direct product, abelian, monomial, 2-elementary
Aliases: C23×C40, SmallGroup(320,1567)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C40 |
| C1 — C23×C40 |
| C1 — C23×C40 |
Subgroups: 338, all normal (12 characteristic)
C1, C2, C2 [×14], C4, C4 [×7], C22 [×35], C5, C8 [×8], C2×C4 [×28], C23 [×15], C10, C10 [×14], C2×C8 [×28], C22×C4 [×14], C24, C20, C20 [×7], C2×C10 [×35], C22×C8 [×14], C23×C4, C40 [×8], C2×C20 [×28], C22×C10 [×15], C23×C8, C2×C40 [×28], C22×C20 [×14], C23×C10, C22×C40 [×14], C23×C20, C23×C40
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C8 [×8], C2×C4 [×28], C23 [×15], C10 [×15], C2×C8 [×28], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C22×C8 [×14], C23×C4, C40 [×8], C2×C20 [×28], C22×C10 [×15], C23×C8, C2×C40 [×28], C22×C20 [×14], C23×C10, C22×C40 [×14], C23×C20, C23×C40
Generators and relations
G = < a,b,c,d | a2=b2=c2=d40=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
►Regular action on 320 points
Generators in S320
(1 154)(2 155)(3 156)(4 157)(5 158)(6 159)(7 160)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 129)(17 130)(18 131)(19 132)(20 133)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(41 282)(42 283)(43 284)(44 285)(45 286)(46 287)(47 288)(48 289)(49 290)(50 291)(51 292)(52 293)(53 294)(54 295)(55 296)(56 297)(57 298)(58 299)(59 300)(60 301)(61 302)(62 303)(63 304)(64 305)(65 306)(66 307)(67 308)(68 309)(69 310)(70 311)(71 312)(72 313)(73 314)(74 315)(75 316)(76 317)(77 318)(78 319)(79 320)(80 281)(81 219)(82 220)(83 221)(84 222)(85 223)(86 224)(87 225)(88 226)(89 227)(90 228)(91 229)(92 230)(93 231)(94 232)(95 233)(96 234)(97 235)(98 236)(99 237)(100 238)(101 239)(102 240)(103 201)(104 202)(105 203)(106 204)(107 205)(108 206)(109 207)(110 208)(111 209)(112 210)(113 211)(114 212)(115 213)(116 214)(117 215)(118 216)(119 217)(120 218)(161 248)(162 249)(163 250)(164 251)(165 252)(166 253)(167 254)(168 255)(169 256)(170 257)(171 258)(172 259)(173 260)(174 261)(175 262)(176 263)(177 264)(178 265)(179 266)(180 267)(181 268)(182 269)(183 270)(184 271)(185 272)(186 273)(187 274)(188 275)(189 276)(190 277)(191 278)(192 279)(193 280)(194 241)(195 242)(196 243)(197 244)(198 245)(199 246)(200 247)
(1 234)(2 235)(3 236)(4 237)(5 238)(6 239)(7 240)(8 201)(9 202)(10 203)(11 204)(12 205)(13 206)(14 207)(15 208)(16 209)(17 210)(18 211)(19 212)(20 213)(21 214)(22 215)(23 216)(24 217)(25 218)(26 219)(27 220)(28 221)(29 222)(30 223)(31 224)(32 225)(33 226)(34 227)(35 228)(36 229)(37 230)(38 231)(39 232)(40 233)(41 253)(42 254)(43 255)(44 256)(45 257)(46 258)(47 259)(48 260)(49 261)(50 262)(51 263)(52 264)(53 265)(54 266)(55 267)(56 268)(57 269)(58 270)(59 271)(60 272)(61 273)(62 274)(63 275)(64 276)(65 277)(66 278)(67 279)(68 280)(69 241)(70 242)(71 243)(72 244)(73 245)(74 246)(75 247)(76 248)(77 249)(78 250)(79 251)(80 252)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(161 317)(162 318)(163 319)(164 320)(165 281)(166 282)(167 283)(168 284)(169 285)(170 286)(171 287)(172 288)(173 289)(174 290)(175 291)(176 292)(177 293)(178 294)(179 295)(180 296)(181 297)(182 298)(183 299)(184 300)(185 301)(186 302)(187 303)(188 304)(189 305)(190 306)(191 307)(192 308)(193 309)(194 310)(195 311)(196 312)(197 313)(198 314)(199 315)(200 316)
(1 293)(2 294)(3 295)(4 296)(5 297)(6 298)(7 299)(8 300)(9 301)(10 302)(11 303)(12 304)(13 305)(14 306)(15 307)(16 308)(17 309)(18 310)(19 311)(20 312)(21 313)(22 314)(23 315)(24 316)(25 317)(26 318)(27 319)(28 320)(29 281)(30 282)(31 283)(32 284)(33 285)(34 286)(35 287)(36 288)(37 289)(38 290)(39 291)(40 292)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)(81 249)(82 250)(83 251)(84 252)(85 253)(86 254)(87 255)(88 256)(89 257)(90 258)(91 259)(92 260)(93 261)(94 262)(95 263)(96 264)(97 265)(98 266)(99 267)(100 268)(101 269)(102 270)(103 271)(104 272)(105 273)(106 274)(107 275)(108 276)(109 277)(110 278)(111 279)(112 280)(113 241)(114 242)(115 243)(116 244)(117 245)(118 246)(119 247)(120 248)(161 218)(162 219)(163 220)(164 221)(165 222)(166 223)(167 224)(168 225)(169 226)(170 227)(171 228)(172 229)(173 230)(174 231)(175 232)(176 233)(177 234)(178 235)(179 236)(180 237)(181 238)(182 239)(183 240)(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 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)| (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,282)(42,283)(43,284)(44,285)(45,286)(46,287)(47,288)(48,289)(49,290)(50,291)(51,292)(52,293)(53,294)(54,295)(55,296)(56,297)(57,298)(58,299)(59,300)(60,301)(61,302)(62,303)(63,304)(64,305)(65,306)(66,307)(67,308)(68,309)(69,310)(70,311)(71,312)(72,313)(73,314)(74,315)(75,316)(76,317)(77,318)(78,319)(79,320)(80,281)(81,219)(82,220)(83,221)(84,222)(85,223)(86,224)(87,225)(88,226)(89,227)(90,228)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,201)(104,202)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)(113,211)(114,212)(115,213)(116,214)(117,215)(118,216)(119,217)(120,218)(161,248)(162,249)(163,250)(164,251)(165,252)(166,253)(167,254)(168,255)(169,256)(170,257)(171,258)(172,259)(173,260)(174,261)(175,262)(176,263)(177,264)(178,265)(179,266)(180,267)(181,268)(182,269)(183,270)(184,271)(185,272)(186,273)(187,274)(188,275)(189,276)(190,277)(191,278)(192,279)(193,280)(194,241)(195,242)(196,243)(197,244)(198,245)(199,246)(200,247), (1,234)(2,235)(3,236)(4,237)(5,238)(6,239)(7,240)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,222)(30,223)(31,224)(32,225)(33,226)(34,227)(35,228)(36,229)(37,230)(38,231)(39,232)(40,233)(41,253)(42,254)(43,255)(44,256)(45,257)(46,258)(47,259)(48,260)(49,261)(50,262)(51,263)(52,264)(53,265)(54,266)(55,267)(56,268)(57,269)(58,270)(59,271)(60,272)(61,273)(62,274)(63,275)(64,276)(65,277)(66,278)(67,279)(68,280)(69,241)(70,242)(71,243)(72,244)(73,245)(74,246)(75,247)(76,248)(77,249)(78,250)(79,251)(80,252)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(161,317)(162,318)(163,319)(164,320)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,309)(194,310)(195,311)(196,312)(197,313)(198,314)(199,315)(200,316), (1,293)(2,294)(3,295)(4,296)(5,297)(6,298)(7,299)(8,300)(9,301)(10,302)(11,303)(12,304)(13,305)(14,306)(15,307)(16,308)(17,309)(18,310)(19,311)(20,312)(21,313)(22,314)(23,315)(24,316)(25,317)(26,318)(27,319)(28,320)(29,281)(30,282)(31,283)(32,284)(33,285)(34,286)(35,287)(36,288)(37,289)(38,290)(39,291)(40,292)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,249)(82,250)(83,251)(84,252)(85,253)(86,254)(87,255)(88,256)(89,257)(90,258)(91,259)(92,260)(93,261)(94,262)(95,263)(96,264)(97,265)(98,266)(99,267)(100,268)(101,269)(102,270)(103,271)(104,272)(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)(112,280)(113,241)(114,242)(115,243)(116,244)(117,245)(118,246)(119,247)(120,248)(161,218)(162,219)(163,220)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)(171,228)(172,229)(173,230)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237)(181,238)(182,239)(183,240)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(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( (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,282)(42,283)(43,284)(44,285)(45,286)(46,287)(47,288)(48,289)(49,290)(50,291)(51,292)(52,293)(53,294)(54,295)(55,296)(56,297)(57,298)(58,299)(59,300)(60,301)(61,302)(62,303)(63,304)(64,305)(65,306)(66,307)(67,308)(68,309)(69,310)(70,311)(71,312)(72,313)(73,314)(74,315)(75,316)(76,317)(77,318)(78,319)(79,320)(80,281)(81,219)(82,220)(83,221)(84,222)(85,223)(86,224)(87,225)(88,226)(89,227)(90,228)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,201)(104,202)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)(113,211)(114,212)(115,213)(116,214)(117,215)(118,216)(119,217)(120,218)(161,248)(162,249)(163,250)(164,251)(165,252)(166,253)(167,254)(168,255)(169,256)(170,257)(171,258)(172,259)(173,260)(174,261)(175,262)(176,263)(177,264)(178,265)(179,266)(180,267)(181,268)(182,269)(183,270)(184,271)(185,272)(186,273)(187,274)(188,275)(189,276)(190,277)(191,278)(192,279)(193,280)(194,241)(195,242)(196,243)(197,244)(198,245)(199,246)(200,247), (1,234)(2,235)(3,236)(4,237)(5,238)(6,239)(7,240)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,222)(30,223)(31,224)(32,225)(33,226)(34,227)(35,228)(36,229)(37,230)(38,231)(39,232)(40,233)(41,253)(42,254)(43,255)(44,256)(45,257)(46,258)(47,259)(48,260)(49,261)(50,262)(51,263)(52,264)(53,265)(54,266)(55,267)(56,268)(57,269)(58,270)(59,271)(60,272)(61,273)(62,274)(63,275)(64,276)(65,277)(66,278)(67,279)(68,280)(69,241)(70,242)(71,243)(72,244)(73,245)(74,246)(75,247)(76,248)(77,249)(78,250)(79,251)(80,252)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(161,317)(162,318)(163,319)(164,320)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,309)(194,310)(195,311)(196,312)(197,313)(198,314)(199,315)(200,316), (1,293)(2,294)(3,295)(4,296)(5,297)(6,298)(7,299)(8,300)(9,301)(10,302)(11,303)(12,304)(13,305)(14,306)(15,307)(16,308)(17,309)(18,310)(19,311)(20,312)(21,313)(22,314)(23,315)(24,316)(25,317)(26,318)(27,319)(28,320)(29,281)(30,282)(31,283)(32,284)(33,285)(34,286)(35,287)(36,288)(37,289)(38,290)(39,291)(40,292)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,249)(82,250)(83,251)(84,252)(85,253)(86,254)(87,255)(88,256)(89,257)(90,258)(91,259)(92,260)(93,261)(94,262)(95,263)(96,264)(97,265)(98,266)(99,267)(100,268)(101,269)(102,270)(103,271)(104,272)(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)(112,280)(113,241)(114,242)(115,243)(116,244)(117,245)(118,246)(119,247)(120,248)(161,218)(162,219)(163,220)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)(171,228)(172,229)(173,230)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237)(181,238)(182,239)(183,240)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(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([(1,154),(2,155),(3,156),(4,157),(5,158),(6,159),(7,160),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,129),(17,130),(18,131),(19,132),(20,133),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,148),(36,149),(37,150),(38,151),(39,152),(40,153),(41,282),(42,283),(43,284),(44,285),(45,286),(46,287),(47,288),(48,289),(49,290),(50,291),(51,292),(52,293),(53,294),(54,295),(55,296),(56,297),(57,298),(58,299),(59,300),(60,301),(61,302),(62,303),(63,304),(64,305),(65,306),(66,307),(67,308),(68,309),(69,310),(70,311),(71,312),(72,313),(73,314),(74,315),(75,316),(76,317),(77,318),(78,319),(79,320),(80,281),(81,219),(82,220),(83,221),(84,222),(85,223),(86,224),(87,225),(88,226),(89,227),(90,228),(91,229),(92,230),(93,231),(94,232),(95,233),(96,234),(97,235),(98,236),(99,237),(100,238),(101,239),(102,240),(103,201),(104,202),(105,203),(106,204),(107,205),(108,206),(109,207),(110,208),(111,209),(112,210),(113,211),(114,212),(115,213),(116,214),(117,215),(118,216),(119,217),(120,218),(161,248),(162,249),(163,250),(164,251),(165,252),(166,253),(167,254),(168,255),(169,256),(170,257),(171,258),(172,259),(173,260),(174,261),(175,262),(176,263),(177,264),(178,265),(179,266),(180,267),(181,268),(182,269),(183,270),(184,271),(185,272),(186,273),(187,274),(188,275),(189,276),(190,277),(191,278),(192,279),(193,280),(194,241),(195,242),(196,243),(197,244),(198,245),(199,246),(200,247)], [(1,234),(2,235),(3,236),(4,237),(5,238),(6,239),(7,240),(8,201),(9,202),(10,203),(11,204),(12,205),(13,206),(14,207),(15,208),(16,209),(17,210),(18,211),(19,212),(20,213),(21,214),(22,215),(23,216),(24,217),(25,218),(26,219),(27,220),(28,221),(29,222),(30,223),(31,224),(32,225),(33,226),(34,227),(35,228),(36,229),(37,230),(38,231),(39,232),(40,233),(41,253),(42,254),(43,255),(44,256),(45,257),(46,258),(47,259),(48,260),(49,261),(50,262),(51,263),(52,264),(53,265),(54,266),(55,267),(56,268),(57,269),(58,270),(59,271),(60,272),(61,273),(62,274),(63,275),(64,276),(65,277),(66,278),(67,279),(68,280),(69,241),(70,242),(71,243),(72,244),(73,245),(74,246),(75,247),(76,248),(77,249),(78,250),(79,251),(80,252),(81,139),(82,140),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(161,317),(162,318),(163,319),(164,320),(165,281),(166,282),(167,283),(168,284),(169,285),(170,286),(171,287),(172,288),(173,289),(174,290),(175,291),(176,292),(177,293),(178,294),(179,295),(180,296),(181,297),(182,298),(183,299),(184,300),(185,301),(186,302),(187,303),(188,304),(189,305),(190,306),(191,307),(192,308),(193,309),(194,310),(195,311),(196,312),(197,313),(198,314),(199,315),(200,316)], [(1,293),(2,294),(3,295),(4,296),(5,297),(6,298),(7,299),(8,300),(9,301),(10,302),(11,303),(12,304),(13,305),(14,306),(15,307),(16,308),(17,309),(18,310),(19,311),(20,312),(21,313),(22,314),(23,315),(24,316),(25,317),(26,318),(27,319),(28,320),(29,281),(30,282),(31,283),(32,284),(33,285),(34,286),(35,287),(36,288),(37,289),(38,290),(39,291),(40,292),(41,143),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,159),(58,160),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142),(81,249),(82,250),(83,251),(84,252),(85,253),(86,254),(87,255),(88,256),(89,257),(90,258),(91,259),(92,260),(93,261),(94,262),(95,263),(96,264),(97,265),(98,266),(99,267),(100,268),(101,269),(102,270),(103,271),(104,272),(105,273),(106,274),(107,275),(108,276),(109,277),(110,278),(111,279),(112,280),(113,241),(114,242),(115,243),(116,244),(117,245),(118,246),(119,247),(120,248),(161,218),(162,219),(163,220),(164,221),(165,222),(166,223),(167,224),(168,225),(169,226),(170,227),(171,228),(172,229),(173,230),(174,231),(175,232),(176,233),(177,234),(178,235),(179,236),(180,237),(181,238),(182,239),(183,240),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 29 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,31,0,0,0,0,29] >;
320 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 5A | 5B | 5C | 5D | 8A | ··· | 8AF | 10A | ··· | 10BH | 20A | ··· | 20BL | 40A | ··· | 40DX |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 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 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 |
| kernel | C23×C40 | C22×C40 | C23×C20 | C22×C20 | C23×C10 | C23×C8 | C22×C10 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 14 | 1 | 14 | 2 | 4 | 32 | 56 | 4 | 56 | 8 | 128 |
In GAP, Magma, Sage, TeX
C_2^3\times C_{40} % in TeX
G:=Group("C2^3xC40"); // GroupNames label
G:=SmallGroup(320,1567);
// by ID
G=gap.SmallGroup(320,1567);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^40=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
𝔽
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