metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic79, C79⋊C4, C2.D79, C158.C2, SmallGroup(316,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C79 — Dic79 |
Generators and relations for Dic79
G = < a,b | a158=1, b2=a79, bab-1=a-1 >
Smallest permutation representation of Dic79
►Regular action on 316 points
Generators in S316
(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)(159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316)
(1 298 80 219)(2 297 81 218)(3 296 82 217)(4 295 83 216)(5 294 84 215)(6 293 85 214)(7 292 86 213)(8 291 87 212)(9 290 88 211)(10 289 89 210)(11 288 90 209)(12 287 91 208)(13 286 92 207)(14 285 93 206)(15 284 94 205)(16 283 95 204)(17 282 96 203)(18 281 97 202)(19 280 98 201)(20 279 99 200)(21 278 100 199)(22 277 101 198)(23 276 102 197)(24 275 103 196)(25 274 104 195)(26 273 105 194)(27 272 106 193)(28 271 107 192)(29 270 108 191)(30 269 109 190)(31 268 110 189)(32 267 111 188)(33 266 112 187)(34 265 113 186)(35 264 114 185)(36 263 115 184)(37 262 116 183)(38 261 117 182)(39 260 118 181)(40 259 119 180)(41 258 120 179)(42 257 121 178)(43 256 122 177)(44 255 123 176)(45 254 124 175)(46 253 125 174)(47 252 126 173)(48 251 127 172)(49 250 128 171)(50 249 129 170)(51 248 130 169)(52 247 131 168)(53 246 132 167)(54 245 133 166)(55 244 134 165)(56 243 135 164)(57 242 136 163)(58 241 137 162)(59 240 138 161)(60 239 139 160)(61 238 140 159)(62 237 141 316)(63 236 142 315)(64 235 143 314)(65 234 144 313)(66 233 145 312)(67 232 146 311)(68 231 147 310)(69 230 148 309)(70 229 149 308)(71 228 150 307)(72 227 151 306)(73 226 152 305)(74 225 153 304)(75 224 154 303)(76 223 155 302)(77 222 156 301)(78 221 157 300)(79 220 158 299)
G:=sub<Sym(316)| (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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316), (1,298,80,219)(2,297,81,218)(3,296,82,217)(4,295,83,216)(5,294,84,215)(6,293,85,214)(7,292,86,213)(8,291,87,212)(9,290,88,211)(10,289,89,210)(11,288,90,209)(12,287,91,208)(13,286,92,207)(14,285,93,206)(15,284,94,205)(16,283,95,204)(17,282,96,203)(18,281,97,202)(19,280,98,201)(20,279,99,200)(21,278,100,199)(22,277,101,198)(23,276,102,197)(24,275,103,196)(25,274,104,195)(26,273,105,194)(27,272,106,193)(28,271,107,192)(29,270,108,191)(30,269,109,190)(31,268,110,189)(32,267,111,188)(33,266,112,187)(34,265,113,186)(35,264,114,185)(36,263,115,184)(37,262,116,183)(38,261,117,182)(39,260,118,181)(40,259,119,180)(41,258,120,179)(42,257,121,178)(43,256,122,177)(44,255,123,176)(45,254,124,175)(46,253,125,174)(47,252,126,173)(48,251,127,172)(49,250,128,171)(50,249,129,170)(51,248,130,169)(52,247,131,168)(53,246,132,167)(54,245,133,166)(55,244,134,165)(56,243,135,164)(57,242,136,163)(58,241,137,162)(59,240,138,161)(60,239,139,160)(61,238,140,159)(62,237,141,316)(63,236,142,315)(64,235,143,314)(65,234,144,313)(66,233,145,312)(67,232,146,311)(68,231,147,310)(69,230,148,309)(70,229,149,308)(71,228,150,307)(72,227,151,306)(73,226,152,305)(74,225,153,304)(75,224,154,303)(76,223,155,302)(77,222,156,301)(78,221,157,300)(79,220,158,299)>;
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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316), (1,298,80,219)(2,297,81,218)(3,296,82,217)(4,295,83,216)(5,294,84,215)(6,293,85,214)(7,292,86,213)(8,291,87,212)(9,290,88,211)(10,289,89,210)(11,288,90,209)(12,287,91,208)(13,286,92,207)(14,285,93,206)(15,284,94,205)(16,283,95,204)(17,282,96,203)(18,281,97,202)(19,280,98,201)(20,279,99,200)(21,278,100,199)(22,277,101,198)(23,276,102,197)(24,275,103,196)(25,274,104,195)(26,273,105,194)(27,272,106,193)(28,271,107,192)(29,270,108,191)(30,269,109,190)(31,268,110,189)(32,267,111,188)(33,266,112,187)(34,265,113,186)(35,264,114,185)(36,263,115,184)(37,262,116,183)(38,261,117,182)(39,260,118,181)(40,259,119,180)(41,258,120,179)(42,257,121,178)(43,256,122,177)(44,255,123,176)(45,254,124,175)(46,253,125,174)(47,252,126,173)(48,251,127,172)(49,250,128,171)(50,249,129,170)(51,248,130,169)(52,247,131,168)(53,246,132,167)(54,245,133,166)(55,244,134,165)(56,243,135,164)(57,242,136,163)(58,241,137,162)(59,240,138,161)(60,239,139,160)(61,238,140,159)(62,237,141,316)(63,236,142,315)(64,235,143,314)(65,234,144,313)(66,233,145,312)(67,232,146,311)(68,231,147,310)(69,230,148,309)(70,229,149,308)(71,228,150,307)(72,227,151,306)(73,226,152,305)(74,225,153,304)(75,224,154,303)(76,223,155,302)(77,222,156,301)(78,221,157,300)(79,220,158,299) );
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),(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316)], [(1,298,80,219),(2,297,81,218),(3,296,82,217),(4,295,83,216),(5,294,84,215),(6,293,85,214),(7,292,86,213),(8,291,87,212),(9,290,88,211),(10,289,89,210),(11,288,90,209),(12,287,91,208),(13,286,92,207),(14,285,93,206),(15,284,94,205),(16,283,95,204),(17,282,96,203),(18,281,97,202),(19,280,98,201),(20,279,99,200),(21,278,100,199),(22,277,101,198),(23,276,102,197),(24,275,103,196),(25,274,104,195),(26,273,105,194),(27,272,106,193),(28,271,107,192),(29,270,108,191),(30,269,109,190),(31,268,110,189),(32,267,111,188),(33,266,112,187),(34,265,113,186),(35,264,114,185),(36,263,115,184),(37,262,116,183),(38,261,117,182),(39,260,118,181),(40,259,119,180),(41,258,120,179),(42,257,121,178),(43,256,122,177),(44,255,123,176),(45,254,124,175),(46,253,125,174),(47,252,126,173),(48,251,127,172),(49,250,128,171),(50,249,129,170),(51,248,130,169),(52,247,131,168),(53,246,132,167),(54,245,133,166),(55,244,134,165),(56,243,135,164),(57,242,136,163),(58,241,137,162),(59,240,138,161),(60,239,139,160),(61,238,140,159),(62,237,141,316),(63,236,142,315),(64,235,143,314),(65,234,144,313),(66,233,145,312),(67,232,146,311),(68,231,147,310),(69,230,148,309),(70,229,149,308),(71,228,150,307),(72,227,151,306),(73,226,152,305),(74,225,153,304),(75,224,154,303),(76,223,155,302),(77,222,156,301),(78,221,157,300),(79,220,158,299)]])
82 conjugacy classes
| class | 1 | 2 | 4A | 4B | 79A | ··· | 79AM | 158A | ··· | 158AM |
| order | 1 | 2 | 4 | 4 | 79 | ··· | 79 | 158 | ··· | 158 |
| size | 1 | 1 | 79 | 79 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D79 | Dic79 |
| kernel | Dic79 | C158 | C79 | C2 | C1 |
| # reps | 1 | 1 | 2 | 39 | 39 |
Matrix representation of Dic79 ►in GL3(𝔽317) generated by
| 316 | 0 | 0 |
| 0 | 261 | 316 |
| 0 | 1 | 0 |
| 203 | 0 | 0 |
| 0 | 291 | 148 |
| 0 | 19 | 26 |
G:=sub<GL(3,GF(317))| [316,0,0,0,261,1,0,316,0],[203,0,0,0,291,19,0,148,26] >;
Dic79 in GAP, Magma, Sage, TeX
{\rm Dic}_{79} % in TeX
G:=Group("Dic79"); // GroupNames label
G:=SmallGroup(316,1);
// by ID
G=gap.SmallGroup(316,1);
# by ID
G:=PCGroup([3,-2,-2,-79,6,2810]);
// Polycyclic
G:=Group<a,b|a^158=1,b^2=a^79,b*a*b^-1=a^-1>;
// generators/relations
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