metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic19⋊C4, C38.5D4, C38.1Q8, C2.1Dic38, C22.4D38, C19⋊1(C4⋊C4), C38.4(C2×C4), (C2×C76).1C2, (C2×C4).1D19, C2.4(C4×D19), C2.1(C19⋊D4), (C2×C38).4C22, (C2×Dic19).1C2, SmallGroup(304,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic19⋊C4
G = < a,b,c | a38=c4=1, b2=a19, bab-1=a-1, ac=ca, cbc-1=a19b >
Smallest permutation representation of Dic19⋊C4
►Regular action on 304 points
Generators in S304
(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)
(1 210 20 191)(2 209 21 228)(3 208 22 227)(4 207 23 226)(5 206 24 225)(6 205 25 224)(7 204 26 223)(8 203 27 222)(9 202 28 221)(10 201 29 220)(11 200 30 219)(12 199 31 218)(13 198 32 217)(14 197 33 216)(15 196 34 215)(16 195 35 214)(17 194 36 213)(18 193 37 212)(19 192 38 211)(39 164 58 183)(40 163 59 182)(41 162 60 181)(42 161 61 180)(43 160 62 179)(44 159 63 178)(45 158 64 177)(46 157 65 176)(47 156 66 175)(48 155 67 174)(49 154 68 173)(50 153 69 172)(51 190 70 171)(52 189 71 170)(53 188 72 169)(54 187 73 168)(55 186 74 167)(56 185 75 166)(57 184 76 165)(77 275 96 294)(78 274 97 293)(79 273 98 292)(80 272 99 291)(81 271 100 290)(82 270 101 289)(83 269 102 288)(84 268 103 287)(85 267 104 286)(86 304 105 285)(87 303 106 284)(88 302 107 283)(89 301 108 282)(90 300 109 281)(91 299 110 280)(92 298 111 279)(93 297 112 278)(94 296 113 277)(95 295 114 276)(115 258 134 239)(116 257 135 238)(117 256 136 237)(118 255 137 236)(119 254 138 235)(120 253 139 234)(121 252 140 233)(122 251 141 232)(123 250 142 231)(124 249 143 230)(125 248 144 229)(126 247 145 266)(127 246 146 265)(128 245 147 264)(129 244 148 263)(130 243 149 262)(131 242 150 261)(132 241 151 260)(133 240 152 259)
(1 144 50 85)(2 145 51 86)(3 146 52 87)(4 147 53 88)(5 148 54 89)(6 149 55 90)(7 150 56 91)(8 151 57 92)(9 152 58 93)(10 115 59 94)(11 116 60 95)(12 117 61 96)(13 118 62 97)(14 119 63 98)(15 120 64 99)(16 121 65 100)(17 122 66 101)(18 123 67 102)(19 124 68 103)(20 125 69 104)(21 126 70 105)(22 127 71 106)(23 128 72 107)(24 129 73 108)(25 130 74 109)(26 131 75 110)(27 132 76 111)(28 133 39 112)(29 134 40 113)(30 135 41 114)(31 136 42 77)(32 137 43 78)(33 138 44 79)(34 139 45 80)(35 140 46 81)(36 141 47 82)(37 142 48 83)(38 143 49 84)(153 286 210 248)(154 287 211 249)(155 288 212 250)(156 289 213 251)(157 290 214 252)(158 291 215 253)(159 292 216 254)(160 293 217 255)(161 294 218 256)(162 295 219 257)(163 296 220 258)(164 297 221 259)(165 298 222 260)(166 299 223 261)(167 300 224 262)(168 301 225 263)(169 302 226 264)(170 303 227 265)(171 304 228 266)(172 267 191 229)(173 268 192 230)(174 269 193 231)(175 270 194 232)(176 271 195 233)(177 272 196 234)(178 273 197 235)(179 274 198 236)(180 275 199 237)(181 276 200 238)(182 277 201 239)(183 278 202 240)(184 279 203 241)(185 280 204 242)(186 281 205 243)(187 282 206 244)(188 283 207 245)(189 284 208 246)(190 285 209 247)
G:=sub<Sym(304)| (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), (1,210,20,191)(2,209,21,228)(3,208,22,227)(4,207,23,226)(5,206,24,225)(6,205,25,224)(7,204,26,223)(8,203,27,222)(9,202,28,221)(10,201,29,220)(11,200,30,219)(12,199,31,218)(13,198,32,217)(14,197,33,216)(15,196,34,215)(16,195,35,214)(17,194,36,213)(18,193,37,212)(19,192,38,211)(39,164,58,183)(40,163,59,182)(41,162,60,181)(42,161,61,180)(43,160,62,179)(44,159,63,178)(45,158,64,177)(46,157,65,176)(47,156,66,175)(48,155,67,174)(49,154,68,173)(50,153,69,172)(51,190,70,171)(52,189,71,170)(53,188,72,169)(54,187,73,168)(55,186,74,167)(56,185,75,166)(57,184,76,165)(77,275,96,294)(78,274,97,293)(79,273,98,292)(80,272,99,291)(81,271,100,290)(82,270,101,289)(83,269,102,288)(84,268,103,287)(85,267,104,286)(86,304,105,285)(87,303,106,284)(88,302,107,283)(89,301,108,282)(90,300,109,281)(91,299,110,280)(92,298,111,279)(93,297,112,278)(94,296,113,277)(95,295,114,276)(115,258,134,239)(116,257,135,238)(117,256,136,237)(118,255,137,236)(119,254,138,235)(120,253,139,234)(121,252,140,233)(122,251,141,232)(123,250,142,231)(124,249,143,230)(125,248,144,229)(126,247,145,266)(127,246,146,265)(128,245,147,264)(129,244,148,263)(130,243,149,262)(131,242,150,261)(132,241,151,260)(133,240,152,259), (1,144,50,85)(2,145,51,86)(3,146,52,87)(4,147,53,88)(5,148,54,89)(6,149,55,90)(7,150,56,91)(8,151,57,92)(9,152,58,93)(10,115,59,94)(11,116,60,95)(12,117,61,96)(13,118,62,97)(14,119,63,98)(15,120,64,99)(16,121,65,100)(17,122,66,101)(18,123,67,102)(19,124,68,103)(20,125,69,104)(21,126,70,105)(22,127,71,106)(23,128,72,107)(24,129,73,108)(25,130,74,109)(26,131,75,110)(27,132,76,111)(28,133,39,112)(29,134,40,113)(30,135,41,114)(31,136,42,77)(32,137,43,78)(33,138,44,79)(34,139,45,80)(35,140,46,81)(36,141,47,82)(37,142,48,83)(38,143,49,84)(153,286,210,248)(154,287,211,249)(155,288,212,250)(156,289,213,251)(157,290,214,252)(158,291,215,253)(159,292,216,254)(160,293,217,255)(161,294,218,256)(162,295,219,257)(163,296,220,258)(164,297,221,259)(165,298,222,260)(166,299,223,261)(167,300,224,262)(168,301,225,263)(169,302,226,264)(170,303,227,265)(171,304,228,266)(172,267,191,229)(173,268,192,230)(174,269,193,231)(175,270,194,232)(176,271,195,233)(177,272,196,234)(178,273,197,235)(179,274,198,236)(180,275,199,237)(181,276,200,238)(182,277,201,239)(183,278,202,240)(184,279,203,241)(185,280,204,242)(186,281,205,243)(187,282,206,244)(188,283,207,245)(189,284,208,246)(190,285,209,247)>;
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), (1,210,20,191)(2,209,21,228)(3,208,22,227)(4,207,23,226)(5,206,24,225)(6,205,25,224)(7,204,26,223)(8,203,27,222)(9,202,28,221)(10,201,29,220)(11,200,30,219)(12,199,31,218)(13,198,32,217)(14,197,33,216)(15,196,34,215)(16,195,35,214)(17,194,36,213)(18,193,37,212)(19,192,38,211)(39,164,58,183)(40,163,59,182)(41,162,60,181)(42,161,61,180)(43,160,62,179)(44,159,63,178)(45,158,64,177)(46,157,65,176)(47,156,66,175)(48,155,67,174)(49,154,68,173)(50,153,69,172)(51,190,70,171)(52,189,71,170)(53,188,72,169)(54,187,73,168)(55,186,74,167)(56,185,75,166)(57,184,76,165)(77,275,96,294)(78,274,97,293)(79,273,98,292)(80,272,99,291)(81,271,100,290)(82,270,101,289)(83,269,102,288)(84,268,103,287)(85,267,104,286)(86,304,105,285)(87,303,106,284)(88,302,107,283)(89,301,108,282)(90,300,109,281)(91,299,110,280)(92,298,111,279)(93,297,112,278)(94,296,113,277)(95,295,114,276)(115,258,134,239)(116,257,135,238)(117,256,136,237)(118,255,137,236)(119,254,138,235)(120,253,139,234)(121,252,140,233)(122,251,141,232)(123,250,142,231)(124,249,143,230)(125,248,144,229)(126,247,145,266)(127,246,146,265)(128,245,147,264)(129,244,148,263)(130,243,149,262)(131,242,150,261)(132,241,151,260)(133,240,152,259), (1,144,50,85)(2,145,51,86)(3,146,52,87)(4,147,53,88)(5,148,54,89)(6,149,55,90)(7,150,56,91)(8,151,57,92)(9,152,58,93)(10,115,59,94)(11,116,60,95)(12,117,61,96)(13,118,62,97)(14,119,63,98)(15,120,64,99)(16,121,65,100)(17,122,66,101)(18,123,67,102)(19,124,68,103)(20,125,69,104)(21,126,70,105)(22,127,71,106)(23,128,72,107)(24,129,73,108)(25,130,74,109)(26,131,75,110)(27,132,76,111)(28,133,39,112)(29,134,40,113)(30,135,41,114)(31,136,42,77)(32,137,43,78)(33,138,44,79)(34,139,45,80)(35,140,46,81)(36,141,47,82)(37,142,48,83)(38,143,49,84)(153,286,210,248)(154,287,211,249)(155,288,212,250)(156,289,213,251)(157,290,214,252)(158,291,215,253)(159,292,216,254)(160,293,217,255)(161,294,218,256)(162,295,219,257)(163,296,220,258)(164,297,221,259)(165,298,222,260)(166,299,223,261)(167,300,224,262)(168,301,225,263)(169,302,226,264)(170,303,227,265)(171,304,228,266)(172,267,191,229)(173,268,192,230)(174,269,193,231)(175,270,194,232)(176,271,195,233)(177,272,196,234)(178,273,197,235)(179,274,198,236)(180,275,199,237)(181,276,200,238)(182,277,201,239)(183,278,202,240)(184,279,203,241)(185,280,204,242)(186,281,205,243)(187,282,206,244)(188,283,207,245)(189,284,208,246)(190,285,209,247) );
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)], [(1,210,20,191),(2,209,21,228),(3,208,22,227),(4,207,23,226),(5,206,24,225),(6,205,25,224),(7,204,26,223),(8,203,27,222),(9,202,28,221),(10,201,29,220),(11,200,30,219),(12,199,31,218),(13,198,32,217),(14,197,33,216),(15,196,34,215),(16,195,35,214),(17,194,36,213),(18,193,37,212),(19,192,38,211),(39,164,58,183),(40,163,59,182),(41,162,60,181),(42,161,61,180),(43,160,62,179),(44,159,63,178),(45,158,64,177),(46,157,65,176),(47,156,66,175),(48,155,67,174),(49,154,68,173),(50,153,69,172),(51,190,70,171),(52,189,71,170),(53,188,72,169),(54,187,73,168),(55,186,74,167),(56,185,75,166),(57,184,76,165),(77,275,96,294),(78,274,97,293),(79,273,98,292),(80,272,99,291),(81,271,100,290),(82,270,101,289),(83,269,102,288),(84,268,103,287),(85,267,104,286),(86,304,105,285),(87,303,106,284),(88,302,107,283),(89,301,108,282),(90,300,109,281),(91,299,110,280),(92,298,111,279),(93,297,112,278),(94,296,113,277),(95,295,114,276),(115,258,134,239),(116,257,135,238),(117,256,136,237),(118,255,137,236),(119,254,138,235),(120,253,139,234),(121,252,140,233),(122,251,141,232),(123,250,142,231),(124,249,143,230),(125,248,144,229),(126,247,145,266),(127,246,146,265),(128,245,147,264),(129,244,148,263),(130,243,149,262),(131,242,150,261),(132,241,151,260),(133,240,152,259)], [(1,144,50,85),(2,145,51,86),(3,146,52,87),(4,147,53,88),(5,148,54,89),(6,149,55,90),(7,150,56,91),(8,151,57,92),(9,152,58,93),(10,115,59,94),(11,116,60,95),(12,117,61,96),(13,118,62,97),(14,119,63,98),(15,120,64,99),(16,121,65,100),(17,122,66,101),(18,123,67,102),(19,124,68,103),(20,125,69,104),(21,126,70,105),(22,127,71,106),(23,128,72,107),(24,129,73,108),(25,130,74,109),(26,131,75,110),(27,132,76,111),(28,133,39,112),(29,134,40,113),(30,135,41,114),(31,136,42,77),(32,137,43,78),(33,138,44,79),(34,139,45,80),(35,140,46,81),(36,141,47,82),(37,142,48,83),(38,143,49,84),(153,286,210,248),(154,287,211,249),(155,288,212,250),(156,289,213,251),(157,290,214,252),(158,291,215,253),(159,292,216,254),(160,293,217,255),(161,294,218,256),(162,295,219,257),(163,296,220,258),(164,297,221,259),(165,298,222,260),(166,299,223,261),(167,300,224,262),(168,301,225,263),(169,302,226,264),(170,303,227,265),(171,304,228,266),(172,267,191,229),(173,268,192,230),(174,269,193,231),(175,270,194,232),(176,271,195,233),(177,272,196,234),(178,273,197,235),(179,274,198,236),(180,275,199,237),(181,276,200,238),(182,277,201,239),(183,278,202,240),(184,279,203,241),(185,280,204,242),(186,281,205,243),(187,282,206,244),(188,283,207,245),(189,284,208,246),(190,285,209,247)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D19 | D38 | Dic38 | C4×D19 | C19⋊D4 |
| kernel | Dic19⋊C4 | C2×Dic19 | C2×C76 | Dic19 | C38 | C38 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 9 | 9 | 18 | 18 | 18 |
Matrix representation of Dic19⋊C4 ►in GL4(𝔽229) generated by
| 0 | 1 | 0 | 0 |
| 228 | 93 | 0 | 0 |
| 0 | 0 | 94 | 228 |
| 0 | 0 | 224 | 117 |
| 107 | 89 | 0 | 0 |
| 193 | 122 | 0 | 0 |
| 0 | 0 | 5 | 188 |
| 0 | 0 | 90 | 224 |
| 107 | 0 | 0 | 0 |
| 0 | 107 | 0 | 0 |
| 0 | 0 | 27 | 82 |
| 0 | 0 | 181 | 202 |
G:=sub<GL(4,GF(229))| [0,228,0,0,1,93,0,0,0,0,94,224,0,0,228,117],[107,193,0,0,89,122,0,0,0,0,5,90,0,0,188,224],[107,0,0,0,0,107,0,0,0,0,27,181,0,0,82,202] >;
Dic19⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{19}\rtimes C_4 % in TeX
G:=Group("Dic19:C4"); // GroupNames label
G:=SmallGroup(304,11);
// by ID
G=gap.SmallGroup(304,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,40,101,26,7204]);
// Polycyclic
G:=Group<a,b,c|a^38=c^4=1,b^2=a^19,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^19*b>;
// generators/relations
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