direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×Dic20, C20.57C24, C40.61C23, C23.62D20, Dic10.21C23, C10⋊1(C2×Q16), (C2×C10)⋊6Q16, C5⋊1(C22×Q16), C4.47(C2×D20), (C2×C8).310D10, (C2×C4).102D20, (C2×C20).392D4, C20.292(C2×D4), (C22×C8).10D5, C4.54(C23×D5), C8.52(C22×D5), (C22×C40).16C2, C22.72(C2×D20), C10.24(C22×D4), C2.26(C22×D20), (C2×C40).382C22, (C2×C20).788C23, (C22×C10).147D4, (C22×C4).445D10, (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
Subgroups: 862 in 258 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C5, C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×20], C23, C10, C10 [×6], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×18], Dic5 [×8], C20, C20 [×3], C2×C10 [×7], C22×C8, C2×Q16 [×12], C22×Q8 [×2], C40 [×4], Dic10 [×8], Dic10 [×12], C2×Dic5 [×12], C2×C20 [×6], C22×C10, C22×Q16, Dic20 [×16], C2×C40 [×6], C2×Dic10 [×12], C2×Dic10 [×6], C22×Dic5 [×2], C22×C20, C2×Dic20 [×12], C22×C40, C22×Dic10 [×2], C22×Dic20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, Q16 [×4], C2×D4 [×6], C24, D10 [×7], C2×Q16 [×6], C22×D4, D20 [×4], C22×D5 [×7], C22×Q16, Dic20 [×4], C2×D20 [×6], C23×D5, C2×Dic20 [×6], C22×D20, C22×Dic20
Generators and relations
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 >
►Regular action on 320 points
Generators in S320
(1 117)(2 118)(3 119)(4 120)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 197)(42 198)(43 199)(44 200)(45 161)(46 162)(47 163)(48 164)(49 165)(50 166)(51 167)(52 168)(53 169)(54 170)(55 171)(56 172)(57 173)(58 174)(59 175)(60 176)(61 177)(62 178)(63 179)(64 180)(65 181)(66 182)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 189)(74 190)(75 191)(76 192)(77 193)(78 194)(79 195)(80 196)(121 285)(122 286)(123 287)(124 288)(125 289)(126 290)(127 291)(128 292)(129 293)(130 294)(131 295)(132 296)(133 297)(134 298)(135 299)(136 300)(137 301)(138 302)(139 303)(140 304)(141 305)(142 306)(143 307)(144 308)(145 309)(146 310)(147 311)(148 312)(149 313)(150 314)(151 315)(152 316)(153 317)(154 318)(155 319)(156 320)(157 281)(158 282)(159 283)(160 284)(201 273)(202 274)(203 275)(204 276)(205 277)(206 278)(207 279)(208 280)(209 241)(210 242)(211 243)(212 244)(213 245)(214 246)(215 247)(216 248)(217 249)(218 250)(219 251)(220 252)(221 253)(222 254)(223 255)(224 256)(225 257)(226 258)(227 259)(228 260)(229 261)(230 262)(231 263)(232 264)(233 265)(234 266)(235 267)(236 268)(237 269)(238 270)(239 271)(240 272)
(1 209)(2 210)(3 211)(4 212)(5 213)(6 214)(7 215)(8 216)(9 217)(10 218)(11 219)(12 220)(13 221)(14 222)(15 223)(16 224)(17 225)(18 226)(19 227)(20 228)(21 229)(22 230)(23 231)(24 232)(25 233)(26 234)(27 235)(28 236)(29 237)(30 238)(31 239)(32 240)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 245)(82 246)(83 247)(84 248)(85 249)(86 250)(87 251)(88 252)(89 253)(90 254)(91 255)(92 256)(93 257)(94 258)(95 259)(96 260)(97 261)(98 262)(99 263)(100 264)(101 265)(102 266)(103 267)(104 268)(105 269)(106 270)(107 271)(108 272)(109 273)(110 274)(111 275)(112 276)(113 277)(114 278)(115 279)(116 280)(117 241)(118 242)(119 243)(120 244)(161 298)(162 299)(163 300)(164 301)(165 302)(166 303)(167 304)(168 305)(169 306)(170 307)(171 308)(172 309)(173 310)(174 311)(175 312)(176 313)(177 314)(178 315)(179 316)(180 317)(181 318)(182 319)(183 320)(184 281)(185 282)(186 283)(187 284)(188 285)(189 286)(190 287)(191 288)(192 289)(193 290)(194 291)(195 292)(196 293)(197 294)(198 295)(199 296)(200 297)
(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)
(1 163 21 183)(2 162 22 182)(3 161 23 181)(4 200 24 180)(5 199 25 179)(6 198 26 178)(7 197 27 177)(8 196 28 176)(9 195 29 175)(10 194 30 174)(11 193 31 173)(12 192 32 172)(13 191 33 171)(14 190 34 170)(15 189 35 169)(16 188 36 168)(17 187 37 167)(18 186 38 166)(19 185 39 165)(20 184 40 164)(41 103 61 83)(42 102 62 82)(43 101 63 81)(44 100 64 120)(45 99 65 119)(46 98 66 118)(47 97 67 117)(48 96 68 116)(49 95 69 115)(50 94 70 114)(51 93 71 113)(52 92 72 112)(53 91 73 111)(54 90 74 110)(55 89 75 109)(56 88 76 108)(57 87 77 107)(58 86 78 106)(59 85 79 105)(60 84 80 104)(121 276 141 256)(122 275 142 255)(123 274 143 254)(124 273 144 253)(125 272 145 252)(126 271 146 251)(127 270 147 250)(128 269 148 249)(129 268 149 248)(130 267 150 247)(131 266 151 246)(132 265 152 245)(133 264 153 244)(134 263 154 243)(135 262 155 242)(136 261 156 241)(137 260 157 280)(138 259 158 279)(139 258 159 278)(140 257 160 277)(201 308 221 288)(202 307 222 287)(203 306 223 286)(204 305 224 285)(205 304 225 284)(206 303 226 283)(207 302 227 282)(208 301 228 281)(209 300 229 320)(210 299 230 319)(211 298 231 318)(212 297 232 317)(213 296 233 316)(214 295 234 315)(215 294 235 314)(216 293 236 313)(217 292 237 312)(218 291 238 311)(219 290 239 310)(220 289 240 309)
G:=sub<Sym(320)| (1,117)(2,118)(3,119)(4,120)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,197)(42,198)(43,199)(44,200)(45,161)(46,162)(47,163)(48,164)(49,165)(50,166)(51,167)(52,168)(53,169)(54,170)(55,171)(56,172)(57,173)(58,174)(59,175)(60,176)(61,177)(62,178)(63,179)(64,180)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,195)(80,196)(121,285)(122,286)(123,287)(124,288)(125,289)(126,290)(127,291)(128,292)(129,293)(130,294)(131,295)(132,296)(133,297)(134,298)(135,299)(136,300)(137,301)(138,302)(139,303)(140,304)(141,305)(142,306)(143,307)(144,308)(145,309)(146,310)(147,311)(148,312)(149,313)(150,314)(151,315)(152,316)(153,317)(154,318)(155,319)(156,320)(157,281)(158,282)(159,283)(160,284)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)(217,249)(218,250)(219,251)(220,252)(221,253)(222,254)(223,255)(224,256)(225,257)(226,258)(227,259)(228,260)(229,261)(230,262)(231,263)(232,264)(233,265)(234,266)(235,267)(236,268)(237,269)(238,270)(239,271)(240,272), (1,209)(2,210)(3,211)(4,212)(5,213)(6,214)(7,215)(8,216)(9,217)(10,218)(11,219)(12,220)(13,221)(14,222)(15,223)(16,224)(17,225)(18,226)(19,227)(20,228)(21,229)(22,230)(23,231)(24,232)(25,233)(26,234)(27,235)(28,236)(29,237)(30,238)(31,239)(32,240)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,245)(82,246)(83,247)(84,248)(85,249)(86,250)(87,251)(88,252)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,261)(98,262)(99,263)(100,264)(101,265)(102,266)(103,267)(104,268)(105,269)(106,270)(107,271)(108,272)(109,273)(110,274)(111,275)(112,276)(113,277)(114,278)(115,279)(116,280)(117,241)(118,242)(119,243)(120,244)(161,298)(162,299)(163,300)(164,301)(165,302)(166,303)(167,304)(168,305)(169,306)(170,307)(171,308)(172,309)(173,310)(174,311)(175,312)(176,313)(177,314)(178,315)(179,316)(180,317)(181,318)(182,319)(183,320)(184,281)(185,282)(186,283)(187,284)(188,285)(189,286)(190,287)(191,288)(192,289)(193,290)(194,291)(195,292)(196,293)(197,294)(198,295)(199,296)(200,297), (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), (1,163,21,183)(2,162,22,182)(3,161,23,181)(4,200,24,180)(5,199,25,179)(6,198,26,178)(7,197,27,177)(8,196,28,176)(9,195,29,175)(10,194,30,174)(11,193,31,173)(12,192,32,172)(13,191,33,171)(14,190,34,170)(15,189,35,169)(16,188,36,168)(17,187,37,167)(18,186,38,166)(19,185,39,165)(20,184,40,164)(41,103,61,83)(42,102,62,82)(43,101,63,81)(44,100,64,120)(45,99,65,119)(46,98,66,118)(47,97,67,117)(48,96,68,116)(49,95,69,115)(50,94,70,114)(51,93,71,113)(52,92,72,112)(53,91,73,111)(54,90,74,110)(55,89,75,109)(56,88,76,108)(57,87,77,107)(58,86,78,106)(59,85,79,105)(60,84,80,104)(121,276,141,256)(122,275,142,255)(123,274,143,254)(124,273,144,253)(125,272,145,252)(126,271,146,251)(127,270,147,250)(128,269,148,249)(129,268,149,248)(130,267,150,247)(131,266,151,246)(132,265,152,245)(133,264,153,244)(134,263,154,243)(135,262,155,242)(136,261,156,241)(137,260,157,280)(138,259,158,279)(139,258,159,278)(140,257,160,277)(201,308,221,288)(202,307,222,287)(203,306,223,286)(204,305,224,285)(205,304,225,284)(206,303,226,283)(207,302,227,282)(208,301,228,281)(209,300,229,320)(210,299,230,319)(211,298,231,318)(212,297,232,317)(213,296,233,316)(214,295,234,315)(215,294,235,314)(216,293,236,313)(217,292,237,312)(218,291,238,311)(219,290,239,310)(220,289,240,309)>;
G:=Group( (1,117)(2,118)(3,119)(4,120)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,197)(42,198)(43,199)(44,200)(45,161)(46,162)(47,163)(48,164)(49,165)(50,166)(51,167)(52,168)(53,169)(54,170)(55,171)(56,172)(57,173)(58,174)(59,175)(60,176)(61,177)(62,178)(63,179)(64,180)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,195)(80,196)(121,285)(122,286)(123,287)(124,288)(125,289)(126,290)(127,291)(128,292)(129,293)(130,294)(131,295)(132,296)(133,297)(134,298)(135,299)(136,300)(137,301)(138,302)(139,303)(140,304)(141,305)(142,306)(143,307)(144,308)(145,309)(146,310)(147,311)(148,312)(149,313)(150,314)(151,315)(152,316)(153,317)(154,318)(155,319)(156,320)(157,281)(158,282)(159,283)(160,284)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)(217,249)(218,250)(219,251)(220,252)(221,253)(222,254)(223,255)(224,256)(225,257)(226,258)(227,259)(228,260)(229,261)(230,262)(231,263)(232,264)(233,265)(234,266)(235,267)(236,268)(237,269)(238,270)(239,271)(240,272), (1,209)(2,210)(3,211)(4,212)(5,213)(6,214)(7,215)(8,216)(9,217)(10,218)(11,219)(12,220)(13,221)(14,222)(15,223)(16,224)(17,225)(18,226)(19,227)(20,228)(21,229)(22,230)(23,231)(24,232)(25,233)(26,234)(27,235)(28,236)(29,237)(30,238)(31,239)(32,240)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,245)(82,246)(83,247)(84,248)(85,249)(86,250)(87,251)(88,252)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,261)(98,262)(99,263)(100,264)(101,265)(102,266)(103,267)(104,268)(105,269)(106,270)(107,271)(108,272)(109,273)(110,274)(111,275)(112,276)(113,277)(114,278)(115,279)(116,280)(117,241)(118,242)(119,243)(120,244)(161,298)(162,299)(163,300)(164,301)(165,302)(166,303)(167,304)(168,305)(169,306)(170,307)(171,308)(172,309)(173,310)(174,311)(175,312)(176,313)(177,314)(178,315)(179,316)(180,317)(181,318)(182,319)(183,320)(184,281)(185,282)(186,283)(187,284)(188,285)(189,286)(190,287)(191,288)(192,289)(193,290)(194,291)(195,292)(196,293)(197,294)(198,295)(199,296)(200,297), (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), (1,163,21,183)(2,162,22,182)(3,161,23,181)(4,200,24,180)(5,199,25,179)(6,198,26,178)(7,197,27,177)(8,196,28,176)(9,195,29,175)(10,194,30,174)(11,193,31,173)(12,192,32,172)(13,191,33,171)(14,190,34,170)(15,189,35,169)(16,188,36,168)(17,187,37,167)(18,186,38,166)(19,185,39,165)(20,184,40,164)(41,103,61,83)(42,102,62,82)(43,101,63,81)(44,100,64,120)(45,99,65,119)(46,98,66,118)(47,97,67,117)(48,96,68,116)(49,95,69,115)(50,94,70,114)(51,93,71,113)(52,92,72,112)(53,91,73,111)(54,90,74,110)(55,89,75,109)(56,88,76,108)(57,87,77,107)(58,86,78,106)(59,85,79,105)(60,84,80,104)(121,276,141,256)(122,275,142,255)(123,274,143,254)(124,273,144,253)(125,272,145,252)(126,271,146,251)(127,270,147,250)(128,269,148,249)(129,268,149,248)(130,267,150,247)(131,266,151,246)(132,265,152,245)(133,264,153,244)(134,263,154,243)(135,262,155,242)(136,261,156,241)(137,260,157,280)(138,259,158,279)(139,258,159,278)(140,257,160,277)(201,308,221,288)(202,307,222,287)(203,306,223,286)(204,305,224,285)(205,304,225,284)(206,303,226,283)(207,302,227,282)(208,301,228,281)(209,300,229,320)(210,299,230,319)(211,298,231,318)(212,297,232,317)(213,296,233,316)(214,295,234,315)(215,294,235,314)(216,293,236,313)(217,292,237,312)(218,291,238,311)(219,290,239,310)(220,289,240,309) );
G=PermutationGroup([(1,117),(2,118),(3,119),(4,120),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,197),(42,198),(43,199),(44,200),(45,161),(46,162),(47,163),(48,164),(49,165),(50,166),(51,167),(52,168),(53,169),(54,170),(55,171),(56,172),(57,173),(58,174),(59,175),(60,176),(61,177),(62,178),(63,179),(64,180),(65,181),(66,182),(67,183),(68,184),(69,185),(70,186),(71,187),(72,188),(73,189),(74,190),(75,191),(76,192),(77,193),(78,194),(79,195),(80,196),(121,285),(122,286),(123,287),(124,288),(125,289),(126,290),(127,291),(128,292),(129,293),(130,294),(131,295),(132,296),(133,297),(134,298),(135,299),(136,300),(137,301),(138,302),(139,303),(140,304),(141,305),(142,306),(143,307),(144,308),(145,309),(146,310),(147,311),(148,312),(149,313),(150,314),(151,315),(152,316),(153,317),(154,318),(155,319),(156,320),(157,281),(158,282),(159,283),(160,284),(201,273),(202,274),(203,275),(204,276),(205,277),(206,278),(207,279),(208,280),(209,241),(210,242),(211,243),(212,244),(213,245),(214,246),(215,247),(216,248),(217,249),(218,250),(219,251),(220,252),(221,253),(222,254),(223,255),(224,256),(225,257),(226,258),(227,259),(228,260),(229,261),(230,262),(231,263),(232,264),(233,265),(234,266),(235,267),(236,268),(237,269),(238,270),(239,271),(240,272)], [(1,209),(2,210),(3,211),(4,212),(5,213),(6,214),(7,215),(8,216),(9,217),(10,218),(11,219),(12,220),(13,221),(14,222),(15,223),(16,224),(17,225),(18,226),(19,227),(20,228),(21,229),(22,230),(23,231),(24,232),(25,233),(26,234),(27,235),(28,236),(29,237),(30,238),(31,239),(32,240),(33,201),(34,202),(35,203),(36,204),(37,205),(38,206),(39,207),(40,208),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,121),(73,122),(74,123),(75,124),(76,125),(77,126),(78,127),(79,128),(80,129),(81,245),(82,246),(83,247),(84,248),(85,249),(86,250),(87,251),(88,252),(89,253),(90,254),(91,255),(92,256),(93,257),(94,258),(95,259),(96,260),(97,261),(98,262),(99,263),(100,264),(101,265),(102,266),(103,267),(104,268),(105,269),(106,270),(107,271),(108,272),(109,273),(110,274),(111,275),(112,276),(113,277),(114,278),(115,279),(116,280),(117,241),(118,242),(119,243),(120,244),(161,298),(162,299),(163,300),(164,301),(165,302),(166,303),(167,304),(168,305),(169,306),(170,307),(171,308),(172,309),(173,310),(174,311),(175,312),(176,313),(177,314),(178,315),(179,316),(180,317),(181,318),(182,319),(183,320),(184,281),(185,282),(186,283),(187,284),(188,285),(189,286),(190,287),(191,288),(192,289),(193,290),(194,291),(195,292),(196,293),(197,294),(198,295),(199,296),(200,297)], [(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)], [(1,163,21,183),(2,162,22,182),(3,161,23,181),(4,200,24,180),(5,199,25,179),(6,198,26,178),(7,197,27,177),(8,196,28,176),(9,195,29,175),(10,194,30,174),(11,193,31,173),(12,192,32,172),(13,191,33,171),(14,190,34,170),(15,189,35,169),(16,188,36,168),(17,187,37,167),(18,186,38,166),(19,185,39,165),(20,184,40,164),(41,103,61,83),(42,102,62,82),(43,101,63,81),(44,100,64,120),(45,99,65,119),(46,98,66,118),(47,97,67,117),(48,96,68,116),(49,95,69,115),(50,94,70,114),(51,93,71,113),(52,92,72,112),(53,91,73,111),(54,90,74,110),(55,89,75,109),(56,88,76,108),(57,87,77,107),(58,86,78,106),(59,85,79,105),(60,84,80,104),(121,276,141,256),(122,275,142,255),(123,274,143,254),(124,273,144,253),(125,272,145,252),(126,271,146,251),(127,270,147,250),(128,269,148,249),(129,268,149,248),(130,267,150,247),(131,266,151,246),(132,265,152,245),(133,264,153,244),(134,263,154,243),(135,262,155,242),(136,261,156,241),(137,260,157,280),(138,259,158,279),(139,258,159,278),(140,257,160,277),(201,308,221,288),(202,307,222,287),(203,306,223,286),(204,305,224,285),(205,304,225,284),(206,303,226,283),(207,302,227,282),(208,301,228,281),(209,300,229,320),(210,299,230,319),(211,298,231,318),(212,297,232,317),(213,296,233,316),(214,295,234,315),(215,294,235,314),(216,293,236,313),(217,292,237,312),(218,291,238,311),(219,290,239,310),(220,289,240,309)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2^2\times 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
×
⋊
ℤ
𝔽
○
≀