direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C19, C4⋊C76, C76⋊3C4, C38.3Q8, C38.13D4, C2.(Q8×C19), C2.2(C2×C76), (C2×C76).2C2, (C2×C4).1C38, C2.2(D4×C19), C38.11(C2×C4), C22.3(C2×C38), (C2×C38).14C22, SmallGroup(304,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C19
G = < a,b,c | a19=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4⋊C4×C19
►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)
(1 294 225 243)(2 295 226 244)(3 296 227 245)(4 297 228 246)(5 298 210 247)(6 299 211 229)(7 300 212 230)(8 301 213 231)(9 302 214 232)(10 303 215 233)(11 304 216 234)(12 286 217 235)(13 287 218 236)(14 288 219 237)(15 289 220 238)(16 290 221 239)(17 291 222 240)(18 292 223 241)(19 293 224 242)(20 118 66 90)(21 119 67 91)(22 120 68 92)(23 121 69 93)(24 122 70 94)(25 123 71 95)(26 124 72 77)(27 125 73 78)(28 126 74 79)(29 127 75 80)(30 128 76 81)(31 129 58 82)(32 130 59 83)(33 131 60 84)(34 132 61 85)(35 133 62 86)(36 115 63 87)(37 116 64 88)(38 117 65 89)(39 209 110 270)(40 191 111 271)(41 192 112 272)(42 193 113 273)(43 194 114 274)(44 195 96 275)(45 196 97 276)(46 197 98 277)(47 198 99 278)(48 199 100 279)(49 200 101 280)(50 201 102 281)(51 202 103 282)(52 203 104 283)(53 204 105 284)(54 205 106 285)(55 206 107 267)(56 207 108 268)(57 208 109 269)(134 253 173 156)(135 254 174 157)(136 255 175 158)(137 256 176 159)(138 257 177 160)(139 258 178 161)(140 259 179 162)(141 260 180 163)(142 261 181 164)(143 262 182 165)(144 263 183 166)(145 264 184 167)(146 265 185 168)(147 266 186 169)(148 248 187 170)(149 249 188 171)(150 250 189 153)(151 251 190 154)(152 252 172 155)
(1 130 109 154)(2 131 110 155)(3 132 111 156)(4 133 112 157)(5 115 113 158)(6 116 114 159)(7 117 96 160)(8 118 97 161)(9 119 98 162)(10 120 99 163)(11 121 100 164)(12 122 101 165)(13 123 102 166)(14 124 103 167)(15 125 104 168)(16 126 105 169)(17 127 106 170)(18 128 107 171)(19 129 108 153)(20 276 178 301)(21 277 179 302)(22 278 180 303)(23 279 181 304)(24 280 182 286)(25 281 183 287)(26 282 184 288)(27 283 185 289)(28 284 186 290)(29 285 187 291)(30 267 188 292)(31 268 189 293)(32 269 190 294)(33 270 172 295)(34 271 173 296)(35 272 174 297)(36 273 175 298)(37 274 176 299)(38 275 177 300)(39 252 226 84)(40 253 227 85)(41 254 228 86)(42 255 210 87)(43 256 211 88)(44 257 212 89)(45 258 213 90)(46 259 214 91)(47 260 215 92)(48 261 216 93)(49 262 217 94)(50 263 218 95)(51 264 219 77)(52 265 220 78)(53 266 221 79)(54 248 222 80)(55 249 223 81)(56 250 224 82)(57 251 225 83)(58 207 150 242)(59 208 151 243)(60 209 152 244)(61 191 134 245)(62 192 135 246)(63 193 136 247)(64 194 137 229)(65 195 138 230)(66 196 139 231)(67 197 140 232)(68 198 141 233)(69 199 142 234)(70 200 143 235)(71 201 144 236)(72 202 145 237)(73 203 146 238)(74 204 147 239)(75 205 148 240)(76 206 149 241)
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), (1,294,225,243)(2,295,226,244)(3,296,227,245)(4,297,228,246)(5,298,210,247)(6,299,211,229)(7,300,212,230)(8,301,213,231)(9,302,214,232)(10,303,215,233)(11,304,216,234)(12,286,217,235)(13,287,218,236)(14,288,219,237)(15,289,220,238)(16,290,221,239)(17,291,222,240)(18,292,223,241)(19,293,224,242)(20,118,66,90)(21,119,67,91)(22,120,68,92)(23,121,69,93)(24,122,70,94)(25,123,71,95)(26,124,72,77)(27,125,73,78)(28,126,74,79)(29,127,75,80)(30,128,76,81)(31,129,58,82)(32,130,59,83)(33,131,60,84)(34,132,61,85)(35,133,62,86)(36,115,63,87)(37,116,64,88)(38,117,65,89)(39,209,110,270)(40,191,111,271)(41,192,112,272)(42,193,113,273)(43,194,114,274)(44,195,96,275)(45,196,97,276)(46,197,98,277)(47,198,99,278)(48,199,100,279)(49,200,101,280)(50,201,102,281)(51,202,103,282)(52,203,104,283)(53,204,105,284)(54,205,106,285)(55,206,107,267)(56,207,108,268)(57,208,109,269)(134,253,173,156)(135,254,174,157)(136,255,175,158)(137,256,176,159)(138,257,177,160)(139,258,178,161)(140,259,179,162)(141,260,180,163)(142,261,181,164)(143,262,182,165)(144,263,183,166)(145,264,184,167)(146,265,185,168)(147,266,186,169)(148,248,187,170)(149,249,188,171)(150,250,189,153)(151,251,190,154)(152,252,172,155), (1,130,109,154)(2,131,110,155)(3,132,111,156)(4,133,112,157)(5,115,113,158)(6,116,114,159)(7,117,96,160)(8,118,97,161)(9,119,98,162)(10,120,99,163)(11,121,100,164)(12,122,101,165)(13,123,102,166)(14,124,103,167)(15,125,104,168)(16,126,105,169)(17,127,106,170)(18,128,107,171)(19,129,108,153)(20,276,178,301)(21,277,179,302)(22,278,180,303)(23,279,181,304)(24,280,182,286)(25,281,183,287)(26,282,184,288)(27,283,185,289)(28,284,186,290)(29,285,187,291)(30,267,188,292)(31,268,189,293)(32,269,190,294)(33,270,172,295)(34,271,173,296)(35,272,174,297)(36,273,175,298)(37,274,176,299)(38,275,177,300)(39,252,226,84)(40,253,227,85)(41,254,228,86)(42,255,210,87)(43,256,211,88)(44,257,212,89)(45,258,213,90)(46,259,214,91)(47,260,215,92)(48,261,216,93)(49,262,217,94)(50,263,218,95)(51,264,219,77)(52,265,220,78)(53,266,221,79)(54,248,222,80)(55,249,223,81)(56,250,224,82)(57,251,225,83)(58,207,150,242)(59,208,151,243)(60,209,152,244)(61,191,134,245)(62,192,135,246)(63,193,136,247)(64,194,137,229)(65,195,138,230)(66,196,139,231)(67,197,140,232)(68,198,141,233)(69,199,142,234)(70,200,143,235)(71,201,144,236)(72,202,145,237)(73,203,146,238)(74,204,147,239)(75,205,148,240)(76,206,149,241)>;
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), (1,294,225,243)(2,295,226,244)(3,296,227,245)(4,297,228,246)(5,298,210,247)(6,299,211,229)(7,300,212,230)(8,301,213,231)(9,302,214,232)(10,303,215,233)(11,304,216,234)(12,286,217,235)(13,287,218,236)(14,288,219,237)(15,289,220,238)(16,290,221,239)(17,291,222,240)(18,292,223,241)(19,293,224,242)(20,118,66,90)(21,119,67,91)(22,120,68,92)(23,121,69,93)(24,122,70,94)(25,123,71,95)(26,124,72,77)(27,125,73,78)(28,126,74,79)(29,127,75,80)(30,128,76,81)(31,129,58,82)(32,130,59,83)(33,131,60,84)(34,132,61,85)(35,133,62,86)(36,115,63,87)(37,116,64,88)(38,117,65,89)(39,209,110,270)(40,191,111,271)(41,192,112,272)(42,193,113,273)(43,194,114,274)(44,195,96,275)(45,196,97,276)(46,197,98,277)(47,198,99,278)(48,199,100,279)(49,200,101,280)(50,201,102,281)(51,202,103,282)(52,203,104,283)(53,204,105,284)(54,205,106,285)(55,206,107,267)(56,207,108,268)(57,208,109,269)(134,253,173,156)(135,254,174,157)(136,255,175,158)(137,256,176,159)(138,257,177,160)(139,258,178,161)(140,259,179,162)(141,260,180,163)(142,261,181,164)(143,262,182,165)(144,263,183,166)(145,264,184,167)(146,265,185,168)(147,266,186,169)(148,248,187,170)(149,249,188,171)(150,250,189,153)(151,251,190,154)(152,252,172,155), (1,130,109,154)(2,131,110,155)(3,132,111,156)(4,133,112,157)(5,115,113,158)(6,116,114,159)(7,117,96,160)(8,118,97,161)(9,119,98,162)(10,120,99,163)(11,121,100,164)(12,122,101,165)(13,123,102,166)(14,124,103,167)(15,125,104,168)(16,126,105,169)(17,127,106,170)(18,128,107,171)(19,129,108,153)(20,276,178,301)(21,277,179,302)(22,278,180,303)(23,279,181,304)(24,280,182,286)(25,281,183,287)(26,282,184,288)(27,283,185,289)(28,284,186,290)(29,285,187,291)(30,267,188,292)(31,268,189,293)(32,269,190,294)(33,270,172,295)(34,271,173,296)(35,272,174,297)(36,273,175,298)(37,274,176,299)(38,275,177,300)(39,252,226,84)(40,253,227,85)(41,254,228,86)(42,255,210,87)(43,256,211,88)(44,257,212,89)(45,258,213,90)(46,259,214,91)(47,260,215,92)(48,261,216,93)(49,262,217,94)(50,263,218,95)(51,264,219,77)(52,265,220,78)(53,266,221,79)(54,248,222,80)(55,249,223,81)(56,250,224,82)(57,251,225,83)(58,207,150,242)(59,208,151,243)(60,209,152,244)(61,191,134,245)(62,192,135,246)(63,193,136,247)(64,194,137,229)(65,195,138,230)(66,196,139,231)(67,197,140,232)(68,198,141,233)(69,199,142,234)(70,200,143,235)(71,201,144,236)(72,202,145,237)(73,203,146,238)(74,204,147,239)(75,205,148,240)(76,206,149,241) );
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)], [(1,294,225,243),(2,295,226,244),(3,296,227,245),(4,297,228,246),(5,298,210,247),(6,299,211,229),(7,300,212,230),(8,301,213,231),(9,302,214,232),(10,303,215,233),(11,304,216,234),(12,286,217,235),(13,287,218,236),(14,288,219,237),(15,289,220,238),(16,290,221,239),(17,291,222,240),(18,292,223,241),(19,293,224,242),(20,118,66,90),(21,119,67,91),(22,120,68,92),(23,121,69,93),(24,122,70,94),(25,123,71,95),(26,124,72,77),(27,125,73,78),(28,126,74,79),(29,127,75,80),(30,128,76,81),(31,129,58,82),(32,130,59,83),(33,131,60,84),(34,132,61,85),(35,133,62,86),(36,115,63,87),(37,116,64,88),(38,117,65,89),(39,209,110,270),(40,191,111,271),(41,192,112,272),(42,193,113,273),(43,194,114,274),(44,195,96,275),(45,196,97,276),(46,197,98,277),(47,198,99,278),(48,199,100,279),(49,200,101,280),(50,201,102,281),(51,202,103,282),(52,203,104,283),(53,204,105,284),(54,205,106,285),(55,206,107,267),(56,207,108,268),(57,208,109,269),(134,253,173,156),(135,254,174,157),(136,255,175,158),(137,256,176,159),(138,257,177,160),(139,258,178,161),(140,259,179,162),(141,260,180,163),(142,261,181,164),(143,262,182,165),(144,263,183,166),(145,264,184,167),(146,265,185,168),(147,266,186,169),(148,248,187,170),(149,249,188,171),(150,250,189,153),(151,251,190,154),(152,252,172,155)], [(1,130,109,154),(2,131,110,155),(3,132,111,156),(4,133,112,157),(5,115,113,158),(6,116,114,159),(7,117,96,160),(8,118,97,161),(9,119,98,162),(10,120,99,163),(11,121,100,164),(12,122,101,165),(13,123,102,166),(14,124,103,167),(15,125,104,168),(16,126,105,169),(17,127,106,170),(18,128,107,171),(19,129,108,153),(20,276,178,301),(21,277,179,302),(22,278,180,303),(23,279,181,304),(24,280,182,286),(25,281,183,287),(26,282,184,288),(27,283,185,289),(28,284,186,290),(29,285,187,291),(30,267,188,292),(31,268,189,293),(32,269,190,294),(33,270,172,295),(34,271,173,296),(35,272,174,297),(36,273,175,298),(37,274,176,299),(38,275,177,300),(39,252,226,84),(40,253,227,85),(41,254,228,86),(42,255,210,87),(43,256,211,88),(44,257,212,89),(45,258,213,90),(46,259,214,91),(47,260,215,92),(48,261,216,93),(49,262,217,94),(50,263,218,95),(51,264,219,77),(52,265,220,78),(53,266,221,79),(54,248,222,80),(55,249,223,81),(56,250,224,82),(57,251,225,83),(58,207,150,242),(59,208,151,243),(60,209,152,244),(61,191,134,245),(62,192,135,246),(63,193,136,247),(64,194,137,229),(65,195,138,230),(66,196,139,231),(67,197,140,232),(68,198,141,233),(69,199,142,234),(70,200,143,235),(71,201,144,236),(72,202,145,237),(73,203,146,238),(74,204,147,239),(75,205,148,240),(76,206,149,241)]])
190 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 19A | ··· | 19R | 38A | ··· | 38BB | 76A | ··· | 76DD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
190 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C19 | C38 | C76 | D4 | Q8 | D4×C19 | Q8×C19 |
| kernel | C4⋊C4×C19 | C2×C76 | C76 | C4⋊C4 | C2×C4 | C4 | C38 | C38 | C2 | C2 |
| # reps | 1 | 3 | 4 | 18 | 54 | 72 | 1 | 1 | 18 | 18 |
Matrix representation of C4⋊C4×C19 ►in GL4(𝔽229) generated by
| 203 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 0 | 228 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 107 | 0 | 0 |
| 0 | 0 | 121 | 181 |
| 0 | 0 | 181 | 108 |
G:=sub<GL(4,GF(229))| [203,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[228,0,0,0,0,228,0,0,0,0,0,1,0,0,228,0],[1,0,0,0,0,107,0,0,0,0,121,181,0,0,181,108] >;
C4⋊C4×C19 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{19} % in TeX
G:=Group("C4:C4xC19"); // GroupNames label
G:=SmallGroup(304,21);
// by ID
G=gap.SmallGroup(304,21);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,760,781,386]);
// Polycyclic
G:=Group<a,b,c|a^19=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export