C2^2xDic20 - GroupNames

G = C₂²×Dic₂₀ order 320 = 2⁶·5

Direct product of C₂² and Dic₂₀

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

Aliases: C₂²×Dic₂₀, C₄₀.₆₁C₂³, C₂₀.₅₇C₂⁴, C₂³.₆₂D₂₀, Dic₁₀.₂₁C₂³, C₁₀⋊₁(C₂×Q₁₆), (C₂×C₁₀)⋊₆Q₁₆, C₅⋊₁(C₂²×Q₁₆), C₄.₄₇(C₂×D₂₀), C₂₀.₂₉₂(C₂×D₄), (C₂×C₂₀).₃₉₂D₄, (C₂×C₈).₃₁₀D₁₀, (C₂×C₄).₁₀₂D₂₀, (C₂²×C₈).₁₀D₅, C₄.₅₄(C₂³×D₅), C₈.₅₂(C₂²×D₅), (C₂²×C₄₀).₁₆C₂, C₂².₇₂(C₂×D₂₀), C₂.₂₆(C₂²×D₂₀), C₁₀.₂₄(C₂²×D₄), (C₂×C₄₀).₃₈₂C₂², (C₂×C₂₀).₇₈₈C₂³, (C₂²×C₄).₄₄₅D₁₀, (C₂²×C₁₀).₁₄₇D₄, (C₂²×C₂₀).₅₂₇C₂², (C₂²×Dic₁₀).₁₀C₂, (C₂×Dic₁₀).₂₆₆C₂², (C₂×C₁₀).₁₈₀(C₂×D₄), (C₂×C₄).₇₃₈(C₂²×D₅), SmallGroup(320,1414)

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, C₄, C₄, C₄, C₂², C₅, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₀, C₁₀, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₅, C₂₀, C₂₀, C₂×C₁₀, C₂²×C₈, C₂×Q₁₆, C₂²×Q₈, C₄₀, Dic₁₀, Dic₁₀, C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, C₂²×Q₁₆, Dic₂₀, C₂×C₄₀, C₂×Dic₁₀, C₂×Dic₁₀, C₂²×Dic₅, C₂²×C₂₀, C₂×Dic₂₀, C₂²×C₄₀, C₂²×Dic₁₀, C₂²×Dic₂₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, Q₁₆, C₂×D₄, C₂⁴, D₁₀, C₂×Q₁₆, C₂²×D₄, D₂₀, C₂²×D₅, C₂²×Q₁₆, Dic₂₀, C₂×D₂₀, C₂³×D₅, C₂×Dic₂₀, C₂²×D₂₀, C₂²×Dic₂₀

Smallest permutation representation of C₂²×Dic₂₀
►Regular action on 320 points

Generators in S₃₂₀

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

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

(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)

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

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

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

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

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

G = C22×Dic20 order 320 = 26·5

Direct product of C22 and Dic20

G = C₂²×Dic₂₀ order 320 = 2⁶·5

Direct product of C₂² and Dic₂₀