direct product, abelian, monomial, 2-elementary
Aliases: C22×C4×C20, SmallGroup(320,1513)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C4×C20 |
| C1 — C22×C4×C20 |
| C1 — C22×C4×C20 |
Generators and relations for C22×C4×C20
G = < a,b,c,d | a2=b2=c4=d20=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 498, all normal (8 characteristic)
C1, C2, C4, C22, C22, C5, C2×C4, C23, C10, C42, C22×C4, C24, C20, C2×C10, C2×C10, C2×C42, C23×C4, C2×C20, C22×C10, C22×C42, C4×C20, C22×C20, C23×C10, C2×C4×C20, C23×C20, C22×C4×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, C22×C4, C24, C20, C2×C10, C2×C42, C23×C4, C2×C20, C22×C10, C22×C42, C4×C20, C22×C20, C23×C10, C2×C4×C20, C23×C20, C22×C4×C20
►Regular action on 320 points
Generators in S320
(1 299)(2 300)(3 281)(4 282)(5 283)(6 284)(7 285)(8 286)(9 287)(10 288)(11 289)(12 290)(13 291)(14 292)(15 293)(16 294)(17 295)(18 296)(19 297)(20 298)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 121)(51 122)(52 123)(53 124)(54 125)(55 126)(56 127)(57 128)(58 129)(59 130)(60 131)(61 231)(62 232)(63 233)(64 234)(65 235)(66 236)(67 237)(68 238)(69 239)(70 240)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 227)(78 228)(79 229)(80 230)(81 186)(82 187)(83 188)(84 189)(85 190)(86 191)(87 192)(88 193)(89 194)(90 195)(91 196)(92 197)(93 198)(94 199)(95 200)(96 181)(97 182)(98 183)(99 184)(100 185)(141 244)(142 245)(143 246)(144 247)(145 248)(146 249)(147 250)(148 251)(149 252)(150 253)(151 254)(152 255)(153 256)(154 257)(155 258)(156 259)(157 260)(158 241)(159 242)(160 243)(161 267)(162 268)(163 269)(164 270)(165 271)(166 272)(167 273)(168 274)(169 275)(170 276)(171 277)(172 278)(173 279)(174 280)(175 261)(176 262)(177 263)(178 264)(179 265)(180 266)(201 310)(202 311)(203 312)(204 313)(205 314)(206 315)(207 316)(208 317)(209 318)(210 319)(211 320)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)
(1 153)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 267)(42 268)(43 269)(44 270)(45 271)(46 272)(47 273)(48 274)(49 275)(50 276)(51 277)(52 278)(53 279)(54 280)(55 261)(56 262)(57 263)(58 264)(59 265)(60 266)(81 304)(82 305)(83 306)(84 307)(85 308)(86 309)(87 310)(88 311)(89 312)(90 313)(91 314)(92 315)(93 316)(94 317)(95 318)(96 319)(97 320)(98 301)(99 302)(100 303)(101 234)(102 235)(103 236)(104 237)(105 238)(106 239)(107 240)(108 221)(109 222)(110 223)(111 224)(112 225)(113 226)(114 227)(115 228)(116 229)(117 230)(118 231)(119 232)(120 233)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(181 210)(182 211)(183 212)(184 213)(185 214)(186 215)(187 216)(188 217)(189 218)(190 219)(191 220)(192 201)(193 202)(194 203)(195 204)(196 205)(197 206)(198 207)(199 208)(200 209)(241 284)(242 285)(243 286)(244 287)(245 288)(246 289)(247 290)(248 291)(249 292)(250 293)(251 294)(252 295)(253 296)(254 297)(255 298)(256 299)(257 300)(258 281)(259 282)(260 283)
(1 313 135 67)(2 314 136 68)(3 315 137 69)(4 316 138 70)(5 317 139 71)(6 318 140 72)(7 319 121 73)(8 320 122 74)(9 301 123 75)(10 302 124 76)(11 303 125 77)(12 304 126 78)(13 305 127 79)(14 306 128 80)(15 307 129 61)(16 308 130 62)(17 309 131 63)(18 310 132 64)(19 311 133 65)(20 312 134 66)(21 141 98 172)(22 142 99 173)(23 143 100 174)(24 144 81 175)(25 145 82 176)(26 146 83 177)(27 147 84 178)(28 148 85 179)(29 149 86 180)(30 150 87 161)(31 151 88 162)(32 152 89 163)(33 153 90 164)(34 154 91 165)(35 155 92 166)(36 156 93 167)(37 157 94 168)(38 158 95 169)(39 159 96 170)(40 160 97 171)(41 234 296 201)(42 235 297 202)(43 236 298 203)(44 237 299 204)(45 238 300 205)(46 239 281 206)(47 240 282 207)(48 221 283 208)(49 222 284 209)(50 223 285 210)(51 224 286 211)(52 225 287 212)(53 226 288 213)(54 227 289 214)(55 228 290 215)(56 229 291 216)(57 230 292 217)(58 231 293 218)(59 232 294 219)(60 233 295 220)(101 253 192 267)(102 254 193 268)(103 255 194 269)(104 256 195 270)(105 257 196 271)(106 258 197 272)(107 259 198 273)(108 260 199 274)(109 241 200 275)(110 242 181 276)(111 243 182 277)(112 244 183 278)(113 245 184 279)(114 246 185 280)(115 247 186 261)(116 248 187 262)(117 249 188 263)(118 250 189 264)(119 251 190 265)(120 252 191 266)
(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)
G:=sub<Sym(320)| (1,299)(2,300)(3,281)(4,282)(5,283)(6,284)(7,285)(8,286)(9,287)(10,288)(11,289)(12,290)(13,291)(14,292)(15,293)(16,294)(17,295)(18,296)(19,297)(20,298)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,131)(61,231)(62,232)(63,233)(64,234)(65,235)(66,236)(67,237)(68,238)(69,239)(70,240)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,196)(92,197)(93,198)(94,199)(95,200)(96,181)(97,182)(98,183)(99,184)(100,185)(141,244)(142,245)(143,246)(144,247)(145,248)(146,249)(147,250)(148,251)(149,252)(150,253)(151,254)(152,255)(153,256)(154,257)(155,258)(156,259)(157,260)(158,241)(159,242)(160,243)(161,267)(162,268)(163,269)(164,270)(165,271)(166,272)(167,273)(168,274)(169,275)(170,276)(171,277)(172,278)(173,279)(174,280)(175,261)(176,262)(177,263)(178,264)(179,265)(180,266)(201,310)(202,311)(203,312)(204,313)(205,314)(206,315)(207,316)(208,317)(209,318)(210,319)(211,320)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309), (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,267)(42,268)(43,269)(44,270)(45,271)(46,272)(47,273)(48,274)(49,275)(50,276)(51,277)(52,278)(53,279)(54,280)(55,261)(56,262)(57,263)(58,264)(59,265)(60,266)(81,304)(82,305)(83,306)(84,307)(85,308)(86,309)(87,310)(88,311)(89,312)(90,313)(91,314)(92,315)(93,316)(94,317)(95,318)(96,319)(97,320)(98,301)(99,302)(100,303)(101,234)(102,235)(103,236)(104,237)(105,238)(106,239)(107,240)(108,221)(109,222)(110,223)(111,224)(112,225)(113,226)(114,227)(115,228)(116,229)(117,230)(118,231)(119,232)(120,233)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(181,210)(182,211)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)(199,208)(200,209)(241,284)(242,285)(243,286)(244,287)(245,288)(246,289)(247,290)(248,291)(249,292)(250,293)(251,294)(252,295)(253,296)(254,297)(255,298)(256,299)(257,300)(258,281)(259,282)(260,283), (1,313,135,67)(2,314,136,68)(3,315,137,69)(4,316,138,70)(5,317,139,71)(6,318,140,72)(7,319,121,73)(8,320,122,74)(9,301,123,75)(10,302,124,76)(11,303,125,77)(12,304,126,78)(13,305,127,79)(14,306,128,80)(15,307,129,61)(16,308,130,62)(17,309,131,63)(18,310,132,64)(19,311,133,65)(20,312,134,66)(21,141,98,172)(22,142,99,173)(23,143,100,174)(24,144,81,175)(25,145,82,176)(26,146,83,177)(27,147,84,178)(28,148,85,179)(29,149,86,180)(30,150,87,161)(31,151,88,162)(32,152,89,163)(33,153,90,164)(34,154,91,165)(35,155,92,166)(36,156,93,167)(37,157,94,168)(38,158,95,169)(39,159,96,170)(40,160,97,171)(41,234,296,201)(42,235,297,202)(43,236,298,203)(44,237,299,204)(45,238,300,205)(46,239,281,206)(47,240,282,207)(48,221,283,208)(49,222,284,209)(50,223,285,210)(51,224,286,211)(52,225,287,212)(53,226,288,213)(54,227,289,214)(55,228,290,215)(56,229,291,216)(57,230,292,217)(58,231,293,218)(59,232,294,219)(60,233,295,220)(101,253,192,267)(102,254,193,268)(103,255,194,269)(104,256,195,270)(105,257,196,271)(106,258,197,272)(107,259,198,273)(108,260,199,274)(109,241,200,275)(110,242,181,276)(111,243,182,277)(112,244,183,278)(113,245,184,279)(114,246,185,280)(115,247,186,261)(116,248,187,262)(117,249,188,263)(118,250,189,264)(119,251,190,265)(120,252,191,266), (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)>;
G:=Group( (1,299)(2,300)(3,281)(4,282)(5,283)(6,284)(7,285)(8,286)(9,287)(10,288)(11,289)(12,290)(13,291)(14,292)(15,293)(16,294)(17,295)(18,296)(19,297)(20,298)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,131)(61,231)(62,232)(63,233)(64,234)(65,235)(66,236)(67,237)(68,238)(69,239)(70,240)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,196)(92,197)(93,198)(94,199)(95,200)(96,181)(97,182)(98,183)(99,184)(100,185)(141,244)(142,245)(143,246)(144,247)(145,248)(146,249)(147,250)(148,251)(149,252)(150,253)(151,254)(152,255)(153,256)(154,257)(155,258)(156,259)(157,260)(158,241)(159,242)(160,243)(161,267)(162,268)(163,269)(164,270)(165,271)(166,272)(167,273)(168,274)(169,275)(170,276)(171,277)(172,278)(173,279)(174,280)(175,261)(176,262)(177,263)(178,264)(179,265)(180,266)(201,310)(202,311)(203,312)(204,313)(205,314)(206,315)(207,316)(208,317)(209,318)(210,319)(211,320)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309), (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,267)(42,268)(43,269)(44,270)(45,271)(46,272)(47,273)(48,274)(49,275)(50,276)(51,277)(52,278)(53,279)(54,280)(55,261)(56,262)(57,263)(58,264)(59,265)(60,266)(81,304)(82,305)(83,306)(84,307)(85,308)(86,309)(87,310)(88,311)(89,312)(90,313)(91,314)(92,315)(93,316)(94,317)(95,318)(96,319)(97,320)(98,301)(99,302)(100,303)(101,234)(102,235)(103,236)(104,237)(105,238)(106,239)(107,240)(108,221)(109,222)(110,223)(111,224)(112,225)(113,226)(114,227)(115,228)(116,229)(117,230)(118,231)(119,232)(120,233)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(181,210)(182,211)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)(199,208)(200,209)(241,284)(242,285)(243,286)(244,287)(245,288)(246,289)(247,290)(248,291)(249,292)(250,293)(251,294)(252,295)(253,296)(254,297)(255,298)(256,299)(257,300)(258,281)(259,282)(260,283), (1,313,135,67)(2,314,136,68)(3,315,137,69)(4,316,138,70)(5,317,139,71)(6,318,140,72)(7,319,121,73)(8,320,122,74)(9,301,123,75)(10,302,124,76)(11,303,125,77)(12,304,126,78)(13,305,127,79)(14,306,128,80)(15,307,129,61)(16,308,130,62)(17,309,131,63)(18,310,132,64)(19,311,133,65)(20,312,134,66)(21,141,98,172)(22,142,99,173)(23,143,100,174)(24,144,81,175)(25,145,82,176)(26,146,83,177)(27,147,84,178)(28,148,85,179)(29,149,86,180)(30,150,87,161)(31,151,88,162)(32,152,89,163)(33,153,90,164)(34,154,91,165)(35,155,92,166)(36,156,93,167)(37,157,94,168)(38,158,95,169)(39,159,96,170)(40,160,97,171)(41,234,296,201)(42,235,297,202)(43,236,298,203)(44,237,299,204)(45,238,300,205)(46,239,281,206)(47,240,282,207)(48,221,283,208)(49,222,284,209)(50,223,285,210)(51,224,286,211)(52,225,287,212)(53,226,288,213)(54,227,289,214)(55,228,290,215)(56,229,291,216)(57,230,292,217)(58,231,293,218)(59,232,294,219)(60,233,295,220)(101,253,192,267)(102,254,193,268)(103,255,194,269)(104,256,195,270)(105,257,196,271)(106,258,197,272)(107,259,198,273)(108,260,199,274)(109,241,200,275)(110,242,181,276)(111,243,182,277)(112,244,183,278)(113,245,184,279)(114,246,185,280)(115,247,186,261)(116,248,187,262)(117,249,188,263)(118,250,189,264)(119,251,190,265)(120,252,191,266), (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) );
G=PermutationGroup([[(1,299),(2,300),(3,281),(4,282),(5,283),(6,284),(7,285),(8,286),(9,287),(10,288),(11,289),(12,290),(13,291),(14,292),(15,293),(16,294),(17,295),(18,296),(19,297),(20,298),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,132),(42,133),(43,134),(44,135),(45,136),(46,137),(47,138),(48,139),(49,140),(50,121),(51,122),(52,123),(53,124),(54,125),(55,126),(56,127),(57,128),(58,129),(59,130),(60,131),(61,231),(62,232),(63,233),(64,234),(65,235),(66,236),(67,237),(68,238),(69,239),(70,240),(71,221),(72,222),(73,223),(74,224),(75,225),(76,226),(77,227),(78,228),(79,229),(80,230),(81,186),(82,187),(83,188),(84,189),(85,190),(86,191),(87,192),(88,193),(89,194),(90,195),(91,196),(92,197),(93,198),(94,199),(95,200),(96,181),(97,182),(98,183),(99,184),(100,185),(141,244),(142,245),(143,246),(144,247),(145,248),(146,249),(147,250),(148,251),(149,252),(150,253),(151,254),(152,255),(153,256),(154,257),(155,258),(156,259),(157,260),(158,241),(159,242),(160,243),(161,267),(162,268),(163,269),(164,270),(165,271),(166,272),(167,273),(168,274),(169,275),(170,276),(171,277),(172,278),(173,279),(174,280),(175,261),(176,262),(177,263),(178,264),(179,265),(180,266),(201,310),(202,311),(203,312),(204,313),(205,314),(206,315),(207,316),(208,317),(209,318),(210,319),(211,320),(212,301),(213,302),(214,303),(215,304),(216,305),(217,306),(218,307),(219,308),(220,309)], [(1,153),(2,154),(3,155),(4,156),(5,157),(6,158),(7,159),(8,160),(9,141),(10,142),(11,143),(12,144),(13,145),(14,146),(15,147),(16,148),(17,149),(18,150),(19,151),(20,152),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,267),(42,268),(43,269),(44,270),(45,271),(46,272),(47,273),(48,274),(49,275),(50,276),(51,277),(52,278),(53,279),(54,280),(55,261),(56,262),(57,263),(58,264),(59,265),(60,266),(81,304),(82,305),(83,306),(84,307),(85,308),(86,309),(87,310),(88,311),(89,312),(90,313),(91,314),(92,315),(93,316),(94,317),(95,318),(96,319),(97,320),(98,301),(99,302),(100,303),(101,234),(102,235),(103,236),(104,237),(105,238),(106,239),(107,240),(108,221),(109,222),(110,223),(111,224),(112,225),(113,226),(114,227),(115,228),(116,229),(117,230),(118,231),(119,232),(120,233),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,177),(129,178),(130,179),(131,180),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(181,210),(182,211),(183,212),(184,213),(185,214),(186,215),(187,216),(188,217),(189,218),(190,219),(191,220),(192,201),(193,202),(194,203),(195,204),(196,205),(197,206),(198,207),(199,208),(200,209),(241,284),(242,285),(243,286),(244,287),(245,288),(246,289),(247,290),(248,291),(249,292),(250,293),(251,294),(252,295),(253,296),(254,297),(255,298),(256,299),(257,300),(258,281),(259,282),(260,283)], [(1,313,135,67),(2,314,136,68),(3,315,137,69),(4,316,138,70),(5,317,139,71),(6,318,140,72),(7,319,121,73),(8,320,122,74),(9,301,123,75),(10,302,124,76),(11,303,125,77),(12,304,126,78),(13,305,127,79),(14,306,128,80),(15,307,129,61),(16,308,130,62),(17,309,131,63),(18,310,132,64),(19,311,133,65),(20,312,134,66),(21,141,98,172),(22,142,99,173),(23,143,100,174),(24,144,81,175),(25,145,82,176),(26,146,83,177),(27,147,84,178),(28,148,85,179),(29,149,86,180),(30,150,87,161),(31,151,88,162),(32,152,89,163),(33,153,90,164),(34,154,91,165),(35,155,92,166),(36,156,93,167),(37,157,94,168),(38,158,95,169),(39,159,96,170),(40,160,97,171),(41,234,296,201),(42,235,297,202),(43,236,298,203),(44,237,299,204),(45,238,300,205),(46,239,281,206),(47,240,282,207),(48,221,283,208),(49,222,284,209),(50,223,285,210),(51,224,286,211),(52,225,287,212),(53,226,288,213),(54,227,289,214),(55,228,290,215),(56,229,291,216),(57,230,292,217),(58,231,293,218),(59,232,294,219),(60,233,295,220),(101,253,192,267),(102,254,193,268),(103,255,194,269),(104,256,195,270),(105,257,196,271),(106,258,197,272),(107,259,198,273),(108,260,199,274),(109,241,200,275),(110,242,181,276),(111,243,182,277),(112,244,183,278),(113,245,184,279),(114,246,185,280),(115,247,186,261),(116,248,187,262),(117,249,188,263),(118,250,189,264),(119,251,190,265),(120,252,191,266)], [(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)]])
320 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4AV | 5A | 5B | 5C | 5D | 10A | ··· | 10BH | 20A | ··· | 20GJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
320 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 |
| kernel | C22×C4×C20 | C2×C4×C20 | C23×C20 | C22×C20 | C22×C42 | C2×C42 | C23×C4 | C22×C4 |
| # reps | 1 | 12 | 3 | 48 | 4 | 48 | 12 | 192 |
Matrix representation of C22×C4×C20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 39 | 0 |
| 0 | 0 | 0 | 20 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,9,0,0,0,0,32,0,0,0,0,32],[39,0,0,0,0,8,0,0,0,0,39,0,0,0,0,20] >;
C22×C4×C20 in GAP, Magma, Sage, TeX
C_2^2\times C_4\times C_{20} % in TeX
G:=Group("C2^2xC4xC20"); // GroupNames label
G:=SmallGroup(320,1513);
// by ID
G=gap.SmallGroup(320,1513);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1128]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^20=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations