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G = C4×Dic20order 320 = 26·5

Direct product of C4 and Dic20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×Dic20, C205Q16, C42.265D10, C52(C4×Q16), (C4×C8).8D5, C8.21(C4×D5), (C4×C40).10C2, C40.92(C2×C4), C2.11(C4×D20), (C2×C4).64D20, C10.38(C4×D4), C10.3(C2×Q16), C10.6(C4○D8), (C2×C20).354D4, (C2×C8).289D10, C405C4.18C2, C2.1(C2×Dic20), (C4×Dic10).3C2, C22.29(C2×D20), C4.104(C4○D20), C20.220(C4○D4), C2.3(D407C2), C20.162(C22×C4), (C2×C40).349C22, (C2×C20).727C23, (C4×C20).327C22, (C2×Dic20).14C2, Dic10.27(C2×C4), C20.44D4.17C2, C4⋊Dic5.264C22, (C2×Dic10).213C22, C4.61(C2×C4×D5), (C2×C10).110(C2×D4), (C2×C4).670(C22×D5), SmallGroup(320,325)

Series: Derived Chief Lower central Upper central

C1C20 — C4×Dic20
C1C5C10C2×C10C2×C20C2×Dic10C2×Dic20 — C4×Dic20
C5C10C20 — C4×Dic20
C1C2×C4C42C4×C8

Generators and relations for C4×Dic20
 G = < a,b,c | a4=b40=1, c2=b20, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 374 in 110 conjugacy classes, 55 normal (31 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×7], C22, C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×6], C10 [×3], C42, C42 [×2], C4⋊C4 [×4], C2×C8 [×2], Q16 [×4], C2×Q8 [×2], Dic5 [×6], C20 [×2], C20 [×2], C20, C2×C10, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C40 [×2], C40, Dic10 [×4], Dic10 [×2], C2×Dic5 [×4], C2×C20 [×3], C4×Q16, Dic20 [×4], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C20.44D4 [×2], C405C4, C4×C40, C4×Dic10 [×2], C2×Dic20, C4×Dic20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, Q16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×Q16, C4○D8, C4×D5 [×2], D20 [×2], C22×D5, C4×Q16, Dic20 [×2], C2×C4×D5, C2×D20, C4○D20, C4×D20, D407C2, C2×Dic20, C4×Dic20

Smallest permutation representation of C4×Dic20
Regular action on 320 points
Generators in S320
(1 146 67 301)(2 147 68 302)(3 148 69 303)(4 149 70 304)(5 150 71 305)(6 151 72 306)(7 152 73 307)(8 153 74 308)(9 154 75 309)(10 155 76 310)(11 156 77 311)(12 157 78 312)(13 158 79 313)(14 159 80 314)(15 160 41 315)(16 121 42 316)(17 122 43 317)(18 123 44 318)(19 124 45 319)(20 125 46 320)(21 126 47 281)(22 127 48 282)(23 128 49 283)(24 129 50 284)(25 130 51 285)(26 131 52 286)(27 132 53 287)(28 133 54 288)(29 134 55 289)(30 135 56 290)(31 136 57 291)(32 137 58 292)(33 138 59 293)(34 139 60 294)(35 140 61 295)(36 141 62 296)(37 142 63 297)(38 143 64 298)(39 144 65 299)(40 145 66 300)(81 215 165 265)(82 216 166 266)(83 217 167 267)(84 218 168 268)(85 219 169 269)(86 220 170 270)(87 221 171 271)(88 222 172 272)(89 223 173 273)(90 224 174 274)(91 225 175 275)(92 226 176 276)(93 227 177 277)(94 228 178 278)(95 229 179 279)(96 230 180 280)(97 231 181 241)(98 232 182 242)(99 233 183 243)(100 234 184 244)(101 235 185 245)(102 236 186 246)(103 237 187 247)(104 238 188 248)(105 239 189 249)(106 240 190 250)(107 201 191 251)(108 202 192 252)(109 203 193 253)(110 204 194 254)(111 205 195 255)(112 206 196 256)(113 207 197 257)(114 208 198 258)(115 209 199 259)(116 210 200 260)(117 211 161 261)(118 212 162 262)(119 213 163 263)(120 214 164 264)
(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)
(1 280 21 260)(2 279 22 259)(3 278 23 258)(4 277 24 257)(5 276 25 256)(6 275 26 255)(7 274 27 254)(8 273 28 253)(9 272 29 252)(10 271 30 251)(11 270 31 250)(12 269 32 249)(13 268 33 248)(14 267 34 247)(15 266 35 246)(16 265 36 245)(17 264 37 244)(18 263 38 243)(19 262 39 242)(20 261 40 241)(41 216 61 236)(42 215 62 235)(43 214 63 234)(44 213 64 233)(45 212 65 232)(46 211 66 231)(47 210 67 230)(48 209 68 229)(49 208 69 228)(50 207 70 227)(51 206 71 226)(52 205 72 225)(53 204 73 224)(54 203 74 223)(55 202 75 222)(56 201 76 221)(57 240 77 220)(58 239 78 219)(59 238 79 218)(60 237 80 217)(81 141 101 121)(82 140 102 160)(83 139 103 159)(84 138 104 158)(85 137 105 157)(86 136 106 156)(87 135 107 155)(88 134 108 154)(89 133 109 153)(90 132 110 152)(91 131 111 151)(92 130 112 150)(93 129 113 149)(94 128 114 148)(95 127 115 147)(96 126 116 146)(97 125 117 145)(98 124 118 144)(99 123 119 143)(100 122 120 142)(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)

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

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

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

92 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I···4P5A5B8A···8H10A···10F20A···20X40A···40AF
order1222444444444···4558···810···1020···2040···40
size11111111222220···20222···22···22···22···2

92 irreducible representations

dim1111111222222222222
type++++++++-+++-
imageC1C2C2C2C2C2C4D4D5Q16C4○D4D10D10C4○D8C4×D5D20Dic20C4○D20D407C2
kernelC4×Dic20C20.44D4C405C4C4×C40C4×Dic10C2×Dic20Dic20C2×C20C4×C8C20C20C42C2×C8C10C8C2×C4C4C4C2
# reps121121822422448816816

Matrix representation of C4×Dic20 in GL4(𝔽41) generated by

9000
0900
0010
0001
,
40100
53500
00153
003835
,
253000
121600
001738
00124
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[40,5,0,0,1,35,0,0,0,0,15,38,0,0,3,35],[25,12,0,0,30,16,0,0,0,0,17,1,0,0,38,24] >;

C4×Dic20 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{20}
% in TeX

G:=Group("C4xDic20");
// GroupNames label

G:=SmallGroup(320,325);
// by ID

G=gap.SmallGroup(320,325);
# by ID

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

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

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