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