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G = C17×Dic6order 408 = 23·3·17

Direct product of C17 and Dic6

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C17×Dic6, C514Q8, C68.3S3, C12.1C34, C204.5C2, C34.13D6, Dic3.C34, C102.18C22, C3⋊(Q8×C17), C4.(S3×C17), C2.3(S3×C34), C6.1(C2×C34), (Dic3×C17).2C2, SmallGroup(408,20)

Series: Derived Chief Lower central Upper central

C1C6 — C17×Dic6
C1C3C6C102Dic3×C17 — C17×Dic6
C3C6 — C17×Dic6
C1C34C68

Generators and relations for C17×Dic6
 G = < a,b,c | a17=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3Q8
3C68
3C68
3Q8×C17

Smallest permutation representation of C17×Dic6
Regular action on 408 points
Generators in S408
(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)
(1 149 19 69 306 391 57 258 351 123 157 277)(2 150 20 70 290 375 58 259 352 124 158 278)(3 151 21 71 291 376 59 260 353 125 159 279)(4 152 22 72 292 377 60 261 354 126 160 280)(5 153 23 73 293 378 61 262 355 127 161 281)(6 137 24 74 294 379 62 263 356 128 162 282)(7 138 25 75 295 380 63 264 357 129 163 283)(8 139 26 76 296 381 64 265 341 130 164 284)(9 140 27 77 297 382 65 266 342 131 165 285)(10 141 28 78 298 383 66 267 343 132 166 286)(11 142 29 79 299 384 67 268 344 133 167 287)(12 143 30 80 300 385 68 269 345 134 168 288)(13 144 31 81 301 386 52 270 346 135 169 289)(14 145 32 82 302 387 53 271 347 136 170 273)(15 146 33 83 303 388 54 272 348 120 154 274)(16 147 34 84 304 389 55 256 349 121 155 275)(17 148 18 85 305 390 56 257 350 122 156 276)(35 245 226 316 402 325 372 181 207 114 195 101)(36 246 227 317 403 326 373 182 208 115 196 102)(37 247 228 318 404 327 374 183 209 116 197 86)(38 248 229 319 405 328 358 184 210 117 198 87)(39 249 230 320 406 329 359 185 211 118 199 88)(40 250 231 321 407 330 360 186 212 119 200 89)(41 251 232 322 408 331 361 187 213 103 201 90)(42 252 233 323 392 332 362 171 214 104 202 91)(43 253 234 307 393 333 363 172 215 105 203 92)(44 254 235 308 394 334 364 173 216 106 204 93)(45 255 236 309 395 335 365 174 217 107 188 94)(46 239 237 310 396 336 366 175 218 108 189 95)(47 240 238 311 397 337 367 176 219 109 190 96)(48 241 222 312 398 338 368 177 220 110 191 97)(49 242 223 313 399 339 369 178 221 111 192 98)(50 243 224 314 400 340 370 179 205 112 193 99)(51 244 225 315 401 324 371 180 206 113 194 100)
(1 101 57 325)(2 102 58 326)(3 86 59 327)(4 87 60 328)(5 88 61 329)(6 89 62 330)(7 90 63 331)(8 91 64 332)(9 92 65 333)(10 93 66 334)(11 94 67 335)(12 95 68 336)(13 96 52 337)(14 97 53 338)(15 98 54 339)(16 99 55 340)(17 100 56 324)(18 113 350 315)(19 114 351 316)(20 115 352 317)(21 116 353 318)(22 117 354 319)(23 118 355 320)(24 119 356 321)(25 103 357 322)(26 104 341 323)(27 105 342 307)(28 106 343 308)(29 107 344 309)(30 108 345 310)(31 109 346 311)(32 110 347 312)(33 111 348 313)(34 112 349 314)(35 391 372 277)(36 375 373 278)(37 376 374 279)(38 377 358 280)(39 378 359 281)(40 379 360 282)(41 380 361 283)(42 381 362 284)(43 382 363 285)(44 383 364 286)(45 384 365 287)(46 385 366 288)(47 386 367 289)(48 387 368 273)(49 388 369 274)(50 389 370 275)(51 390 371 276)(69 207 123 226)(70 208 124 227)(71 209 125 228)(72 210 126 229)(73 211 127 230)(74 212 128 231)(75 213 129 232)(76 214 130 233)(77 215 131 234)(78 216 132 235)(79 217 133 236)(80 218 134 237)(81 219 135 238)(82 220 136 222)(83 221 120 223)(84 205 121 224)(85 206 122 225)(137 200 263 407)(138 201 264 408)(139 202 265 392)(140 203 266 393)(141 204 267 394)(142 188 268 395)(143 189 269 396)(144 190 270 397)(145 191 271 398)(146 192 272 399)(147 193 256 400)(148 194 257 401)(149 195 258 402)(150 196 259 403)(151 197 260 404)(152 198 261 405)(153 199 262 406)(154 242 303 178)(155 243 304 179)(156 244 305 180)(157 245 306 181)(158 246 290 182)(159 247 291 183)(160 248 292 184)(161 249 293 185)(162 250 294 186)(163 251 295 187)(164 252 296 171)(165 253 297 172)(166 254 298 173)(167 255 299 174)(168 239 300 175)(169 240 301 176)(170 241 302 177)

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

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

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

153 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B17A···17P34A···34P51A···51P68A···68P68Q···68AV102A···102P204A···204AF
order1234446121217···1734···3451···5168···6868···68102···102204···204
size1122662221···11···12···22···26···62···22···2

153 irreducible representations

dim11111122222222
type++++-+-
imageC1C2C2C17C34C34S3Q8D6Dic6S3×C17Q8×C17S3×C34C17×Dic6
kernelC17×Dic6Dic3×C17C204Dic6Dic3C12C68C51C34C17C4C3C2C1
# reps121163216111216161632

Matrix representation of C17×Dic6 in GL4(𝔽409) generated by

341000
034100
0010
0001
,
040800
1100
004083
002721
,
18916600
38622000
00288168
0093121
G:=sub<GL(4,GF(409))| [341,0,0,0,0,341,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,408,1,0,0,0,0,408,272,0,0,3,1],[189,386,0,0,166,220,0,0,0,0,288,93,0,0,168,121] >;

C17×Dic6 in GAP, Magma, Sage, TeX

C_{17}\times {\rm Dic}_6
% in TeX

G:=Group("C17xDic6");
// GroupNames label

G:=SmallGroup(408,20);
// by ID

G=gap.SmallGroup(408,20);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-3,340,701,346,6804]);
// Polycyclic

G:=Group<a,b,c|a^17=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C17×Dic6 in TeX

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