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## G = Dic20⋊9C4order 320 = 26·5

### 3rd semidirect product of Dic20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic20⋊9C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C2×Dic20 — Dic20⋊9C4
 Lower central C5 — C10 — C20 — Dic20⋊9C4
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for Dic209C4
G = < a,b,c | a40=c4=1, b2=a20, bab-1=a-1, cac-1=a21, bc=cb >

Subgroups: 374 in 108 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C40, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, Q16⋊C4, Dic20, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C20.44D4, C406C4, C5×C8⋊C4, C4×Dic10, C2×Dic20, Dic209C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8.C22, C4×D5, D20, C22×D5, Q16⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8.D10, Dic209C4

Smallest permutation representation of Dic209C4
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)
(1 157 21 137)(2 156 22 136)(3 155 23 135)(4 154 24 134)(5 153 25 133)(6 152 26 132)(7 151 27 131)(8 150 28 130)(9 149 29 129)(10 148 30 128)(11 147 31 127)(12 146 32 126)(13 145 33 125)(14 144 34 124)(15 143 35 123)(16 142 36 122)(17 141 37 121)(18 140 38 160)(19 139 39 159)(20 138 40 158)(41 201 61 221)(42 240 62 220)(43 239 63 219)(44 238 64 218)(45 237 65 217)(46 236 66 216)(47 235 67 215)(48 234 68 214)(49 233 69 213)(50 232 70 212)(51 231 71 211)(52 230 72 210)(53 229 73 209)(54 228 74 208)(55 227 75 207)(56 226 76 206)(57 225 77 205)(58 224 78 204)(59 223 79 203)(60 222 80 202)(81 279 101 259)(82 278 102 258)(83 277 103 257)(84 276 104 256)(85 275 105 255)(86 274 106 254)(87 273 107 253)(88 272 108 252)(89 271 109 251)(90 270 110 250)(91 269 111 249)(92 268 112 248)(93 267 113 247)(94 266 114 246)(95 265 115 245)(96 264 116 244)(97 263 117 243)(98 262 118 242)(99 261 119 241)(100 260 120 280)(161 300 181 320)(162 299 182 319)(163 298 183 318)(164 297 184 317)(165 296 185 316)(166 295 186 315)(167 294 187 314)(168 293 188 313)(169 292 189 312)(170 291 190 311)(171 290 191 310)(172 289 192 309)(173 288 193 308)(174 287 194 307)(175 286 195 306)(176 285 196 305)(177 284 197 304)(178 283 198 303)(179 282 199 302)(180 281 200 301)
(1 68 114 286)(2 49 115 307)(3 70 116 288)(4 51 117 309)(5 72 118 290)(6 53 119 311)(7 74 120 292)(8 55 81 313)(9 76 82 294)(10 57 83 315)(11 78 84 296)(12 59 85 317)(13 80 86 298)(14 61 87 319)(15 42 88 300)(16 63 89 281)(17 44 90 302)(18 65 91 283)(19 46 92 304)(20 67 93 285)(21 48 94 306)(22 69 95 287)(23 50 96 308)(24 71 97 289)(25 52 98 310)(26 73 99 291)(27 54 100 312)(28 75 101 293)(29 56 102 314)(30 77 103 295)(31 58 104 316)(32 79 105 297)(33 60 106 318)(34 41 107 299)(35 62 108 320)(36 43 109 301)(37 64 110 282)(38 45 111 303)(39 66 112 284)(40 47 113 305)(121 218 250 199)(122 239 251 180)(123 220 252 161)(124 201 253 182)(125 222 254 163)(126 203 255 184)(127 224 256 165)(128 205 257 186)(129 226 258 167)(130 207 259 188)(131 228 260 169)(132 209 261 190)(133 230 262 171)(134 211 263 192)(135 232 264 173)(136 213 265 194)(137 234 266 175)(138 215 267 196)(139 236 268 177)(140 217 269 198)(141 238 270 179)(142 219 271 200)(143 240 272 181)(144 221 273 162)(145 202 274 183)(146 223 275 164)(147 204 276 185)(148 225 277 166)(149 206 278 187)(150 227 279 168)(151 208 280 189)(152 229 241 170)(153 210 242 191)(154 231 243 172)(155 212 244 193)(156 233 245 174)(157 214 246 195)(158 235 247 176)(159 216 248 197)(160 237 249 178)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G ··· 4N 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 4 ··· 4 4 ··· 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 ··· 2 20 ··· 20 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C4×D5 D20 C4○D20 C8.C22 C8.D10 kernel Dic20⋊9C4 C20.44D4 C40⋊6C4 C5×C8⋊C4 C4×Dic10 C2×Dic20 Dic20 C2×C20 C8⋊C4 C20 C42 C2×C8 C8 C2×C4 C4 C10 C2 # reps 1 2 1 1 2 1 8 2 2 2 2 4 8 8 8 2 8

Matrix representation of Dic209C4 in GL6(𝔽41)

 1 23 0 0 0 0 32 40 0 0 0 0 0 0 32 19 34 2 0 0 22 22 39 20 0 0 34 2 9 22 0 0 39 20 19 19
,
 16 27 0 0 0 0 27 25 0 0 0 0 0 0 27 2 35 12 0 0 18 14 13 6 0 0 6 29 27 2 0 0 28 35 18 14
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0

G:=sub<GL(6,GF(41))| [1,32,0,0,0,0,23,40,0,0,0,0,0,0,32,22,34,39,0,0,19,22,2,20,0,0,34,39,9,19,0,0,2,20,22,19],[16,27,0,0,0,0,27,25,0,0,0,0,0,0,27,18,6,28,0,0,2,14,29,35,0,0,35,13,27,18,0,0,12,6,2,14],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

Dic209C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{20}\rtimes_9C_4
% in TeX

G:=Group("Dic20:9C4");
// GroupNames label

G:=SmallGroup(320,343);
// by ID

G=gap.SmallGroup(320,343);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,387,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^4=1,b^2=a^20,b*a*b^-1=a^-1,c*a*c^-1=a^21,b*c=c*b>;
// generators/relations

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