direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19×Dic5, C95⋊5C4, C5⋊2C76, C10.C38, C38.2D5, C190.3C2, C2.(D5×C19), SmallGroup(380,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C19×Dic5 |
Generators and relations for C19×Dic5
G = < a,b,c | a19=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C19×Dic5
►Regular action on 380 points
Generators in S380
(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)
(1 196 353 154 309 236 116 262 103 213)(2 197 354 155 310 237 117 263 104 214)(3 198 355 156 311 238 118 264 105 215)(4 199 356 157 312 239 119 265 106 216)(5 200 357 158 313 240 120 266 107 217)(6 201 358 159 314 241 121 248 108 218)(7 202 359 160 315 242 122 249 109 219)(8 203 360 161 316 243 123 250 110 220)(9 204 361 162 317 244 124 251 111 221)(10 205 343 163 318 245 125 252 112 222)(11 206 344 164 319 246 126 253 113 223)(12 207 345 165 320 247 127 254 114 224)(13 208 346 166 321 229 128 255 96 225)(14 209 347 167 322 230 129 256 97 226)(15 191 348 168 323 231 130 257 98 227)(16 192 349 169 305 232 131 258 99 228)(17 193 350 170 306 233 132 259 100 210)(18 194 351 171 307 234 133 260 101 211)(19 195 352 153 308 235 115 261 102 212)(20 372 61 181 331 77 42 287 274 149)(21 373 62 182 332 78 43 288 275 150)(22 374 63 183 333 79 44 289 276 151)(23 375 64 184 334 80 45 290 277 152)(24 376 65 185 335 81 46 291 278 134)(25 377 66 186 336 82 47 292 279 135)(26 378 67 187 337 83 48 293 280 136)(27 379 68 188 338 84 49 294 281 137)(28 380 69 189 339 85 50 295 282 138)(29 362 70 190 340 86 51 296 283 139)(30 363 71 172 341 87 52 297 284 140)(31 364 72 173 342 88 53 298 285 141)(32 365 73 174 324 89 54 299 267 142)(33 366 74 175 325 90 55 300 268 143)(34 367 75 176 326 91 56 301 269 144)(35 368 76 177 327 92 57 302 270 145)(36 369 58 178 328 93 39 303 271 146)(37 370 59 179 329 94 40 304 272 147)(38 371 60 180 330 95 41 286 273 148)
(1 190 236 283)(2 172 237 284)(3 173 238 285)(4 174 239 267)(5 175 240 268)(6 176 241 269)(7 177 242 270)(8 178 243 271)(9 179 244 272)(10 180 245 273)(11 181 246 274)(12 182 247 275)(13 183 229 276)(14 184 230 277)(15 185 231 278)(16 186 232 279)(17 187 233 280)(18 188 234 281)(19 189 235 282)(20 113 77 164)(21 114 78 165)(22 96 79 166)(23 97 80 167)(24 98 81 168)(25 99 82 169)(26 100 83 170)(27 101 84 171)(28 102 85 153)(29 103 86 154)(30 104 87 155)(31 105 88 156)(32 106 89 157)(33 107 90 158)(34 108 91 159)(35 109 92 160)(36 110 93 161)(37 111 94 162)(38 112 95 163)(39 360 369 250)(40 361 370 251)(41 343 371 252)(42 344 372 253)(43 345 373 254)(44 346 374 255)(45 347 375 256)(46 348 376 257)(47 349 377 258)(48 350 378 259)(49 351 379 260)(50 352 380 261)(51 353 362 262)(52 354 363 263)(53 355 364 264)(54 356 365 265)(55 357 366 266)(56 358 367 248)(57 359 368 249)(58 123 303 203)(59 124 304 204)(60 125 286 205)(61 126 287 206)(62 127 288 207)(63 128 289 208)(64 129 290 209)(65 130 291 191)(66 131 292 192)(67 132 293 193)(68 133 294 194)(69 115 295 195)(70 116 296 196)(71 117 297 197)(72 118 298 198)(73 119 299 199)(74 120 300 200)(75 121 301 201)(76 122 302 202)(134 227 335 323)(135 228 336 305)(136 210 337 306)(137 211 338 307)(138 212 339 308)(139 213 340 309)(140 214 341 310)(141 215 342 311)(142 216 324 312)(143 217 325 313)(144 218 326 314)(145 219 327 315)(146 220 328 316)(147 221 329 317)(148 222 330 318)(149 223 331 319)(150 224 332 320)(151 225 333 321)(152 226 334 322)
G:=sub<Sym(380)| (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), (1,196,353,154,309,236,116,262,103,213)(2,197,354,155,310,237,117,263,104,214)(3,198,355,156,311,238,118,264,105,215)(4,199,356,157,312,239,119,265,106,216)(5,200,357,158,313,240,120,266,107,217)(6,201,358,159,314,241,121,248,108,218)(7,202,359,160,315,242,122,249,109,219)(8,203,360,161,316,243,123,250,110,220)(9,204,361,162,317,244,124,251,111,221)(10,205,343,163,318,245,125,252,112,222)(11,206,344,164,319,246,126,253,113,223)(12,207,345,165,320,247,127,254,114,224)(13,208,346,166,321,229,128,255,96,225)(14,209,347,167,322,230,129,256,97,226)(15,191,348,168,323,231,130,257,98,227)(16,192,349,169,305,232,131,258,99,228)(17,193,350,170,306,233,132,259,100,210)(18,194,351,171,307,234,133,260,101,211)(19,195,352,153,308,235,115,261,102,212)(20,372,61,181,331,77,42,287,274,149)(21,373,62,182,332,78,43,288,275,150)(22,374,63,183,333,79,44,289,276,151)(23,375,64,184,334,80,45,290,277,152)(24,376,65,185,335,81,46,291,278,134)(25,377,66,186,336,82,47,292,279,135)(26,378,67,187,337,83,48,293,280,136)(27,379,68,188,338,84,49,294,281,137)(28,380,69,189,339,85,50,295,282,138)(29,362,70,190,340,86,51,296,283,139)(30,363,71,172,341,87,52,297,284,140)(31,364,72,173,342,88,53,298,285,141)(32,365,73,174,324,89,54,299,267,142)(33,366,74,175,325,90,55,300,268,143)(34,367,75,176,326,91,56,301,269,144)(35,368,76,177,327,92,57,302,270,145)(36,369,58,178,328,93,39,303,271,146)(37,370,59,179,329,94,40,304,272,147)(38,371,60,180,330,95,41,286,273,148), (1,190,236,283)(2,172,237,284)(3,173,238,285)(4,174,239,267)(5,175,240,268)(6,176,241,269)(7,177,242,270)(8,178,243,271)(9,179,244,272)(10,180,245,273)(11,181,246,274)(12,182,247,275)(13,183,229,276)(14,184,230,277)(15,185,231,278)(16,186,232,279)(17,187,233,280)(18,188,234,281)(19,189,235,282)(20,113,77,164)(21,114,78,165)(22,96,79,166)(23,97,80,167)(24,98,81,168)(25,99,82,169)(26,100,83,170)(27,101,84,171)(28,102,85,153)(29,103,86,154)(30,104,87,155)(31,105,88,156)(32,106,89,157)(33,107,90,158)(34,108,91,159)(35,109,92,160)(36,110,93,161)(37,111,94,162)(38,112,95,163)(39,360,369,250)(40,361,370,251)(41,343,371,252)(42,344,372,253)(43,345,373,254)(44,346,374,255)(45,347,375,256)(46,348,376,257)(47,349,377,258)(48,350,378,259)(49,351,379,260)(50,352,380,261)(51,353,362,262)(52,354,363,263)(53,355,364,264)(54,356,365,265)(55,357,366,266)(56,358,367,248)(57,359,368,249)(58,123,303,203)(59,124,304,204)(60,125,286,205)(61,126,287,206)(62,127,288,207)(63,128,289,208)(64,129,290,209)(65,130,291,191)(66,131,292,192)(67,132,293,193)(68,133,294,194)(69,115,295,195)(70,116,296,196)(71,117,297,197)(72,118,298,198)(73,119,299,199)(74,120,300,200)(75,121,301,201)(76,122,302,202)(134,227,335,323)(135,228,336,305)(136,210,337,306)(137,211,338,307)(138,212,339,308)(139,213,340,309)(140,214,341,310)(141,215,342,311)(142,216,324,312)(143,217,325,313)(144,218,326,314)(145,219,327,315)(146,220,328,316)(147,221,329,317)(148,222,330,318)(149,223,331,319)(150,224,332,320)(151,225,333,321)(152,226,334,322)>;
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), (1,196,353,154,309,236,116,262,103,213)(2,197,354,155,310,237,117,263,104,214)(3,198,355,156,311,238,118,264,105,215)(4,199,356,157,312,239,119,265,106,216)(5,200,357,158,313,240,120,266,107,217)(6,201,358,159,314,241,121,248,108,218)(7,202,359,160,315,242,122,249,109,219)(8,203,360,161,316,243,123,250,110,220)(9,204,361,162,317,244,124,251,111,221)(10,205,343,163,318,245,125,252,112,222)(11,206,344,164,319,246,126,253,113,223)(12,207,345,165,320,247,127,254,114,224)(13,208,346,166,321,229,128,255,96,225)(14,209,347,167,322,230,129,256,97,226)(15,191,348,168,323,231,130,257,98,227)(16,192,349,169,305,232,131,258,99,228)(17,193,350,170,306,233,132,259,100,210)(18,194,351,171,307,234,133,260,101,211)(19,195,352,153,308,235,115,261,102,212)(20,372,61,181,331,77,42,287,274,149)(21,373,62,182,332,78,43,288,275,150)(22,374,63,183,333,79,44,289,276,151)(23,375,64,184,334,80,45,290,277,152)(24,376,65,185,335,81,46,291,278,134)(25,377,66,186,336,82,47,292,279,135)(26,378,67,187,337,83,48,293,280,136)(27,379,68,188,338,84,49,294,281,137)(28,380,69,189,339,85,50,295,282,138)(29,362,70,190,340,86,51,296,283,139)(30,363,71,172,341,87,52,297,284,140)(31,364,72,173,342,88,53,298,285,141)(32,365,73,174,324,89,54,299,267,142)(33,366,74,175,325,90,55,300,268,143)(34,367,75,176,326,91,56,301,269,144)(35,368,76,177,327,92,57,302,270,145)(36,369,58,178,328,93,39,303,271,146)(37,370,59,179,329,94,40,304,272,147)(38,371,60,180,330,95,41,286,273,148), (1,190,236,283)(2,172,237,284)(3,173,238,285)(4,174,239,267)(5,175,240,268)(6,176,241,269)(7,177,242,270)(8,178,243,271)(9,179,244,272)(10,180,245,273)(11,181,246,274)(12,182,247,275)(13,183,229,276)(14,184,230,277)(15,185,231,278)(16,186,232,279)(17,187,233,280)(18,188,234,281)(19,189,235,282)(20,113,77,164)(21,114,78,165)(22,96,79,166)(23,97,80,167)(24,98,81,168)(25,99,82,169)(26,100,83,170)(27,101,84,171)(28,102,85,153)(29,103,86,154)(30,104,87,155)(31,105,88,156)(32,106,89,157)(33,107,90,158)(34,108,91,159)(35,109,92,160)(36,110,93,161)(37,111,94,162)(38,112,95,163)(39,360,369,250)(40,361,370,251)(41,343,371,252)(42,344,372,253)(43,345,373,254)(44,346,374,255)(45,347,375,256)(46,348,376,257)(47,349,377,258)(48,350,378,259)(49,351,379,260)(50,352,380,261)(51,353,362,262)(52,354,363,263)(53,355,364,264)(54,356,365,265)(55,357,366,266)(56,358,367,248)(57,359,368,249)(58,123,303,203)(59,124,304,204)(60,125,286,205)(61,126,287,206)(62,127,288,207)(63,128,289,208)(64,129,290,209)(65,130,291,191)(66,131,292,192)(67,132,293,193)(68,133,294,194)(69,115,295,195)(70,116,296,196)(71,117,297,197)(72,118,298,198)(73,119,299,199)(74,120,300,200)(75,121,301,201)(76,122,302,202)(134,227,335,323)(135,228,336,305)(136,210,337,306)(137,211,338,307)(138,212,339,308)(139,213,340,309)(140,214,341,310)(141,215,342,311)(142,216,324,312)(143,217,325,313)(144,218,326,314)(145,219,327,315)(146,220,328,316)(147,221,329,317)(148,222,330,318)(149,223,331,319)(150,224,332,320)(151,225,333,321)(152,226,334,322) );
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)], [(1,196,353,154,309,236,116,262,103,213),(2,197,354,155,310,237,117,263,104,214),(3,198,355,156,311,238,118,264,105,215),(4,199,356,157,312,239,119,265,106,216),(5,200,357,158,313,240,120,266,107,217),(6,201,358,159,314,241,121,248,108,218),(7,202,359,160,315,242,122,249,109,219),(8,203,360,161,316,243,123,250,110,220),(9,204,361,162,317,244,124,251,111,221),(10,205,343,163,318,245,125,252,112,222),(11,206,344,164,319,246,126,253,113,223),(12,207,345,165,320,247,127,254,114,224),(13,208,346,166,321,229,128,255,96,225),(14,209,347,167,322,230,129,256,97,226),(15,191,348,168,323,231,130,257,98,227),(16,192,349,169,305,232,131,258,99,228),(17,193,350,170,306,233,132,259,100,210),(18,194,351,171,307,234,133,260,101,211),(19,195,352,153,308,235,115,261,102,212),(20,372,61,181,331,77,42,287,274,149),(21,373,62,182,332,78,43,288,275,150),(22,374,63,183,333,79,44,289,276,151),(23,375,64,184,334,80,45,290,277,152),(24,376,65,185,335,81,46,291,278,134),(25,377,66,186,336,82,47,292,279,135),(26,378,67,187,337,83,48,293,280,136),(27,379,68,188,338,84,49,294,281,137),(28,380,69,189,339,85,50,295,282,138),(29,362,70,190,340,86,51,296,283,139),(30,363,71,172,341,87,52,297,284,140),(31,364,72,173,342,88,53,298,285,141),(32,365,73,174,324,89,54,299,267,142),(33,366,74,175,325,90,55,300,268,143),(34,367,75,176,326,91,56,301,269,144),(35,368,76,177,327,92,57,302,270,145),(36,369,58,178,328,93,39,303,271,146),(37,370,59,179,329,94,40,304,272,147),(38,371,60,180,330,95,41,286,273,148)], [(1,190,236,283),(2,172,237,284),(3,173,238,285),(4,174,239,267),(5,175,240,268),(6,176,241,269),(7,177,242,270),(8,178,243,271),(9,179,244,272),(10,180,245,273),(11,181,246,274),(12,182,247,275),(13,183,229,276),(14,184,230,277),(15,185,231,278),(16,186,232,279),(17,187,233,280),(18,188,234,281),(19,189,235,282),(20,113,77,164),(21,114,78,165),(22,96,79,166),(23,97,80,167),(24,98,81,168),(25,99,82,169),(26,100,83,170),(27,101,84,171),(28,102,85,153),(29,103,86,154),(30,104,87,155),(31,105,88,156),(32,106,89,157),(33,107,90,158),(34,108,91,159),(35,109,92,160),(36,110,93,161),(37,111,94,162),(38,112,95,163),(39,360,369,250),(40,361,370,251),(41,343,371,252),(42,344,372,253),(43,345,373,254),(44,346,374,255),(45,347,375,256),(46,348,376,257),(47,349,377,258),(48,350,378,259),(49,351,379,260),(50,352,380,261),(51,353,362,262),(52,354,363,263),(53,355,364,264),(54,356,365,265),(55,357,366,266),(56,358,367,248),(57,359,368,249),(58,123,303,203),(59,124,304,204),(60,125,286,205),(61,126,287,206),(62,127,288,207),(63,128,289,208),(64,129,290,209),(65,130,291,191),(66,131,292,192),(67,132,293,193),(68,133,294,194),(69,115,295,195),(70,116,296,196),(71,117,297,197),(72,118,298,198),(73,119,299,199),(74,120,300,200),(75,121,301,201),(76,122,302,202),(134,227,335,323),(135,228,336,305),(136,210,337,306),(137,211,338,307),(138,212,339,308),(139,213,340,309),(140,214,341,310),(141,215,342,311),(142,216,324,312),(143,217,325,313),(144,218,326,314),(145,219,327,315),(146,220,328,316),(147,221,329,317),(148,222,330,318),(149,223,331,319),(150,224,332,320),(151,225,333,321),(152,226,334,322)]])
152 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 19A | ··· | 19R | 38A | ··· | 38R | 76A | ··· | 76AJ | 95A | ··· | 95AJ | 190A | ··· | 190AJ |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 95 | ··· | 95 | 190 | ··· | 190 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
152 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C19 | C38 | C76 | D5 | Dic5 | D5×C19 | C19×Dic5 |
| kernel | C19×Dic5 | C190 | C95 | Dic5 | C10 | C5 | C38 | C19 | C2 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 | 2 | 2 | 36 | 36 |
Matrix representation of C19×Dic5 ►in GL3(𝔽761) generated by
| 1 | 0 | 0 |
| 0 | 680 | 0 |
| 0 | 0 | 680 |
| 760 | 0 | 0 |
| 0 | 760 | 1 |
| 0 | 668 | 92 |
| 39 | 0 | 0 |
| 0 | 706 | 293 |
| 0 | 65 | 55 |
G:=sub<GL(3,GF(761))| [1,0,0,0,680,0,0,0,680],[760,0,0,0,760,668,0,1,92],[39,0,0,0,706,65,0,293,55] >;
C19×Dic5 in GAP, Magma, Sage, TeX
C_{19}\times {\rm Dic}_5 % in TeX
G:=Group("C19xDic5"); // GroupNames label
G:=SmallGroup(380,1);
// by ID
G=gap.SmallGroup(380,1);
# by ID
G:=PCGroup([4,-2,-19,-2,-5,152,4867]);
// Polycyclic
G:=Group<a,b,c|a^19=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export