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G = C22×Dic20order 320 = 26·5

Direct product of C22 and Dic20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C22×Dic20
 Chief series C1 — C5 — C10 — C20 — Dic10 — C2×Dic10 — C22×Dic10 — C22×Dic20
 Lower central C5 — C10 — C20 — C22×Dic20
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C22×Dic20
G = < a,b,c,d | a2=b2=c40=1, d2=c20, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 862 in 258 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C5, C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×20], C23, C10, C10 [×6], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×18], Dic5 [×8], C20, C20 [×3], C2×C10 [×7], C22×C8, C2×Q16 [×12], C22×Q8 [×2], C40 [×4], Dic10 [×8], Dic10 [×12], C2×Dic5 [×12], C2×C20 [×6], C22×C10, C22×Q16, Dic20 [×16], C2×C40 [×6], C2×Dic10 [×12], C2×Dic10 [×6], C22×Dic5 [×2], C22×C20, C2×Dic20 [×12], C22×C40, C22×Dic10 [×2], C22×Dic20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, Q16 [×4], C2×D4 [×6], C24, D10 [×7], C2×Q16 [×6], C22×D4, D20 [×4], C22×D5 [×7], C22×Q16, Dic20 [×4], C2×D20 [×6], C23×D5, C2×Dic20 [×6], C22×D20, C22×Dic20

Smallest permutation representation of C22×Dic20
Regular action on 320 points
Generators in S320
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 160)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 289)(42 290)(43 291)(44 292)(45 293)(46 294)(47 295)(48 296)(49 297)(50 298)(51 299)(52 300)(53 301)(54 302)(55 303)(56 304)(57 305)(58 306)(59 307)(60 308)(61 309)(62 310)(63 311)(64 312)(65 313)(66 314)(67 315)(68 316)(69 317)(70 318)(71 319)(72 320)(73 281)(74 282)(75 283)(76 284)(77 285)(78 286)(79 287)(80 288)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 197)(98 198)(99 199)(100 200)(101 161)(102 162)(103 163)(104 164)(105 165)(106 166)(107 167)(108 168)(109 169)(110 170)(111 171)(112 172)(113 173)(114 174)(115 175)(116 176)(117 177)(118 178)(119 179)(120 180)(201 265)(202 266)(203 267)(204 268)(205 269)(206 270)(207 271)(208 272)(209 273)(210 274)(211 275)(212 276)(213 277)(214 278)(215 279)(216 280)(217 241)(218 242)(219 243)(220 244)(221 245)(222 246)(223 247)(224 248)(225 249)(226 250)(227 251)(228 252)(229 253)(230 254)(231 255)(232 256)(233 257)(234 258)(235 259)(236 260)(237 261)(238 262)(239 263)(240 264)
(1 196)(2 197)(3 198)(4 199)(5 200)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 169)(15 170)(16 171)(17 172)(18 173)(19 174)(20 175)(21 176)(22 177)(23 178)(24 179)(25 180)(26 181)(27 182)(28 183)(29 184)(30 185)(31 186)(32 187)(33 188)(34 189)(35 190)(36 191)(37 192)(38 193)(39 194)(40 195)(41 226)(42 227)(43 228)(44 229)(45 230)(46 231)(47 232)(48 233)(49 234)(50 235)(51 236)(52 237)(53 238)(54 239)(55 240)(56 201)(57 202)(58 203)(59 204)(60 205)(61 206)(62 207)(63 208)(64 209)(65 210)(66 211)(67 212)(68 213)(69 214)(70 215)(71 216)(72 217)(73 218)(74 219)(75 220)(76 221)(77 222)(78 223)(79 224)(80 225)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(241 320)(242 281)(243 282)(244 283)(245 284)(246 285)(247 286)(248 287)(249 288)(250 289)(251 290)(252 291)(253 292)(254 293)(255 294)(256 295)(257 296)(258 297)(259 298)(260 299)(261 300)(262 301)(263 302)(264 303)(265 304)(266 305)(267 306)(268 307)(269 308)(270 309)(271 310)(272 311)(273 312)(274 313)(275 314)(276 315)(277 316)(278 317)(279 318)(280 319)
(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 50 21 70)(2 49 22 69)(3 48 23 68)(4 47 24 67)(5 46 25 66)(6 45 26 65)(7 44 27 64)(8 43 28 63)(9 42 29 62)(10 41 30 61)(11 80 31 60)(12 79 32 59)(13 78 33 58)(14 77 34 57)(15 76 35 56)(16 75 36 55)(17 74 37 54)(18 73 38 53)(19 72 39 52)(20 71 40 51)(81 274 101 254)(82 273 102 253)(83 272 103 252)(84 271 104 251)(85 270 105 250)(86 269 106 249)(87 268 107 248)(88 267 108 247)(89 266 109 246)(90 265 110 245)(91 264 111 244)(92 263 112 243)(93 262 113 242)(94 261 114 241)(95 260 115 280)(96 259 116 279)(97 258 117 278)(98 257 118 277)(99 256 119 276)(100 255 120 275)(121 305 141 285)(122 304 142 284)(123 303 143 283)(124 302 144 282)(125 301 145 281)(126 300 146 320)(127 299 147 319)(128 298 148 318)(129 297 149 317)(130 296 150 316)(131 295 151 315)(132 294 152 314)(133 293 153 313)(134 292 154 312)(135 291 155 311)(136 290 156 310)(137 289 157 309)(138 288 158 308)(139 287 159 307)(140 286 160 306)(161 230 181 210)(162 229 182 209)(163 228 183 208)(164 227 184 207)(165 226 185 206)(166 225 186 205)(167 224 187 204)(168 223 188 203)(169 222 189 202)(170 221 190 201)(171 220 191 240)(172 219 192 239)(173 218 193 238)(174 217 194 237)(175 216 195 236)(176 215 196 235)(177 214 197 234)(178 213 198 233)(179 212 199 232)(180 211 200 231)

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

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

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

92 conjugacy classes

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

92 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + + - image C1 C2 C2 C2 D4 D4 D5 Q16 D10 D10 D20 D20 Dic20 kernel C22×Dic20 C2×Dic20 C22×C40 C22×Dic10 C2×C20 C22×C10 C22×C8 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 12 1 2 3 1 2 8 12 2 12 4 32

Matrix representation of C22×Dic20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 33 0 0 0 0 0 0 5 0 0 0 0 0 0 33 0 0 0 0 0 0 5 0 0 0 0 0 0 15 38 0 0 0 0 11 17
,
 0 32 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 32 0 0 0 0 0 0 0 21 28 0 0 0 0 34 20

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[33,0,0,0,0,0,0,5,0,0,0,0,0,0,33,0,0,0,0,0,0,5,0,0,0,0,0,0,15,11,0,0,0,0,38,17],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,0,0,21,34,0,0,0,0,28,20] >;

C22×Dic20 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{20}
% in TeX

G:=Group("C2^2xDic20");
// GroupNames label

G:=SmallGroup(320,1414);
// by ID

G=gap.SmallGroup(320,1414);
# by ID

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

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

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