metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic77, C77⋊1C4, C2.D77, C22.D7, C11⋊Dic7, C7⋊Dic11, C14.D11, C154.1C2, SmallGroup(308,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C77 — Dic77 |
Generators and relations for Dic77
G = < a,b | a154=1, b2=a77, bab-1=a-1 >
Smallest permutation representation of Dic77
►Regular action on 308 points
Generators in S308
(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)
(1 197 78 274)(2 196 79 273)(3 195 80 272)(4 194 81 271)(5 193 82 270)(6 192 83 269)(7 191 84 268)(8 190 85 267)(9 189 86 266)(10 188 87 265)(11 187 88 264)(12 186 89 263)(13 185 90 262)(14 184 91 261)(15 183 92 260)(16 182 93 259)(17 181 94 258)(18 180 95 257)(19 179 96 256)(20 178 97 255)(21 177 98 254)(22 176 99 253)(23 175 100 252)(24 174 101 251)(25 173 102 250)(26 172 103 249)(27 171 104 248)(28 170 105 247)(29 169 106 246)(30 168 107 245)(31 167 108 244)(32 166 109 243)(33 165 110 242)(34 164 111 241)(35 163 112 240)(36 162 113 239)(37 161 114 238)(38 160 115 237)(39 159 116 236)(40 158 117 235)(41 157 118 234)(42 156 119 233)(43 155 120 232)(44 308 121 231)(45 307 122 230)(46 306 123 229)(47 305 124 228)(48 304 125 227)(49 303 126 226)(50 302 127 225)(51 301 128 224)(52 300 129 223)(53 299 130 222)(54 298 131 221)(55 297 132 220)(56 296 133 219)(57 295 134 218)(58 294 135 217)(59 293 136 216)(60 292 137 215)(61 291 138 214)(62 290 139 213)(63 289 140 212)(64 288 141 211)(65 287 142 210)(66 286 143 209)(67 285 144 208)(68 284 145 207)(69 283 146 206)(70 282 147 205)(71 281 148 204)(72 280 149 203)(73 279 150 202)(74 278 151 201)(75 277 152 200)(76 276 153 199)(77 275 154 198)
G:=sub<Sym(308)| (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), (1,197,78,274)(2,196,79,273)(3,195,80,272)(4,194,81,271)(5,193,82,270)(6,192,83,269)(7,191,84,268)(8,190,85,267)(9,189,86,266)(10,188,87,265)(11,187,88,264)(12,186,89,263)(13,185,90,262)(14,184,91,261)(15,183,92,260)(16,182,93,259)(17,181,94,258)(18,180,95,257)(19,179,96,256)(20,178,97,255)(21,177,98,254)(22,176,99,253)(23,175,100,252)(24,174,101,251)(25,173,102,250)(26,172,103,249)(27,171,104,248)(28,170,105,247)(29,169,106,246)(30,168,107,245)(31,167,108,244)(32,166,109,243)(33,165,110,242)(34,164,111,241)(35,163,112,240)(36,162,113,239)(37,161,114,238)(38,160,115,237)(39,159,116,236)(40,158,117,235)(41,157,118,234)(42,156,119,233)(43,155,120,232)(44,308,121,231)(45,307,122,230)(46,306,123,229)(47,305,124,228)(48,304,125,227)(49,303,126,226)(50,302,127,225)(51,301,128,224)(52,300,129,223)(53,299,130,222)(54,298,131,221)(55,297,132,220)(56,296,133,219)(57,295,134,218)(58,294,135,217)(59,293,136,216)(60,292,137,215)(61,291,138,214)(62,290,139,213)(63,289,140,212)(64,288,141,211)(65,287,142,210)(66,286,143,209)(67,285,144,208)(68,284,145,207)(69,283,146,206)(70,282,147,205)(71,281,148,204)(72,280,149,203)(73,279,150,202)(74,278,151,201)(75,277,152,200)(76,276,153,199)(77,275,154,198)>;
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), (1,197,78,274)(2,196,79,273)(3,195,80,272)(4,194,81,271)(5,193,82,270)(6,192,83,269)(7,191,84,268)(8,190,85,267)(9,189,86,266)(10,188,87,265)(11,187,88,264)(12,186,89,263)(13,185,90,262)(14,184,91,261)(15,183,92,260)(16,182,93,259)(17,181,94,258)(18,180,95,257)(19,179,96,256)(20,178,97,255)(21,177,98,254)(22,176,99,253)(23,175,100,252)(24,174,101,251)(25,173,102,250)(26,172,103,249)(27,171,104,248)(28,170,105,247)(29,169,106,246)(30,168,107,245)(31,167,108,244)(32,166,109,243)(33,165,110,242)(34,164,111,241)(35,163,112,240)(36,162,113,239)(37,161,114,238)(38,160,115,237)(39,159,116,236)(40,158,117,235)(41,157,118,234)(42,156,119,233)(43,155,120,232)(44,308,121,231)(45,307,122,230)(46,306,123,229)(47,305,124,228)(48,304,125,227)(49,303,126,226)(50,302,127,225)(51,301,128,224)(52,300,129,223)(53,299,130,222)(54,298,131,221)(55,297,132,220)(56,296,133,219)(57,295,134,218)(58,294,135,217)(59,293,136,216)(60,292,137,215)(61,291,138,214)(62,290,139,213)(63,289,140,212)(64,288,141,211)(65,287,142,210)(66,286,143,209)(67,285,144,208)(68,284,145,207)(69,283,146,206)(70,282,147,205)(71,281,148,204)(72,280,149,203)(73,279,150,202)(74,278,151,201)(75,277,152,200)(76,276,153,199)(77,275,154,198) );
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)], [(1,197,78,274),(2,196,79,273),(3,195,80,272),(4,194,81,271),(5,193,82,270),(6,192,83,269),(7,191,84,268),(8,190,85,267),(9,189,86,266),(10,188,87,265),(11,187,88,264),(12,186,89,263),(13,185,90,262),(14,184,91,261),(15,183,92,260),(16,182,93,259),(17,181,94,258),(18,180,95,257),(19,179,96,256),(20,178,97,255),(21,177,98,254),(22,176,99,253),(23,175,100,252),(24,174,101,251),(25,173,102,250),(26,172,103,249),(27,171,104,248),(28,170,105,247),(29,169,106,246),(30,168,107,245),(31,167,108,244),(32,166,109,243),(33,165,110,242),(34,164,111,241),(35,163,112,240),(36,162,113,239),(37,161,114,238),(38,160,115,237),(39,159,116,236),(40,158,117,235),(41,157,118,234),(42,156,119,233),(43,155,120,232),(44,308,121,231),(45,307,122,230),(46,306,123,229),(47,305,124,228),(48,304,125,227),(49,303,126,226),(50,302,127,225),(51,301,128,224),(52,300,129,223),(53,299,130,222),(54,298,131,221),(55,297,132,220),(56,296,133,219),(57,295,134,218),(58,294,135,217),(59,293,136,216),(60,292,137,215),(61,291,138,214),(62,290,139,213),(63,289,140,212),(64,288,141,211),(65,287,142,210),(66,286,143,209),(67,285,144,208),(68,284,145,207),(69,283,146,206),(70,282,147,205),(71,281,148,204),(72,280,149,203),(73,279,150,202),(74,278,151,201),(75,277,152,200),(76,276,153,199),(77,275,154,198)]])
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 11A | ··· | 11E | 14A | 14B | 14C | 22A | ··· | 22E | 77A | ··· | 77AD | 154A | ··· | 154AD |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 11 | ··· | 11 | 14 | 14 | 14 | 22 | ··· | 22 | 77 | ··· | 77 | 154 | ··· | 154 |
| size | 1 | 1 | 77 | 77 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C4 | D7 | D11 | Dic7 | Dic11 | D77 | Dic77 |
| kernel | Dic77 | C154 | C77 | C22 | C14 | C11 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 5 | 3 | 5 | 30 | 30 |
Matrix representation of Dic77 ►in GL2(𝔽617) generated by
| 451 | 6 |
| 611 | 108 |
| 339 | 428 |
| 484 | 278 |
G:=sub<GL(2,GF(617))| [451,611,6,108],[339,484,428,278] >;
Dic77 in GAP, Magma, Sage, TeX
{\rm Dic}_{77} % in TeX
G:=Group("Dic77"); // GroupNames label
G:=SmallGroup(308,3);
// by ID
G=gap.SmallGroup(308,3);
# by ID
G:=PCGroup([4,-2,-2,-7,-11,8,290,4483]);
// Polycyclic
G:=Group<a,b|a^154=1,b^2=a^77,b*a*b^-1=a^-1>;
// generators/relations
Export