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G = C3×C172C8order 408 = 23·3·17

Direct product of C3 and C172C8

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

Aliases: C3×C172C8, C514C8, C172C24, C34.C12, C102.2C4, Dic17.2C6, C6.2(C17⋊C4), (C3×Dic17).4C2, C2.(C3×C17⋊C4), SmallGroup(408,5)

Series: Derived Chief Lower central Upper central

C1C17 — C3×C172C8
C1C17C34Dic17C3×Dic17 — C3×C172C8
C17 — C3×C172C8
C1C6

Generators and relations for C3×C172C8
 G = < a,b,c | a3=b17=c8=1, ab=ba, ac=ca, cbc-1=b4 >

17C4
17C8
17C12
17C24

Smallest permutation representation of C3×C172C8
Regular action on 408 points
Generators in S408
(1 137 69)(2 138 70)(3 139 71)(4 140 72)(5 141 73)(6 142 74)(7 143 75)(8 144 76)(9 145 77)(10 146 78)(11 147 79)(12 148 80)(13 149 81)(14 150 82)(15 151 83)(16 152 84)(17 153 85)(18 154 86)(19 155 87)(20 156 88)(21 157 89)(22 158 90)(23 159 91)(24 160 92)(25 161 93)(26 162 94)(27 163 95)(28 164 96)(29 165 97)(30 166 98)(31 167 99)(32 168 100)(33 169 101)(34 170 102)(35 181 106)(36 182 107)(37 183 108)(38 184 109)(39 185 110)(40 186 111)(41 187 112)(42 171 113)(43 172 114)(44 173 115)(45 174 116)(46 175 117)(47 176 118)(48 177 119)(49 178 103)(50 179 104)(51 180 105)(52 204 132)(53 188 133)(54 189 134)(55 190 135)(56 191 136)(57 192 120)(58 193 121)(59 194 122)(60 195 123)(61 196 124)(62 197 125)(63 198 126)(64 199 127)(65 200 128)(66 201 129)(67 202 130)(68 203 131)(205 348 273)(206 349 274)(207 350 275)(208 351 276)(209 352 277)(210 353 278)(211 354 279)(212 355 280)(213 356 281)(214 357 282)(215 341 283)(216 342 284)(217 343 285)(218 344 286)(219 345 287)(220 346 288)(221 347 289)(222 358 300)(223 359 301)(224 360 302)(225 361 303)(226 362 304)(227 363 305)(228 364 306)(229 365 290)(230 366 291)(231 367 292)(232 368 293)(233 369 294)(234 370 295)(235 371 296)(236 372 297)(237 373 298)(238 374 299)(239 391 310)(240 375 311)(241 376 312)(242 377 313)(243 378 314)(244 379 315)(245 380 316)(246 381 317)(247 382 318)(248 383 319)(249 384 320)(250 385 321)(251 386 322)(252 387 323)(253 388 307)(254 389 308)(255 390 309)(256 393 331)(257 394 332)(258 395 333)(259 396 334)(260 397 335)(261 398 336)(262 399 337)(263 400 338)(264 401 339)(265 402 340)(266 403 324)(267 404 325)(268 405 326)(269 406 327)(270 407 328)(271 408 329)(272 392 330)
(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)
(1 260 59 224 18 241 49 210)(2 256 58 228 19 254 48 214)(3 269 57 232 20 250 47 218)(4 265 56 236 21 246 46 205)(5 261 55 223 22 242 45 209)(6 257 54 227 23 255 44 213)(7 270 53 231 24 251 43 217)(8 266 52 235 25 247 42 221)(9 262 68 222 26 243 41 208)(10 258 67 226 27 239 40 212)(11 271 66 230 28 252 39 216)(12 267 65 234 29 248 38 220)(13 263 64 238 30 244 37 207)(14 259 63 225 31 240 36 211)(15 272 62 229 32 253 35 215)(16 268 61 233 33 249 51 219)(17 264 60 237 34 245 50 206)(69 335 122 302 86 312 103 278)(70 331 121 306 87 308 119 282)(71 327 120 293 88 321 118 286)(72 340 136 297 89 317 117 273)(73 336 135 301 90 313 116 277)(74 332 134 305 91 309 115 281)(75 328 133 292 92 322 114 285)(76 324 132 296 93 318 113 289)(77 337 131 300 94 314 112 276)(78 333 130 304 95 310 111 280)(79 329 129 291 96 323 110 284)(80 325 128 295 97 319 109 288)(81 338 127 299 98 315 108 275)(82 334 126 303 99 311 107 279)(83 330 125 290 100 307 106 283)(84 326 124 294 101 320 105 287)(85 339 123 298 102 316 104 274)(137 397 194 360 154 376 178 353)(138 393 193 364 155 389 177 357)(139 406 192 368 156 385 176 344)(140 402 191 372 157 381 175 348)(141 398 190 359 158 377 174 352)(142 394 189 363 159 390 173 356)(143 407 188 367 160 386 172 343)(144 403 204 371 161 382 171 347)(145 399 203 358 162 378 187 351)(146 395 202 362 163 391 186 355)(147 408 201 366 164 387 185 342)(148 404 200 370 165 383 184 346)(149 400 199 374 166 379 183 350)(150 396 198 361 167 375 182 354)(151 392 197 365 168 388 181 341)(152 405 196 369 169 384 180 345)(153 401 195 373 170 380 179 349)

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

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

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

48 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D12A12B12C12D17A17B17C17D24A···24H34A34B34C34D51A···51H102A···102H
order123344668888121212121717171724···243434343451···51102···102
size11111717111717171717171717444417···1744444···44···4

48 irreducible representations

dim111111114444
type+++-
imageC1C2C3C4C6C8C12C24C17⋊C4C172C8C3×C17⋊C4C3×C172C8
kernelC3×C172C8C3×Dic17C172C8C102Dic17C51C34C17C6C3C2C1
# reps112224484488

Matrix representation of C3×C172C8 in GL5(𝔽409)

530000
01000
00100
00010
00001
,
10000
0901590408
01000
00100
00010
,
310000
01793464833
0357114111379
086247264213
01879393261

G:=sub<GL(5,GF(409))| [53,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,90,1,0,0,0,15,0,1,0,0,90,0,0,1,0,408,0,0,0],[31,0,0,0,0,0,179,357,86,18,0,346,114,247,79,0,48,111,264,393,0,33,379,213,261] >;

C3×C172C8 in GAP, Magma, Sage, TeX

C_3\times C_{17}\rtimes_2C_8
% in TeX

G:=Group("C3xC17:2C8");
// GroupNames label

G:=SmallGroup(408,5);
// by ID

G=gap.SmallGroup(408,5);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-17,30,42,7804,1614]);
// Polycyclic

G:=Group<a,b,c|a^3=b^17=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C3×C172C8 in TeX

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