135,321)(119,337,136,320)(137,351,154,368)(138,350,155,367)(139,349,156,366)(140,348,157,365)(141,347,158,364)(142,346,159,363)(143,345,160,362)(144,344,161,361)(145,343,162,360)(146,342,163,359)(147,341,164,358)(148,374,165,357)(149,373,166,356)(150,372,167,355)(151,371,168,354)(152,370,169,353)(153,369,170,352)(171,396,188,379)(172,395,189,378)(173,394,190,377)(174,393,191,376)(175,392,192,375)(176,391,193,408)(177,390,194,407)(178,389,195,406)(179,388,196,405)(180,387,197,404)(181,386,198,403)(182,385,199,402)(183,384,200,401)(184,383,201,400)(185,382,202,399)(186,381,203,398)(187,380,204,397)>;
G:=Group( (1,164,97,115,43,175)(2,165,98,116,44,176)(3,166,99,117,45,177)(4,167,100,118,46,178)(5,168,101,119,47,179)(6,169,102,120,48,180)(7,170,69,121,49,181)(8,137,70,122,50,182)(9,138,71,123,51,183)(10,139,72,124,52,184)(11,140,73,125,53,185)(12,141,74,126,54,186)(13,142,75,127,55,187)(14,143,76,128,56,188)(15,144,77,129,57,189)(16,145,78,130,58,190)(17,146,79,131,59,191)(18,147,80,132,60,192)(19,148,81,133,61,193)(20,149,82,134,62,194)(21,150,83,135,63,195)(22,151,84,136,64,196)(23,152,85,103,65,197)(24,153,86,104,66,198)(25,154,87,105,67,199)(26,155,88,106,68,200)(27,156,89,107,35,201)(28,157,90,108,36,202)(29,158,91,109,37,203)(30,159,92,110,38,204)(31,160,93,111,39,171)(32,161,94,112,40,172)(33,162,95,113,41,173)(34,163,96,114,42,174)(205,358,273,307,256,392)(206,359,274,308,257,393)(207,360,275,309,258,394)(208,361,276,310,259,395)(209,362,277,311,260,396)(210,363,278,312,261,397)(211,364,279,313,262,398)(212,365,280,314,263,399)(213,366,281,315,264,400)(214,367,282,316,265,401)(215,368,283,317,266,402)(216,369,284,318,267,403)(217,370,285,319,268,404)(218,371,286,320,269,405)(219,372,287,321,270,406)(220,373,288,322,271,407)(221,374,289,323,272,408)(222,341,290,324,239,375)(223,342,291,325,240,376)(224,343,292,326,241,377)(225,344,293,327,242,378)(226,345,294,328,243,379)(227,346,295,329,244,380)(228,347,296,330,245,381)(229,348,297,331,246,382)(230,349,298,332,247,383)(231,350,299,333,248,384)(232,351,300,334,249,385)(233,352,301,335,250,386)(234,353,302,336,251,387)(235,354,303,337,252,388)(236,355,304,338,253,389)(237,356,305,339,254,390)(238,357,306,340,255,391), (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), (1,205,18,222)(2,238,19,221)(3,237,20,220)(4,236,21,219)(5,235,22,218)(6,234,23,217)(7,233,24,216)(8,232,25,215)(9,231,26,214)(10,230,27,213)(11,229,28,212)(12,228,29,211)(13,227,30,210)(14,226,31,209)(15,225,32,208)(16,224,33,207)(17,223,34,206)(35,264,52,247)(36,263,53,246)(37,262,54,245)(38,261,55,244)(39,260,56,243)(40,259,57,242)(41,258,58,241)(42,257,59,240)(43,256,60,239)(44,255,61,272)(45,254,62,271)(46,253,63,270)(47,252,64,269)(48,251,65,268)(49,250,66,267)(50,249,67,266)(51,248,68,265)(69,301,86,284)(70,300,87,283)(71,299,88,282)(72,298,89,281)(73,297,90,280)(74,296,91,279)(75,295,92,278)(76,294,93,277)(77,293,94,276)(78,292,95,275)(79,291,96,274)(80,290,97,273)(81,289,98,306)(82,288,99,305)(83,287,100,304)(84,286,101,303)(85,285,102,302)(103,319,120,336)(104,318,121,335)(105,317,122,334)(106,316,123,333)(107,315,124,332)(108,314,125,331)(109,313,126,330)(110,312,127,329)(111,311,128,328)(112,310,129,327)(113,309,130,326)(114,308,131,325)(115,307,132,324)(116,340,133,323)(117,339,134,322)(118,338,135,321)(119,337,136,320)(137,351,154,368)(138,350,155,367)(139,349,156,366)(140,348,157,365)(141,347,158,364)(142,346,159,363)(143,345,160,362)(144,344,161,361)(145,343,162,360)(146,342,163,359)(147,341,164,358)(148,374,165,357)(149,373,166,356)(150,372,167,355)(151,371,168,354)(152,370,169,353)(153,369,170,352)(171,396,188,379)(172,395,189,378)(173,394,190,377)(174,393,191,376)(175,392,192,375)(176,391,193,408)(177,390,194,407)(178,389,195,406)(179,388,196,405)(180,387,197,404)(181,386,198,403)(182,385,199,402)(183,384,200,401)(184,383,201,400)(185,382,202,399)(186,381,203,398)(187,380,204,397) );
G=PermutationGroup([[(1,164,97,115,43,175),(2,165,98,116,44,176),(3,166,99,117,45,177),(4,167,100,118,46,178),(5,168,101,119,47,179),(6,169,102,120,48,180),(7,170,69,121,49,181),(8,137,70,122,50,182),(9,138,71,123,51,183),(10,139,72,124,52,184),(11,140,73,125,53,185),(12,141,74,126,54,186),(13,142,75,127,55,187),(14,143,76,128,56,188),(15,144,77,129,57,189),(16,145,78,130,58,190),(17,146,79,131,59,191),(18,147,80,132,60,192),(19,148,81,133,61,193),(20,149,82,134,62,194),(21,150,83,135,63,195),(22,151,84,136,64,196),(23,152,85,103,65,197),(24,153,86,104,66,198),(25,154,87,105,67,199),(26,155,88,106,68,200),(27,156,89,107,35,201),(28,157,90,108,36,202),(29,158,91,109,37,203),(30,159,92,110,38,204),(31,160,93,111,39,171),(32,161,94,112,40,172),(33,162,95,113,41,173),(34,163,96,114,42,174),(205,358,273,307,256,392),(206,359,274,308,257,393),(207,360,275,309,258,394),(208,361,276,310,259,395),(209,362,277,311,260,396),(210,363,278,312,261,397),(211,364,279,313,262,398),(212,365,280,314,263,399),(213,366,281,315,264,400),(214,367,282,316,265,401),(215,368,283,317,266,402),(216,369,284,318,267,403),(217,370,285,319,268,404),(218,371,286,320,269,405),(219,372,287,321,270,406),(220,373,288,322,271,407),(221,374,289,323,272,408),(222,341,290,324,239,375),(223,342,291,325,240,376),(224,343,292,326,241,377),(225,344,293,327,242,378),(226,345,294,328,243,379),(227,346,295,329,244,380),(228,347,296,330,245,381),(229,348,297,331,246,382),(230,349,298,332,247,383),(231,350,299,333,248,384),(232,351,300,334,249,385),(233,352,301,335,250,386),(234,353,302,336,251,387),(235,354,303,337,252,388),(236,355,304,338,253,389),(237,356,305,339,254,390),(238,357,306,340,255,391)], [(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)], [(1,205,18,222),(2,238,19,221),(3,237,20,220),(4,236,21,219),(5,235,22,218),(6,234,23,217),(7,233,24,216),(8,232,25,215),(9,231,26,214),(10,230,27,213),(11,229,28,212),(12,228,29,211),(13,227,30,210),(14,226,31,209),(15,225,32,208),(16,224,33,207),(17,223,34,206),(35,264,52,247),(36,263,53,246),(37,262,54,245),(38,261,55,244),(39,260,56,243),(40,259,57,242),(41,258,58,241),(42,257,59,240),(43,256,60,239),(44,255,61,272),(45,254,62,271),(46,253,63,270),(47,252,64,269),(48,251,65,268),(49,250,66,267),(50,249,67,266),(51,248,68,265),(69,301,86,284),(70,300,87,283),(71,299,88,282),(72,298,89,281),(73,297,90,280),(74,296,91,279),(75,295,92,278),(76,294,93,277),(77,293,94,276),(78,292,95,275),(79,291,96,274),(80,290,97,273),(81,289,98,306),(82,288,99,305),(83,287,100,304),(84,286,101,303),(85,285,102,302),(103,319,120,336),(104,318,121,335),(105,317,122,334),(106,316,123,333),(107,315,124,332),(108,314,125,331),(109,313,126,330),(110,312,127,329),(111,311,128,328),(112,310,129,327),(113,309,130,326),(114,308,131,325),(115,307,132,324),(116,340,133,323),(117,339,134,322),(118,338,135,321),(119,337,136,320),(137,351,154,368),(138,350,155,367),(139,349,156,366),(140,348,157,365),(141,347,158,364),(142,346,159,363),(143,345,160,362),(144,344,161,361),(145,343,162,360),(146,342,163,359),(147,341,164,358),(148,374,165,357),(149,373,166,356),(150,372,167,355),(151,371,168,354),(152,370,169,353),(153,369,170,352),(171,396,188,379),(172,395,189,378),(173,394,190,377),(174,393,191,376),(175,392,192,375),(176,391,193,408),(177,390,194,407),(178,389,195,406),(179,388,196,405),(180,387,197,404),(181,386,198,403),(182,385,199,402),(183,384,200,401),(184,383,201,400),(185,382,202,399),(186,381,203,398),(187,380,204,397)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 17A | ··· | 17H | 34A | ··· | 34X | 51A | ··· | 51P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 1 | ··· | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D17 | Dic17 | D34 | C3×D17 | C3×Dic17 | C6×D17 |
| kernel | C6×Dic17 | C3×Dic17 | C2×C102 | C2×Dic17 | C102 | Dic17 | C2×C34 | C34 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 8 | 16 | 8 | 16 | 32 | 16 |
Matrix representation of C6×Dic17 ►in GL3(𝔽409) generated by
| 408 | 0 | 0 |
| 0 | 54 | 0 |
| 0 | 0 | 54 |
| 408 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 408 | 205 |
| 143 | 0 | 0 |
| 0 | 403 | 94 |
| 0 | 91 | 6 |
G:=sub<GL(3,GF(409))| [408,0,0,0,54,0,0,0,54],[408,0,0,0,0,408,0,1,205],[143,0,0,0,403,91,0,94,6] >;
C6×Dic17 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{17} % in TeX
G:=Group("C6xDic17"); // GroupNames label
G:=SmallGroup(408,18);
// by ID
G=gap.SmallGroup(408,18);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-17,60,9604]);
// Polycyclic
G:=Group<a,b,c|a^6=b^34=1,c^2=b^17,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export