metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic78, C4.D39, C39⋊2Q8, C52.1S3, C2.3D78, C6.8D26, C26.8D6, C3⋊2Dic26, C13⋊2Dic6, C156.1C2, C12.1D13, C78.8C22, Dic39.1C2, SmallGroup(312,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic78
G = < a,b | a156=1, b2=a78, bab-1=a-1 >
Smallest permutation representation of Dic78
►Regular action on 312 points
Generators in S312
(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)
(1 232 79 310)(2 231 80 309)(3 230 81 308)(4 229 82 307)(5 228 83 306)(6 227 84 305)(7 226 85 304)(8 225 86 303)(9 224 87 302)(10 223 88 301)(11 222 89 300)(12 221 90 299)(13 220 91 298)(14 219 92 297)(15 218 93 296)(16 217 94 295)(17 216 95 294)(18 215 96 293)(19 214 97 292)(20 213 98 291)(21 212 99 290)(22 211 100 289)(23 210 101 288)(24 209 102 287)(25 208 103 286)(26 207 104 285)(27 206 105 284)(28 205 106 283)(29 204 107 282)(30 203 108 281)(31 202 109 280)(32 201 110 279)(33 200 111 278)(34 199 112 277)(35 198 113 276)(36 197 114 275)(37 196 115 274)(38 195 116 273)(39 194 117 272)(40 193 118 271)(41 192 119 270)(42 191 120 269)(43 190 121 268)(44 189 122 267)(45 188 123 266)(46 187 124 265)(47 186 125 264)(48 185 126 263)(49 184 127 262)(50 183 128 261)(51 182 129 260)(52 181 130 259)(53 180 131 258)(54 179 132 257)(55 178 133 256)(56 177 134 255)(57 176 135 254)(58 175 136 253)(59 174 137 252)(60 173 138 251)(61 172 139 250)(62 171 140 249)(63 170 141 248)(64 169 142 247)(65 168 143 246)(66 167 144 245)(67 166 145 244)(68 165 146 243)(69 164 147 242)(70 163 148 241)(71 162 149 240)(72 161 150 239)(73 160 151 238)(74 159 152 237)(75 158 153 236)(76 157 154 235)(77 312 155 234)(78 311 156 233)
G:=sub<Sym(312)| (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), (1,232,79,310)(2,231,80,309)(3,230,81,308)(4,229,82,307)(5,228,83,306)(6,227,84,305)(7,226,85,304)(8,225,86,303)(9,224,87,302)(10,223,88,301)(11,222,89,300)(12,221,90,299)(13,220,91,298)(14,219,92,297)(15,218,93,296)(16,217,94,295)(17,216,95,294)(18,215,96,293)(19,214,97,292)(20,213,98,291)(21,212,99,290)(22,211,100,289)(23,210,101,288)(24,209,102,287)(25,208,103,286)(26,207,104,285)(27,206,105,284)(28,205,106,283)(29,204,107,282)(30,203,108,281)(31,202,109,280)(32,201,110,279)(33,200,111,278)(34,199,112,277)(35,198,113,276)(36,197,114,275)(37,196,115,274)(38,195,116,273)(39,194,117,272)(40,193,118,271)(41,192,119,270)(42,191,120,269)(43,190,121,268)(44,189,122,267)(45,188,123,266)(46,187,124,265)(47,186,125,264)(48,185,126,263)(49,184,127,262)(50,183,128,261)(51,182,129,260)(52,181,130,259)(53,180,131,258)(54,179,132,257)(55,178,133,256)(56,177,134,255)(57,176,135,254)(58,175,136,253)(59,174,137,252)(60,173,138,251)(61,172,139,250)(62,171,140,249)(63,170,141,248)(64,169,142,247)(65,168,143,246)(66,167,144,245)(67,166,145,244)(68,165,146,243)(69,164,147,242)(70,163,148,241)(71,162,149,240)(72,161,150,239)(73,160,151,238)(74,159,152,237)(75,158,153,236)(76,157,154,235)(77,312,155,234)(78,311,156,233)>;
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), (1,232,79,310)(2,231,80,309)(3,230,81,308)(4,229,82,307)(5,228,83,306)(6,227,84,305)(7,226,85,304)(8,225,86,303)(9,224,87,302)(10,223,88,301)(11,222,89,300)(12,221,90,299)(13,220,91,298)(14,219,92,297)(15,218,93,296)(16,217,94,295)(17,216,95,294)(18,215,96,293)(19,214,97,292)(20,213,98,291)(21,212,99,290)(22,211,100,289)(23,210,101,288)(24,209,102,287)(25,208,103,286)(26,207,104,285)(27,206,105,284)(28,205,106,283)(29,204,107,282)(30,203,108,281)(31,202,109,280)(32,201,110,279)(33,200,111,278)(34,199,112,277)(35,198,113,276)(36,197,114,275)(37,196,115,274)(38,195,116,273)(39,194,117,272)(40,193,118,271)(41,192,119,270)(42,191,120,269)(43,190,121,268)(44,189,122,267)(45,188,123,266)(46,187,124,265)(47,186,125,264)(48,185,126,263)(49,184,127,262)(50,183,128,261)(51,182,129,260)(52,181,130,259)(53,180,131,258)(54,179,132,257)(55,178,133,256)(56,177,134,255)(57,176,135,254)(58,175,136,253)(59,174,137,252)(60,173,138,251)(61,172,139,250)(62,171,140,249)(63,170,141,248)(64,169,142,247)(65,168,143,246)(66,167,144,245)(67,166,145,244)(68,165,146,243)(69,164,147,242)(70,163,148,241)(71,162,149,240)(72,161,150,239)(73,160,151,238)(74,159,152,237)(75,158,153,236)(76,157,154,235)(77,312,155,234)(78,311,156,233) );
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)], [(1,232,79,310),(2,231,80,309),(3,230,81,308),(4,229,82,307),(5,228,83,306),(6,227,84,305),(7,226,85,304),(8,225,86,303),(9,224,87,302),(10,223,88,301),(11,222,89,300),(12,221,90,299),(13,220,91,298),(14,219,92,297),(15,218,93,296),(16,217,94,295),(17,216,95,294),(18,215,96,293),(19,214,97,292),(20,213,98,291),(21,212,99,290),(22,211,100,289),(23,210,101,288),(24,209,102,287),(25,208,103,286),(26,207,104,285),(27,206,105,284),(28,205,106,283),(29,204,107,282),(30,203,108,281),(31,202,109,280),(32,201,110,279),(33,200,111,278),(34,199,112,277),(35,198,113,276),(36,197,114,275),(37,196,115,274),(38,195,116,273),(39,194,117,272),(40,193,118,271),(41,192,119,270),(42,191,120,269),(43,190,121,268),(44,189,122,267),(45,188,123,266),(46,187,124,265),(47,186,125,264),(48,185,126,263),(49,184,127,262),(50,183,128,261),(51,182,129,260),(52,181,130,259),(53,180,131,258),(54,179,132,257),(55,178,133,256),(56,177,134,255),(57,176,135,254),(58,175,136,253),(59,174,137,252),(60,173,138,251),(61,172,139,250),(62,171,140,249),(63,170,141,248),(64,169,142,247),(65,168,143,246),(66,167,144,245),(67,166,145,244),(68,165,146,243),(69,164,147,242),(70,163,148,241),(71,162,149,240),(72,161,150,239),(73,160,151,238),(74,159,152,237),(75,158,153,236),(76,157,154,235),(77,312,155,234),(78,311,156,233)]])
81 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 2 | 2 | 78 | 78 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
81 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D6 | Dic6 | D13 | D26 | D39 | Dic26 | D78 | Dic78 |
| kernel | Dic78 | Dic39 | C156 | C52 | C39 | C26 | C13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of Dic78 ►in GL2(𝔽157) generated by
| 119 | 98 |
| 59 | 42 |
| 150 | 103 |
| 30 | 7 |
G:=sub<GL(2,GF(157))| [119,59,98,42],[150,30,103,7] >;
Dic78 in GAP, Magma, Sage, TeX
{\rm Dic}_{78} % in TeX
G:=Group("Dic78"); // GroupNames label
G:=SmallGroup(312,37);
// by ID
G=gap.SmallGroup(312,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,61,26,323,7204]);
// Polycyclic
G:=Group<a,b|a^156=1,b^2=a^78,b*a*b^-1=a^-1>;
// generators/relations
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