metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic76, C8.D19, C19⋊1Q16, C2.5D76, C38.3D4, C152.1C2, C4.10D38, C76.10C22, Dic38.1C2, SmallGroup(304,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic76
G = < a,b | a152=1, b2=a76, bab-1=a-1 >
Smallest permutation representation of Dic76
►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)
(1 230 77 154)(2 229 78 153)(3 228 79 304)(4 227 80 303)(5 226 81 302)(6 225 82 301)(7 224 83 300)(8 223 84 299)(9 222 85 298)(10 221 86 297)(11 220 87 296)(12 219 88 295)(13 218 89 294)(14 217 90 293)(15 216 91 292)(16 215 92 291)(17 214 93 290)(18 213 94 289)(19 212 95 288)(20 211 96 287)(21 210 97 286)(22 209 98 285)(23 208 99 284)(24 207 100 283)(25 206 101 282)(26 205 102 281)(27 204 103 280)(28 203 104 279)(29 202 105 278)(30 201 106 277)(31 200 107 276)(32 199 108 275)(33 198 109 274)(34 197 110 273)(35 196 111 272)(36 195 112 271)(37 194 113 270)(38 193 114 269)(39 192 115 268)(40 191 116 267)(41 190 117 266)(42 189 118 265)(43 188 119 264)(44 187 120 263)(45 186 121 262)(46 185 122 261)(47 184 123 260)(48 183 124 259)(49 182 125 258)(50 181 126 257)(51 180 127 256)(52 179 128 255)(53 178 129 254)(54 177 130 253)(55 176 131 252)(56 175 132 251)(57 174 133 250)(58 173 134 249)(59 172 135 248)(60 171 136 247)(61 170 137 246)(62 169 138 245)(63 168 139 244)(64 167 140 243)(65 166 141 242)(66 165 142 241)(67 164 143 240)(68 163 144 239)(69 162 145 238)(70 161 146 237)(71 160 147 236)(72 159 148 235)(73 158 149 234)(74 157 150 233)(75 156 151 232)(76 155 152 231)
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), (1,230,77,154)(2,229,78,153)(3,228,79,304)(4,227,80,303)(5,226,81,302)(6,225,82,301)(7,224,83,300)(8,223,84,299)(9,222,85,298)(10,221,86,297)(11,220,87,296)(12,219,88,295)(13,218,89,294)(14,217,90,293)(15,216,91,292)(16,215,92,291)(17,214,93,290)(18,213,94,289)(19,212,95,288)(20,211,96,287)(21,210,97,286)(22,209,98,285)(23,208,99,284)(24,207,100,283)(25,206,101,282)(26,205,102,281)(27,204,103,280)(28,203,104,279)(29,202,105,278)(30,201,106,277)(31,200,107,276)(32,199,108,275)(33,198,109,274)(34,197,110,273)(35,196,111,272)(36,195,112,271)(37,194,113,270)(38,193,114,269)(39,192,115,268)(40,191,116,267)(41,190,117,266)(42,189,118,265)(43,188,119,264)(44,187,120,263)(45,186,121,262)(46,185,122,261)(47,184,123,260)(48,183,124,259)(49,182,125,258)(50,181,126,257)(51,180,127,256)(52,179,128,255)(53,178,129,254)(54,177,130,253)(55,176,131,252)(56,175,132,251)(57,174,133,250)(58,173,134,249)(59,172,135,248)(60,171,136,247)(61,170,137,246)(62,169,138,245)(63,168,139,244)(64,167,140,243)(65,166,141,242)(66,165,142,241)(67,164,143,240)(68,163,144,239)(69,162,145,238)(70,161,146,237)(71,160,147,236)(72,159,148,235)(73,158,149,234)(74,157,150,233)(75,156,151,232)(76,155,152,231)>;
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), (1,230,77,154)(2,229,78,153)(3,228,79,304)(4,227,80,303)(5,226,81,302)(6,225,82,301)(7,224,83,300)(8,223,84,299)(9,222,85,298)(10,221,86,297)(11,220,87,296)(12,219,88,295)(13,218,89,294)(14,217,90,293)(15,216,91,292)(16,215,92,291)(17,214,93,290)(18,213,94,289)(19,212,95,288)(20,211,96,287)(21,210,97,286)(22,209,98,285)(23,208,99,284)(24,207,100,283)(25,206,101,282)(26,205,102,281)(27,204,103,280)(28,203,104,279)(29,202,105,278)(30,201,106,277)(31,200,107,276)(32,199,108,275)(33,198,109,274)(34,197,110,273)(35,196,111,272)(36,195,112,271)(37,194,113,270)(38,193,114,269)(39,192,115,268)(40,191,116,267)(41,190,117,266)(42,189,118,265)(43,188,119,264)(44,187,120,263)(45,186,121,262)(46,185,122,261)(47,184,123,260)(48,183,124,259)(49,182,125,258)(50,181,126,257)(51,180,127,256)(52,179,128,255)(53,178,129,254)(54,177,130,253)(55,176,131,252)(56,175,132,251)(57,174,133,250)(58,173,134,249)(59,172,135,248)(60,171,136,247)(61,170,137,246)(62,169,138,245)(63,168,139,244)(64,167,140,243)(65,166,141,242)(66,165,142,241)(67,164,143,240)(68,163,144,239)(69,162,145,238)(70,161,146,237)(71,160,147,236)(72,159,148,235)(73,158,149,234)(74,157,150,233)(75,156,151,232)(76,155,152,231) );
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)], [(1,230,77,154),(2,229,78,153),(3,228,79,304),(4,227,80,303),(5,226,81,302),(6,225,82,301),(7,224,83,300),(8,223,84,299),(9,222,85,298),(10,221,86,297),(11,220,87,296),(12,219,88,295),(13,218,89,294),(14,217,90,293),(15,216,91,292),(16,215,92,291),(17,214,93,290),(18,213,94,289),(19,212,95,288),(20,211,96,287),(21,210,97,286),(22,209,98,285),(23,208,99,284),(24,207,100,283),(25,206,101,282),(26,205,102,281),(27,204,103,280),(28,203,104,279),(29,202,105,278),(30,201,106,277),(31,200,107,276),(32,199,108,275),(33,198,109,274),(34,197,110,273),(35,196,111,272),(36,195,112,271),(37,194,113,270),(38,193,114,269),(39,192,115,268),(40,191,116,267),(41,190,117,266),(42,189,118,265),(43,188,119,264),(44,187,120,263),(45,186,121,262),(46,185,122,261),(47,184,123,260),(48,183,124,259),(49,182,125,258),(50,181,126,257),(51,180,127,256),(52,179,128,255),(53,178,129,254),(54,177,130,253),(55,176,131,252),(56,175,132,251),(57,174,133,250),(58,173,134,249),(59,172,135,248),(60,171,136,247),(61,170,137,246),(62,169,138,245),(63,168,139,244),(64,167,140,243),(65,166,141,242),(66,165,142,241),(67,164,143,240),(68,163,144,239),(69,162,145,238),(70,161,146,237),(71,160,147,236),(72,159,148,235),(73,158,149,234),(74,157,150,233),(75,156,151,232),(76,155,152,231)]])
79 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 2 | 76 | 76 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
79 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D19 | D38 | D76 | Dic76 |
| kernel | Dic76 | C152 | Dic38 | C38 | C19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 9 | 9 | 18 | 36 |
Matrix representation of Dic76 ►in GL2(𝔽457) generated by
| 210 | 291 |
| 166 | 32 |
| 135 | 408 |
| 316 | 322 |
G:=sub<GL(2,GF(457))| [210,166,291,32],[135,316,408,322] >;
Dic76 in GAP, Magma, Sage, TeX
{\rm Dic}_{76} % in TeX
G:=Group("Dic76"); // GroupNames label
G:=SmallGroup(304,7);
// by ID
G=gap.SmallGroup(304,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,40,61,66,182,42,7204]);
// Polycyclic
G:=Group<a,b|a^152=1,b^2=a^76,b*a*b^-1=a^-1>;
// generators/relations
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