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G = Dic77order 308 = 22·7·11

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic77, C771C4, C2.D77, C22.D7, C11⋊Dic7, C7⋊Dic11, C14.D11, C154.1C2, SmallGroup(308,3)

Series: Derived Chief Lower central Upper central

C1C77 — Dic77
C1C11C77C154 — Dic77
C77 — Dic77
C1C2

Generators and relations for Dic77
 G = < a,b | a154=1, b2=a77, bab-1=a-1 >

77C4
11Dic7
7Dic11

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

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

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

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

80 conjugacy classes

class 1  2 4A4B7A7B7C11A···11E14A14B14C22A···22E77A···77AD154A···154AD
order124477711···1114141422···2277···77154···154
size1177772222···22222···22···22···2

80 irreducible representations

dim111222222
type++++--+-
imageC1C2C4D7D11Dic7Dic11D77Dic77
kernelDic77C154C77C22C14C11C7C2C1
# reps11235353030

Matrix representation of Dic77 in GL2(𝔽617) generated by

4516
611108
,
339428
484278
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

Subgroup lattice of Dic77 in TeX

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