metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic74, C37⋊Q8, C4.D37, C2.3D74, C148.1C2, C74.1C22, Dic37.1C2, SmallGroup(296,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic74
G = < a,b | a148=1, b2=a74, bab-1=a-1 >
Smallest permutation representation of Dic74
►Regular action on 296 points
Generators in S296
(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)
(1 263 75 189)(2 262 76 188)(3 261 77 187)(4 260 78 186)(5 259 79 185)(6 258 80 184)(7 257 81 183)(8 256 82 182)(9 255 83 181)(10 254 84 180)(11 253 85 179)(12 252 86 178)(13 251 87 177)(14 250 88 176)(15 249 89 175)(16 248 90 174)(17 247 91 173)(18 246 92 172)(19 245 93 171)(20 244 94 170)(21 243 95 169)(22 242 96 168)(23 241 97 167)(24 240 98 166)(25 239 99 165)(26 238 100 164)(27 237 101 163)(28 236 102 162)(29 235 103 161)(30 234 104 160)(31 233 105 159)(32 232 106 158)(33 231 107 157)(34 230 108 156)(35 229 109 155)(36 228 110 154)(37 227 111 153)(38 226 112 152)(39 225 113 151)(40 224 114 150)(41 223 115 149)(42 222 116 296)(43 221 117 295)(44 220 118 294)(45 219 119 293)(46 218 120 292)(47 217 121 291)(48 216 122 290)(49 215 123 289)(50 214 124 288)(51 213 125 287)(52 212 126 286)(53 211 127 285)(54 210 128 284)(55 209 129 283)(56 208 130 282)(57 207 131 281)(58 206 132 280)(59 205 133 279)(60 204 134 278)(61 203 135 277)(62 202 136 276)(63 201 137 275)(64 200 138 274)(65 199 139 273)(66 198 140 272)(67 197 141 271)(68 196 142 270)(69 195 143 269)(70 194 144 268)(71 193 145 267)(72 192 146 266)(73 191 147 265)(74 190 148 264)
G:=sub<Sym(296)| (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), (1,263,75,189)(2,262,76,188)(3,261,77,187)(4,260,78,186)(5,259,79,185)(6,258,80,184)(7,257,81,183)(8,256,82,182)(9,255,83,181)(10,254,84,180)(11,253,85,179)(12,252,86,178)(13,251,87,177)(14,250,88,176)(15,249,89,175)(16,248,90,174)(17,247,91,173)(18,246,92,172)(19,245,93,171)(20,244,94,170)(21,243,95,169)(22,242,96,168)(23,241,97,167)(24,240,98,166)(25,239,99,165)(26,238,100,164)(27,237,101,163)(28,236,102,162)(29,235,103,161)(30,234,104,160)(31,233,105,159)(32,232,106,158)(33,231,107,157)(34,230,108,156)(35,229,109,155)(36,228,110,154)(37,227,111,153)(38,226,112,152)(39,225,113,151)(40,224,114,150)(41,223,115,149)(42,222,116,296)(43,221,117,295)(44,220,118,294)(45,219,119,293)(46,218,120,292)(47,217,121,291)(48,216,122,290)(49,215,123,289)(50,214,124,288)(51,213,125,287)(52,212,126,286)(53,211,127,285)(54,210,128,284)(55,209,129,283)(56,208,130,282)(57,207,131,281)(58,206,132,280)(59,205,133,279)(60,204,134,278)(61,203,135,277)(62,202,136,276)(63,201,137,275)(64,200,138,274)(65,199,139,273)(66,198,140,272)(67,197,141,271)(68,196,142,270)(69,195,143,269)(70,194,144,268)(71,193,145,267)(72,192,146,266)(73,191,147,265)(74,190,148,264)>;
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), (1,263,75,189)(2,262,76,188)(3,261,77,187)(4,260,78,186)(5,259,79,185)(6,258,80,184)(7,257,81,183)(8,256,82,182)(9,255,83,181)(10,254,84,180)(11,253,85,179)(12,252,86,178)(13,251,87,177)(14,250,88,176)(15,249,89,175)(16,248,90,174)(17,247,91,173)(18,246,92,172)(19,245,93,171)(20,244,94,170)(21,243,95,169)(22,242,96,168)(23,241,97,167)(24,240,98,166)(25,239,99,165)(26,238,100,164)(27,237,101,163)(28,236,102,162)(29,235,103,161)(30,234,104,160)(31,233,105,159)(32,232,106,158)(33,231,107,157)(34,230,108,156)(35,229,109,155)(36,228,110,154)(37,227,111,153)(38,226,112,152)(39,225,113,151)(40,224,114,150)(41,223,115,149)(42,222,116,296)(43,221,117,295)(44,220,118,294)(45,219,119,293)(46,218,120,292)(47,217,121,291)(48,216,122,290)(49,215,123,289)(50,214,124,288)(51,213,125,287)(52,212,126,286)(53,211,127,285)(54,210,128,284)(55,209,129,283)(56,208,130,282)(57,207,131,281)(58,206,132,280)(59,205,133,279)(60,204,134,278)(61,203,135,277)(62,202,136,276)(63,201,137,275)(64,200,138,274)(65,199,139,273)(66,198,140,272)(67,197,141,271)(68,196,142,270)(69,195,143,269)(70,194,144,268)(71,193,145,267)(72,192,146,266)(73,191,147,265)(74,190,148,264) );
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)], [(1,263,75,189),(2,262,76,188),(3,261,77,187),(4,260,78,186),(5,259,79,185),(6,258,80,184),(7,257,81,183),(8,256,82,182),(9,255,83,181),(10,254,84,180),(11,253,85,179),(12,252,86,178),(13,251,87,177),(14,250,88,176),(15,249,89,175),(16,248,90,174),(17,247,91,173),(18,246,92,172),(19,245,93,171),(20,244,94,170),(21,243,95,169),(22,242,96,168),(23,241,97,167),(24,240,98,166),(25,239,99,165),(26,238,100,164),(27,237,101,163),(28,236,102,162),(29,235,103,161),(30,234,104,160),(31,233,105,159),(32,232,106,158),(33,231,107,157),(34,230,108,156),(35,229,109,155),(36,228,110,154),(37,227,111,153),(38,226,112,152),(39,225,113,151),(40,224,114,150),(41,223,115,149),(42,222,116,296),(43,221,117,295),(44,220,118,294),(45,219,119,293),(46,218,120,292),(47,217,121,291),(48,216,122,290),(49,215,123,289),(50,214,124,288),(51,213,125,287),(52,212,126,286),(53,211,127,285),(54,210,128,284),(55,209,129,283),(56,208,130,282),(57,207,131,281),(58,206,132,280),(59,205,133,279),(60,204,134,278),(61,203,135,277),(62,202,136,276),(63,201,137,275),(64,200,138,274),(65,199,139,273),(66,198,140,272),(67,197,141,271),(68,196,142,270),(69,195,143,269),(70,194,144,268),(71,193,145,267),(72,192,146,266),(73,191,147,265),(74,190,148,264)]])
77 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 37A | ··· | 37R | 74A | ··· | 74R | 148A | ··· | 148AJ |
| order | 1 | 2 | 4 | 4 | 4 | 37 | ··· | 37 | 74 | ··· | 74 | 148 | ··· | 148 |
| size | 1 | 1 | 2 | 74 | 74 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D37 | D74 | Dic74 |
| kernel | Dic74 | Dic37 | C148 | C37 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 18 | 18 | 36 |
Matrix representation of Dic74 ►in GL2(𝔽149) generated by
| 29 | 102 |
| 47 | 78 |
| 68 | 58 |
| 41 | 81 |
G:=sub<GL(2,GF(149))| [29,47,102,78],[68,41,58,81] >;
Dic74 in GAP, Magma, Sage, TeX
{\rm Dic}_{74} % in TeX
G:=Group("Dic74"); // GroupNames label
G:=SmallGroup(296,4);
// by ID
G=gap.SmallGroup(296,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-37,16,49,21,4611]);
// Polycyclic
G:=Group<a,b|a^148=1,b^2=a^74,b*a*b^-1=a^-1>;
// generators/relations
Export