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## G = Dic80order 320 = 26·5

### Dicyclic group

Aliases: Dic80, C32.D5, C51Q64, C2.5D80, C4.3D40, C8.7D20, C160.1C2, C40.57D4, C20.28D8, C10.3D16, C16.15D10, C80.16C22, Dic40.1C2, SmallGroup(320,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C80 — Dic80
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — Dic40 — Dic80
 Lower central C5 — C10 — C20 — C40 — C80 — Dic80
 Upper central C1 — C2 — C4 — C8 — C16 — C32

Generators and relations for Dic80
G = < a,b | a160=1, b2=a80, bab-1=a-1 >

Smallest permutation representation of Dic80
Regular action on 320 points
Generators in S320
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 273 81 193)(2 272 82 192)(3 271 83 191)(4 270 84 190)(5 269 85 189)(6 268 86 188)(7 267 87 187)(8 266 88 186)(9 265 89 185)(10 264 90 184)(11 263 91 183)(12 262 92 182)(13 261 93 181)(14 260 94 180)(15 259 95 179)(16 258 96 178)(17 257 97 177)(18 256 98 176)(19 255 99 175)(20 254 100 174)(21 253 101 173)(22 252 102 172)(23 251 103 171)(24 250 104 170)(25 249 105 169)(26 248 106 168)(27 247 107 167)(28 246 108 166)(29 245 109 165)(30 244 110 164)(31 243 111 163)(32 242 112 162)(33 241 113 161)(34 240 114 320)(35 239 115 319)(36 238 116 318)(37 237 117 317)(38 236 118 316)(39 235 119 315)(40 234 120 314)(41 233 121 313)(42 232 122 312)(43 231 123 311)(44 230 124 310)(45 229 125 309)(46 228 126 308)(47 227 127 307)(48 226 128 306)(49 225 129 305)(50 224 130 304)(51 223 131 303)(52 222 132 302)(53 221 133 301)(54 220 134 300)(55 219 135 299)(56 218 136 298)(57 217 137 297)(58 216 138 296)(59 215 139 295)(60 214 140 294)(61 213 141 293)(62 212 142 292)(63 211 143 291)(64 210 144 290)(65 209 145 289)(66 208 146 288)(67 207 147 287)(68 206 148 286)(69 205 149 285)(70 204 150 284)(71 203 151 283)(72 202 152 282)(73 201 153 281)(74 200 154 280)(75 199 155 279)(76 198 156 278)(77 197 157 277)(78 196 158 276)(79 195 159 275)(80 194 160 274)

G:=sub<Sym(320)| (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,273,81,193)(2,272,82,192)(3,271,83,191)(4,270,84,190)(5,269,85,189)(6,268,86,188)(7,267,87,187)(8,266,88,186)(9,265,89,185)(10,264,90,184)(11,263,91,183)(12,262,92,182)(13,261,93,181)(14,260,94,180)(15,259,95,179)(16,258,96,178)(17,257,97,177)(18,256,98,176)(19,255,99,175)(20,254,100,174)(21,253,101,173)(22,252,102,172)(23,251,103,171)(24,250,104,170)(25,249,105,169)(26,248,106,168)(27,247,107,167)(28,246,108,166)(29,245,109,165)(30,244,110,164)(31,243,111,163)(32,242,112,162)(33,241,113,161)(34,240,114,320)(35,239,115,319)(36,238,116,318)(37,237,117,317)(38,236,118,316)(39,235,119,315)(40,234,120,314)(41,233,121,313)(42,232,122,312)(43,231,123,311)(44,230,124,310)(45,229,125,309)(46,228,126,308)(47,227,127,307)(48,226,128,306)(49,225,129,305)(50,224,130,304)(51,223,131,303)(52,222,132,302)(53,221,133,301)(54,220,134,300)(55,219,135,299)(56,218,136,298)(57,217,137,297)(58,216,138,296)(59,215,139,295)(60,214,140,294)(61,213,141,293)(62,212,142,292)(63,211,143,291)(64,210,144,290)(65,209,145,289)(66,208,146,288)(67,207,147,287)(68,206,148,286)(69,205,149,285)(70,204,150,284)(71,203,151,283)(72,202,152,282)(73,201,153,281)(74,200,154,280)(75,199,155,279)(76,198,156,278)(77,197,157,277)(78,196,158,276)(79,195,159,275)(80,194,160,274)>;

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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,273,81,193)(2,272,82,192)(3,271,83,191)(4,270,84,190)(5,269,85,189)(6,268,86,188)(7,267,87,187)(8,266,88,186)(9,265,89,185)(10,264,90,184)(11,263,91,183)(12,262,92,182)(13,261,93,181)(14,260,94,180)(15,259,95,179)(16,258,96,178)(17,257,97,177)(18,256,98,176)(19,255,99,175)(20,254,100,174)(21,253,101,173)(22,252,102,172)(23,251,103,171)(24,250,104,170)(25,249,105,169)(26,248,106,168)(27,247,107,167)(28,246,108,166)(29,245,109,165)(30,244,110,164)(31,243,111,163)(32,242,112,162)(33,241,113,161)(34,240,114,320)(35,239,115,319)(36,238,116,318)(37,237,117,317)(38,236,118,316)(39,235,119,315)(40,234,120,314)(41,233,121,313)(42,232,122,312)(43,231,123,311)(44,230,124,310)(45,229,125,309)(46,228,126,308)(47,227,127,307)(48,226,128,306)(49,225,129,305)(50,224,130,304)(51,223,131,303)(52,222,132,302)(53,221,133,301)(54,220,134,300)(55,219,135,299)(56,218,136,298)(57,217,137,297)(58,216,138,296)(59,215,139,295)(60,214,140,294)(61,213,141,293)(62,212,142,292)(63,211,143,291)(64,210,144,290)(65,209,145,289)(66,208,146,288)(67,207,147,287)(68,206,148,286)(69,205,149,285)(70,204,150,284)(71,203,151,283)(72,202,152,282)(73,201,153,281)(74,200,154,280)(75,199,155,279)(76,198,156,278)(77,197,157,277)(78,196,158,276)(79,195,159,275)(80,194,160,274) );

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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,273,81,193),(2,272,82,192),(3,271,83,191),(4,270,84,190),(5,269,85,189),(6,268,86,188),(7,267,87,187),(8,266,88,186),(9,265,89,185),(10,264,90,184),(11,263,91,183),(12,262,92,182),(13,261,93,181),(14,260,94,180),(15,259,95,179),(16,258,96,178),(17,257,97,177),(18,256,98,176),(19,255,99,175),(20,254,100,174),(21,253,101,173),(22,252,102,172),(23,251,103,171),(24,250,104,170),(25,249,105,169),(26,248,106,168),(27,247,107,167),(28,246,108,166),(29,245,109,165),(30,244,110,164),(31,243,111,163),(32,242,112,162),(33,241,113,161),(34,240,114,320),(35,239,115,319),(36,238,116,318),(37,237,117,317),(38,236,118,316),(39,235,119,315),(40,234,120,314),(41,233,121,313),(42,232,122,312),(43,231,123,311),(44,230,124,310),(45,229,125,309),(46,228,126,308),(47,227,127,307),(48,226,128,306),(49,225,129,305),(50,224,130,304),(51,223,131,303),(52,222,132,302),(53,221,133,301),(54,220,134,300),(55,219,135,299),(56,218,136,298),(57,217,137,297),(58,216,138,296),(59,215,139,295),(60,214,140,294),(61,213,141,293),(62,212,142,292),(63,211,143,291),(64,210,144,290),(65,209,145,289),(66,208,146,288),(67,207,147,287),(68,206,148,286),(69,205,149,285),(70,204,150,284),(71,203,151,283),(72,202,152,282),(73,201,153,281),(74,200,154,280),(75,199,155,279),(76,198,156,278),(77,197,157,277),(78,196,158,276),(79,195,159,275),(80,194,160,274)]])

83 conjugacy classes

 class 1 2 4A 4B 4C 5A 5B 8A 8B 10A 10B 16A 16B 16C 16D 20A 20B 20C 20D 32A ··· 32H 40A ··· 40H 80A ··· 80P 160A ··· 160AF order 1 2 4 4 4 5 5 8 8 10 10 16 16 16 16 20 20 20 20 32 ··· 32 40 ··· 40 80 ··· 80 160 ··· 160 size 1 1 2 80 80 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

83 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - + + - image C1 C2 C2 D4 D5 D8 D10 D16 D20 Q64 D40 D80 Dic80 kernel Dic80 C160 Dic40 C40 C32 C20 C16 C10 C8 C5 C4 C2 C1 # reps 1 1 2 1 2 2 2 4 4 8 8 16 32

Matrix representation of Dic80 in GL2(𝔽641) generated by

 79 177 464 569
,
 4 268 98 637
G:=sub<GL(2,GF(641))| [79,464,177,569],[4,98,268,637] >;

Dic80 in GAP, Magma, Sage, TeX

{\rm Dic}_{80}
% in TeX

G:=Group("Dic80");
// GroupNames label

G:=SmallGroup(320,8);
// by ID

G=gap.SmallGroup(320,8);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,85,92,254,142,675,192,1684,102,12550]);
// Polycyclic

G:=Group<a,b|a^160=1,b^2=a^80,b*a*b^-1=a^-1>;
// generators/relations

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