metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic71, C71⋊C4, C2.D71, C142.C2, SmallGroup(284,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C71 — Dic71 |
Generators and relations for Dic71
G = < a,b | a142=1, b2=a71, bab-1=a-1 >
Smallest permutation representation of Dic71
►Regular action on 284 points
Generators in S284
(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)
(1 274 72 203)(2 273 73 202)(3 272 74 201)(4 271 75 200)(5 270 76 199)(6 269 77 198)(7 268 78 197)(8 267 79 196)(9 266 80 195)(10 265 81 194)(11 264 82 193)(12 263 83 192)(13 262 84 191)(14 261 85 190)(15 260 86 189)(16 259 87 188)(17 258 88 187)(18 257 89 186)(19 256 90 185)(20 255 91 184)(21 254 92 183)(22 253 93 182)(23 252 94 181)(24 251 95 180)(25 250 96 179)(26 249 97 178)(27 248 98 177)(28 247 99 176)(29 246 100 175)(30 245 101 174)(31 244 102 173)(32 243 103 172)(33 242 104 171)(34 241 105 170)(35 240 106 169)(36 239 107 168)(37 238 108 167)(38 237 109 166)(39 236 110 165)(40 235 111 164)(41 234 112 163)(42 233 113 162)(43 232 114 161)(44 231 115 160)(45 230 116 159)(46 229 117 158)(47 228 118 157)(48 227 119 156)(49 226 120 155)(50 225 121 154)(51 224 122 153)(52 223 123 152)(53 222 124 151)(54 221 125 150)(55 220 126 149)(56 219 127 148)(57 218 128 147)(58 217 129 146)(59 216 130 145)(60 215 131 144)(61 214 132 143)(62 213 133 284)(63 212 134 283)(64 211 135 282)(65 210 136 281)(66 209 137 280)(67 208 138 279)(68 207 139 278)(69 206 140 277)(70 205 141 276)(71 204 142 275)
G:=sub<Sym(284)| (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), (1,274,72,203)(2,273,73,202)(3,272,74,201)(4,271,75,200)(5,270,76,199)(6,269,77,198)(7,268,78,197)(8,267,79,196)(9,266,80,195)(10,265,81,194)(11,264,82,193)(12,263,83,192)(13,262,84,191)(14,261,85,190)(15,260,86,189)(16,259,87,188)(17,258,88,187)(18,257,89,186)(19,256,90,185)(20,255,91,184)(21,254,92,183)(22,253,93,182)(23,252,94,181)(24,251,95,180)(25,250,96,179)(26,249,97,178)(27,248,98,177)(28,247,99,176)(29,246,100,175)(30,245,101,174)(31,244,102,173)(32,243,103,172)(33,242,104,171)(34,241,105,170)(35,240,106,169)(36,239,107,168)(37,238,108,167)(38,237,109,166)(39,236,110,165)(40,235,111,164)(41,234,112,163)(42,233,113,162)(43,232,114,161)(44,231,115,160)(45,230,116,159)(46,229,117,158)(47,228,118,157)(48,227,119,156)(49,226,120,155)(50,225,121,154)(51,224,122,153)(52,223,123,152)(53,222,124,151)(54,221,125,150)(55,220,126,149)(56,219,127,148)(57,218,128,147)(58,217,129,146)(59,216,130,145)(60,215,131,144)(61,214,132,143)(62,213,133,284)(63,212,134,283)(64,211,135,282)(65,210,136,281)(66,209,137,280)(67,208,138,279)(68,207,139,278)(69,206,140,277)(70,205,141,276)(71,204,142,275)>;
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), (1,274,72,203)(2,273,73,202)(3,272,74,201)(4,271,75,200)(5,270,76,199)(6,269,77,198)(7,268,78,197)(8,267,79,196)(9,266,80,195)(10,265,81,194)(11,264,82,193)(12,263,83,192)(13,262,84,191)(14,261,85,190)(15,260,86,189)(16,259,87,188)(17,258,88,187)(18,257,89,186)(19,256,90,185)(20,255,91,184)(21,254,92,183)(22,253,93,182)(23,252,94,181)(24,251,95,180)(25,250,96,179)(26,249,97,178)(27,248,98,177)(28,247,99,176)(29,246,100,175)(30,245,101,174)(31,244,102,173)(32,243,103,172)(33,242,104,171)(34,241,105,170)(35,240,106,169)(36,239,107,168)(37,238,108,167)(38,237,109,166)(39,236,110,165)(40,235,111,164)(41,234,112,163)(42,233,113,162)(43,232,114,161)(44,231,115,160)(45,230,116,159)(46,229,117,158)(47,228,118,157)(48,227,119,156)(49,226,120,155)(50,225,121,154)(51,224,122,153)(52,223,123,152)(53,222,124,151)(54,221,125,150)(55,220,126,149)(56,219,127,148)(57,218,128,147)(58,217,129,146)(59,216,130,145)(60,215,131,144)(61,214,132,143)(62,213,133,284)(63,212,134,283)(64,211,135,282)(65,210,136,281)(66,209,137,280)(67,208,138,279)(68,207,139,278)(69,206,140,277)(70,205,141,276)(71,204,142,275) );
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)], [(1,274,72,203),(2,273,73,202),(3,272,74,201),(4,271,75,200),(5,270,76,199),(6,269,77,198),(7,268,78,197),(8,267,79,196),(9,266,80,195),(10,265,81,194),(11,264,82,193),(12,263,83,192),(13,262,84,191),(14,261,85,190),(15,260,86,189),(16,259,87,188),(17,258,88,187),(18,257,89,186),(19,256,90,185),(20,255,91,184),(21,254,92,183),(22,253,93,182),(23,252,94,181),(24,251,95,180),(25,250,96,179),(26,249,97,178),(27,248,98,177),(28,247,99,176),(29,246,100,175),(30,245,101,174),(31,244,102,173),(32,243,103,172),(33,242,104,171),(34,241,105,170),(35,240,106,169),(36,239,107,168),(37,238,108,167),(38,237,109,166),(39,236,110,165),(40,235,111,164),(41,234,112,163),(42,233,113,162),(43,232,114,161),(44,231,115,160),(45,230,116,159),(46,229,117,158),(47,228,118,157),(48,227,119,156),(49,226,120,155),(50,225,121,154),(51,224,122,153),(52,223,123,152),(53,222,124,151),(54,221,125,150),(55,220,126,149),(56,219,127,148),(57,218,128,147),(58,217,129,146),(59,216,130,145),(60,215,131,144),(61,214,132,143),(62,213,133,284),(63,212,134,283),(64,211,135,282),(65,210,136,281),(66,209,137,280),(67,208,138,279),(68,207,139,278),(69,206,140,277),(70,205,141,276),(71,204,142,275)]])
74 conjugacy classes
| class | 1 | 2 | 4A | 4B | 71A | ··· | 71AI | 142A | ··· | 142AI |
| order | 1 | 2 | 4 | 4 | 71 | ··· | 71 | 142 | ··· | 142 |
| size | 1 | 1 | 71 | 71 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D71 | Dic71 |
| kernel | Dic71 | C142 | C71 | C2 | C1 |
| # reps | 1 | 1 | 2 | 35 | 35 |
Matrix representation of Dic71 ►in GL2(𝔽569) generated by
| 504 | 1 |
| 568 | 0 |
| 354 | 493 |
| 174 | 215 |
G:=sub<GL(2,GF(569))| [504,568,1,0],[354,174,493,215] >;
Dic71 in GAP, Magma, Sage, TeX
{\rm Dic}_{71} % in TeX
G:=Group("Dic71"); // GroupNames label
G:=SmallGroup(284,1);
// by ID
G=gap.SmallGroup(284,1);
# by ID
G:=PCGroup([3,-2,-2,-71,6,2522]);
// Polycyclic
G:=Group<a,b|a^142=1,b^2=a^71,b*a*b^-1=a^-1>;
// generators/relations
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