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G = Dic79order 316 = 22·79

Dicyclic group

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

C1C79 — Dic79
C1C79C158 — Dic79
C79 — Dic79
C1C2

Generators and relations for Dic79
 G = < a,b | a158=1, b2=a79, bab-1=a-1 >

79C4

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)
(1 278 80 199)(2 277 81 198)(3 276 82 197)(4 275 83 196)(5 274 84 195)(6 273 85 194)(7 272 86 193)(8 271 87 192)(9 270 88 191)(10 269 89 190)(11 268 90 189)(12 267 91 188)(13 266 92 187)(14 265 93 186)(15 264 94 185)(16 263 95 184)(17 262 96 183)(18 261 97 182)(19 260 98 181)(20 259 99 180)(21 258 100 179)(22 257 101 178)(23 256 102 177)(24 255 103 176)(25 254 104 175)(26 253 105 174)(27 252 106 173)(28 251 107 172)(29 250 108 171)(30 249 109 170)(31 248 110 169)(32 247 111 168)(33 246 112 167)(34 245 113 166)(35 244 114 165)(36 243 115 164)(37 242 116 163)(38 241 117 162)(39 240 118 161)(40 239 119 160)(41 238 120 159)(42 237 121 316)(43 236 122 315)(44 235 123 314)(45 234 124 313)(46 233 125 312)(47 232 126 311)(48 231 127 310)(49 230 128 309)(50 229 129 308)(51 228 130 307)(52 227 131 306)(53 226 132 305)(54 225 133 304)(55 224 134 303)(56 223 135 302)(57 222 136 301)(58 221 137 300)(59 220 138 299)(60 219 139 298)(61 218 140 297)(62 217 141 296)(63 216 142 295)(64 215 143 294)(65 214 144 293)(66 213 145 292)(67 212 146 291)(68 211 147 290)(69 210 148 289)(70 209 149 288)(71 208 150 287)(72 207 151 286)(73 206 152 285)(74 205 153 284)(75 204 154 283)(76 203 155 282)(77 202 156 281)(78 201 157 280)(79 200 158 279)

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), (1,278,80,199)(2,277,81,198)(3,276,82,197)(4,275,83,196)(5,274,84,195)(6,273,85,194)(7,272,86,193)(8,271,87,192)(9,270,88,191)(10,269,89,190)(11,268,90,189)(12,267,91,188)(13,266,92,187)(14,265,93,186)(15,264,94,185)(16,263,95,184)(17,262,96,183)(18,261,97,182)(19,260,98,181)(20,259,99,180)(21,258,100,179)(22,257,101,178)(23,256,102,177)(24,255,103,176)(25,254,104,175)(26,253,105,174)(27,252,106,173)(28,251,107,172)(29,250,108,171)(30,249,109,170)(31,248,110,169)(32,247,111,168)(33,246,112,167)(34,245,113,166)(35,244,114,165)(36,243,115,164)(37,242,116,163)(38,241,117,162)(39,240,118,161)(40,239,119,160)(41,238,120,159)(42,237,121,316)(43,236,122,315)(44,235,123,314)(45,234,124,313)(46,233,125,312)(47,232,126,311)(48,231,127,310)(49,230,128,309)(50,229,129,308)(51,228,130,307)(52,227,131,306)(53,226,132,305)(54,225,133,304)(55,224,134,303)(56,223,135,302)(57,222,136,301)(58,221,137,300)(59,220,138,299)(60,219,139,298)(61,218,140,297)(62,217,141,296)(63,216,142,295)(64,215,143,294)(65,214,144,293)(66,213,145,292)(67,212,146,291)(68,211,147,290)(69,210,148,289)(70,209,149,288)(71,208,150,287)(72,207,151,286)(73,206,152,285)(74,205,153,284)(75,204,154,283)(76,203,155,282)(77,202,156,281)(78,201,157,280)(79,200,158,279)>;

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), (1,278,80,199)(2,277,81,198)(3,276,82,197)(4,275,83,196)(5,274,84,195)(6,273,85,194)(7,272,86,193)(8,271,87,192)(9,270,88,191)(10,269,89,190)(11,268,90,189)(12,267,91,188)(13,266,92,187)(14,265,93,186)(15,264,94,185)(16,263,95,184)(17,262,96,183)(18,261,97,182)(19,260,98,181)(20,259,99,180)(21,258,100,179)(22,257,101,178)(23,256,102,177)(24,255,103,176)(25,254,104,175)(26,253,105,174)(27,252,106,173)(28,251,107,172)(29,250,108,171)(30,249,109,170)(31,248,110,169)(32,247,111,168)(33,246,112,167)(34,245,113,166)(35,244,114,165)(36,243,115,164)(37,242,116,163)(38,241,117,162)(39,240,118,161)(40,239,119,160)(41,238,120,159)(42,237,121,316)(43,236,122,315)(44,235,123,314)(45,234,124,313)(46,233,125,312)(47,232,126,311)(48,231,127,310)(49,230,128,309)(50,229,129,308)(51,228,130,307)(52,227,131,306)(53,226,132,305)(54,225,133,304)(55,224,134,303)(56,223,135,302)(57,222,136,301)(58,221,137,300)(59,220,138,299)(60,219,139,298)(61,218,140,297)(62,217,141,296)(63,216,142,295)(64,215,143,294)(65,214,144,293)(66,213,145,292)(67,212,146,291)(68,211,147,290)(69,210,148,289)(70,209,149,288)(71,208,150,287)(72,207,151,286)(73,206,152,285)(74,205,153,284)(75,204,154,283)(76,203,155,282)(77,202,156,281)(78,201,157,280)(79,200,158,279) );

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)], [(1,278,80,199),(2,277,81,198),(3,276,82,197),(4,275,83,196),(5,274,84,195),(6,273,85,194),(7,272,86,193),(8,271,87,192),(9,270,88,191),(10,269,89,190),(11,268,90,189),(12,267,91,188),(13,266,92,187),(14,265,93,186),(15,264,94,185),(16,263,95,184),(17,262,96,183),(18,261,97,182),(19,260,98,181),(20,259,99,180),(21,258,100,179),(22,257,101,178),(23,256,102,177),(24,255,103,176),(25,254,104,175),(26,253,105,174),(27,252,106,173),(28,251,107,172),(29,250,108,171),(30,249,109,170),(31,248,110,169),(32,247,111,168),(33,246,112,167),(34,245,113,166),(35,244,114,165),(36,243,115,164),(37,242,116,163),(38,241,117,162),(39,240,118,161),(40,239,119,160),(41,238,120,159),(42,237,121,316),(43,236,122,315),(44,235,123,314),(45,234,124,313),(46,233,125,312),(47,232,126,311),(48,231,127,310),(49,230,128,309),(50,229,129,308),(51,228,130,307),(52,227,131,306),(53,226,132,305),(54,225,133,304),(55,224,134,303),(56,223,135,302),(57,222,136,301),(58,221,137,300),(59,220,138,299),(60,219,139,298),(61,218,140,297),(62,217,141,296),(63,216,142,295),(64,215,143,294),(65,214,144,293),(66,213,145,292),(67,212,146,291),(68,211,147,290),(69,210,148,289),(70,209,149,288),(71,208,150,287),(72,207,151,286),(73,206,152,285),(74,205,153,284),(75,204,154,283),(76,203,155,282),(77,202,156,281),(78,201,157,280),(79,200,158,279)])

82 conjugacy classes

class 1  2 4A4B79A···79AM158A···158AM
order124479···79158···158
size1179792···22···2

82 irreducible representations

dim11122
type+++-
imageC1C2C4D79Dic79
kernelDic79C158C79C2C1
# reps1123939

Matrix representation of Dic79 in GL3(𝔽317) generated by

31600
0261316
010
,
20300
0291148
01926
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

Export

Subgroup lattice of Dic79 in TeX

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