direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic10, C20.66(C2×Q8), (C2×C20).30Q8, C4⋊C4.260D10, C10.8(C22×Q8), (C2×C10).43C24, C10⋊3(C42.C2), C4.31(C2×Dic10), (C2×C4).37Dic10, (C2×C20).134C23, (C22×C4).358D10, C22.81(C23×D5), C4⋊Dic5.357C22, (C22×C20).73C22, C22.37(C2×Dic10), C2.10(C22×Dic10), C23.323(C22×D5), C22.72(D4⋊2D5), (C22×C10).392C23, C22.32(Q8⋊2D5), (C2×Dic5).193C23, (C4×Dic5).279C22, C10.D4.104C22, (C22×Dic5).231C22, C5⋊3(C2×C42.C2), (C2×C4⋊C4).26D5, (C10×C4⋊C4).19C2, C10.70(C2×C4○D4), C2.5(C2×Q8⋊2D5), (C2×C10).51(C2×Q8), (C2×C4×Dic5).13C2, C2.14(C2×D4⋊2D5), (C2×C4⋊Dic5).44C2, (C5×C4⋊C4).292C22, (C2×C4).139(C22×D5), (C2×C10).170(C4○D4), (C2×C10.D4).24C2, SmallGroup(320,1171)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 606 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×20], C23, C10 [×3], C10 [×4], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×5], C42.C2 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×4], C22×C10, C2×C42.C2, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×12], C5×C4⋊C4 [×4], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C4.Dic10 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C4⋊Dic5 [×2], C10×C4⋊C4, C2×C4.Dic10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], Dic10 [×4], C22×D5 [×7], C2×C42.C2, C2×Dic10 [×6], D4⋊2D5 [×2], Q8⋊2D5 [×2], C23×D5, C4.Dic10 [×4], C22×Dic10, C2×D4⋊2D5, C2×Q8⋊2D5, C2×C4.Dic10
Generators and relations
G = < a,b,c,d | a2=b4=c20=1, d2=c10, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >
►Regular action on 320 points
Generators in S320
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 205)(42 206)(43 207)(44 208)(45 209)(46 210)(47 211)(48 212)(49 213)(50 214)(51 215)(52 216)(53 217)(54 218)(55 219)(56 220)(57 201)(58 202)(59 203)(60 204)(61 272)(62 273)(63 274)(64 275)(65 276)(66 277)(67 278)(68 279)(69 280)(70 261)(71 262)(72 263)(73 264)(74 265)(75 266)(76 267)(77 268)(78 269)(79 270)(80 271)(81 188)(82 189)(83 190)(84 191)(85 192)(86 193)(87 194)(88 195)(89 196)(90 197)(91 198)(92 199)(93 200)(94 181)(95 182)(96 183)(97 184)(98 185)(99 186)(100 187)(101 315)(102 316)(103 317)(104 318)(105 319)(106 320)(107 301)(108 302)(109 303)(110 304)(111 305)(112 306)(113 307)(114 308)(115 309)(116 310)(117 311)(118 312)(119 313)(120 314)(121 158)(122 159)(123 160)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)(161 246)(162 247)(163 248)(164 249)(165 250)(166 251)(167 252)(168 253)(169 254)(170 255)(171 256)(172 257)(173 258)(174 259)(175 260)(176 241)(177 242)(178 243)(179 244)(180 245)(221 297)(222 298)(223 299)(224 300)(225 281)(226 282)(227 283)(228 284)(229 285)(230 286)(231 287)(232 288)(233 289)(234 290)(235 291)(236 292)(237 293)(238 294)(239 295)(240 296)
(1 163 142 96)(2 97 143 164)(3 165 144 98)(4 99 145 166)(5 167 146 100)(6 81 147 168)(7 169 148 82)(8 83 149 170)(9 171 150 84)(10 85 151 172)(11 173 152 86)(12 87 153 174)(13 175 154 88)(14 89 155 176)(15 177 156 90)(16 91 157 178)(17 179 158 92)(18 93 159 180)(19 161 160 94)(20 95 141 162)(21 258 135 193)(22 194 136 259)(23 260 137 195)(24 196 138 241)(25 242 139 197)(26 198 140 243)(27 244 121 199)(28 200 122 245)(29 246 123 181)(30 182 124 247)(31 248 125 183)(32 184 126 249)(33 250 127 185)(34 186 128 251)(35 252 129 187)(36 188 130 253)(37 254 131 189)(38 190 132 255)(39 256 133 191)(40 192 134 257)(41 303 271 288)(42 289 272 304)(43 305 273 290)(44 291 274 306)(45 307 275 292)(46 293 276 308)(47 309 277 294)(48 295 278 310)(49 311 279 296)(50 297 280 312)(51 313 261 298)(52 299 262 314)(53 315 263 300)(54 281 264 316)(55 317 265 282)(56 283 266 318)(57 319 267 284)(58 285 268 320)(59 301 269 286)(60 287 270 302)(61 110 206 233)(62 234 207 111)(63 112 208 235)(64 236 209 113)(65 114 210 237)(66 238 211 115)(67 116 212 239)(68 240 213 117)(69 118 214 221)(70 222 215 119)(71 120 216 223)(72 224 217 101)(73 102 218 225)(74 226 219 103)(75 104 220 227)(76 228 201 105)(77 106 202 229)(78 230 203 107)(79 108 204 231)(80 232 205 109)
(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)
(1 224 11 234)(2 120 12 110)(3 222 13 232)(4 118 14 108)(5 240 15 230)(6 116 16 106)(7 238 17 228)(8 114 18 104)(9 236 19 226)(10 112 20 102)(21 290 31 300)(22 304 32 314)(23 288 33 298)(24 302 34 312)(25 286 35 296)(26 320 36 310)(27 284 37 294)(28 318 38 308)(29 282 39 292)(30 316 40 306)(41 250 51 260)(42 184 52 194)(43 248 53 258)(44 182 54 192)(45 246 55 256)(46 200 56 190)(47 244 57 254)(48 198 58 188)(49 242 59 252)(50 196 60 186)(61 164 71 174)(62 96 72 86)(63 162 73 172)(64 94 74 84)(65 180 75 170)(66 92 76 82)(67 178 77 168)(68 90 78 100)(69 176 79 166)(70 88 80 98)(81 212 91 202)(83 210 93 220)(85 208 95 218)(87 206 97 216)(89 204 99 214)(101 152 111 142)(103 150 113 160)(105 148 115 158)(107 146 117 156)(109 144 119 154)(121 319 131 309)(122 283 132 293)(123 317 133 307)(124 281 134 291)(125 315 135 305)(126 299 136 289)(127 313 137 303)(128 297 138 287)(129 311 139 301)(130 295 140 285)(141 225 151 235)(143 223 153 233)(145 221 155 231)(147 239 157 229)(149 237 159 227)(161 219 171 209)(163 217 173 207)(165 215 175 205)(167 213 177 203)(169 211 179 201)(181 265 191 275)(183 263 193 273)(185 261 195 271)(187 279 197 269)(189 277 199 267)(241 270 251 280)(243 268 253 278)(245 266 255 276)(247 264 257 274)(249 262 259 272)
G:=sub<Sym(320)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,201)(58,202)(59,203)(60,204)(61,272)(62,273)(63,274)(64,275)(65,276)(66,277)(67,278)(68,279)(69,280)(70,261)(71,262)(72,263)(73,264)(74,265)(75,266)(76,267)(77,268)(78,269)(79,270)(80,271)(81,188)(82,189)(83,190)(84,191)(85,192)(86,193)(87,194)(88,195)(89,196)(90,197)(91,198)(92,199)(93,200)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,315)(102,316)(103,317)(104,318)(105,319)(106,320)(107,301)(108,302)(109,303)(110,304)(111,305)(112,306)(113,307)(114,308)(115,309)(116,310)(117,311)(118,312)(119,313)(120,314)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157)(161,246)(162,247)(163,248)(164,249)(165,250)(166,251)(167,252)(168,253)(169,254)(170,255)(171,256)(172,257)(173,258)(174,259)(175,260)(176,241)(177,242)(178,243)(179,244)(180,245)(221,297)(222,298)(223,299)(224,300)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,287)(232,288)(233,289)(234,290)(235,291)(236,292)(237,293)(238,294)(239,295)(240,296), (1,163,142,96)(2,97,143,164)(3,165,144,98)(4,99,145,166)(5,167,146,100)(6,81,147,168)(7,169,148,82)(8,83,149,170)(9,171,150,84)(10,85,151,172)(11,173,152,86)(12,87,153,174)(13,175,154,88)(14,89,155,176)(15,177,156,90)(16,91,157,178)(17,179,158,92)(18,93,159,180)(19,161,160,94)(20,95,141,162)(21,258,135,193)(22,194,136,259)(23,260,137,195)(24,196,138,241)(25,242,139,197)(26,198,140,243)(27,244,121,199)(28,200,122,245)(29,246,123,181)(30,182,124,247)(31,248,125,183)(32,184,126,249)(33,250,127,185)(34,186,128,251)(35,252,129,187)(36,188,130,253)(37,254,131,189)(38,190,132,255)(39,256,133,191)(40,192,134,257)(41,303,271,288)(42,289,272,304)(43,305,273,290)(44,291,274,306)(45,307,275,292)(46,293,276,308)(47,309,277,294)(48,295,278,310)(49,311,279,296)(50,297,280,312)(51,313,261,298)(52,299,262,314)(53,315,263,300)(54,281,264,316)(55,317,265,282)(56,283,266,318)(57,319,267,284)(58,285,268,320)(59,301,269,286)(60,287,270,302)(61,110,206,233)(62,234,207,111)(63,112,208,235)(64,236,209,113)(65,114,210,237)(66,238,211,115)(67,116,212,239)(68,240,213,117)(69,118,214,221)(70,222,215,119)(71,120,216,223)(72,224,217,101)(73,102,218,225)(74,226,219,103)(75,104,220,227)(76,228,201,105)(77,106,202,229)(78,230,203,107)(79,108,204,231)(80,232,205,109), (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), (1,224,11,234)(2,120,12,110)(3,222,13,232)(4,118,14,108)(5,240,15,230)(6,116,16,106)(7,238,17,228)(8,114,18,104)(9,236,19,226)(10,112,20,102)(21,290,31,300)(22,304,32,314)(23,288,33,298)(24,302,34,312)(25,286,35,296)(26,320,36,310)(27,284,37,294)(28,318,38,308)(29,282,39,292)(30,316,40,306)(41,250,51,260)(42,184,52,194)(43,248,53,258)(44,182,54,192)(45,246,55,256)(46,200,56,190)(47,244,57,254)(48,198,58,188)(49,242,59,252)(50,196,60,186)(61,164,71,174)(62,96,72,86)(63,162,73,172)(64,94,74,84)(65,180,75,170)(66,92,76,82)(67,178,77,168)(68,90,78,100)(69,176,79,166)(70,88,80,98)(81,212,91,202)(83,210,93,220)(85,208,95,218)(87,206,97,216)(89,204,99,214)(101,152,111,142)(103,150,113,160)(105,148,115,158)(107,146,117,156)(109,144,119,154)(121,319,131,309)(122,283,132,293)(123,317,133,307)(124,281,134,291)(125,315,135,305)(126,299,136,289)(127,313,137,303)(128,297,138,287)(129,311,139,301)(130,295,140,285)(141,225,151,235)(143,223,153,233)(145,221,155,231)(147,239,157,229)(149,237,159,227)(161,219,171,209)(163,217,173,207)(165,215,175,205)(167,213,177,203)(169,211,179,201)(181,265,191,275)(183,263,193,273)(185,261,195,271)(187,279,197,269)(189,277,199,267)(241,270,251,280)(243,268,253,278)(245,266,255,276)(247,264,257,274)(249,262,259,272)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,201)(58,202)(59,203)(60,204)(61,272)(62,273)(63,274)(64,275)(65,276)(66,277)(67,278)(68,279)(69,280)(70,261)(71,262)(72,263)(73,264)(74,265)(75,266)(76,267)(77,268)(78,269)(79,270)(80,271)(81,188)(82,189)(83,190)(84,191)(85,192)(86,193)(87,194)(88,195)(89,196)(90,197)(91,198)(92,199)(93,200)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,315)(102,316)(103,317)(104,318)(105,319)(106,320)(107,301)(108,302)(109,303)(110,304)(111,305)(112,306)(113,307)(114,308)(115,309)(116,310)(117,311)(118,312)(119,313)(120,314)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157)(161,246)(162,247)(163,248)(164,249)(165,250)(166,251)(167,252)(168,253)(169,254)(170,255)(171,256)(172,257)(173,258)(174,259)(175,260)(176,241)(177,242)(178,243)(179,244)(180,245)(221,297)(222,298)(223,299)(224,300)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,287)(232,288)(233,289)(234,290)(235,291)(236,292)(237,293)(238,294)(239,295)(240,296), (1,163,142,96)(2,97,143,164)(3,165,144,98)(4,99,145,166)(5,167,146,100)(6,81,147,168)(7,169,148,82)(8,83,149,170)(9,171,150,84)(10,85,151,172)(11,173,152,86)(12,87,153,174)(13,175,154,88)(14,89,155,176)(15,177,156,90)(16,91,157,178)(17,179,158,92)(18,93,159,180)(19,161,160,94)(20,95,141,162)(21,258,135,193)(22,194,136,259)(23,260,137,195)(24,196,138,241)(25,242,139,197)(26,198,140,243)(27,244,121,199)(28,200,122,245)(29,246,123,181)(30,182,124,247)(31,248,125,183)(32,184,126,249)(33,250,127,185)(34,186,128,251)(35,252,129,187)(36,188,130,253)(37,254,131,189)(38,190,132,255)(39,256,133,191)(40,192,134,257)(41,303,271,288)(42,289,272,304)(43,305,273,290)(44,291,274,306)(45,307,275,292)(46,293,276,308)(47,309,277,294)(48,295,278,310)(49,311,279,296)(50,297,280,312)(51,313,261,298)(52,299,262,314)(53,315,263,300)(54,281,264,316)(55,317,265,282)(56,283,266,318)(57,319,267,284)(58,285,268,320)(59,301,269,286)(60,287,270,302)(61,110,206,233)(62,234,207,111)(63,112,208,235)(64,236,209,113)(65,114,210,237)(66,238,211,115)(67,116,212,239)(68,240,213,117)(69,118,214,221)(70,222,215,119)(71,120,216,223)(72,224,217,101)(73,102,218,225)(74,226,219,103)(75,104,220,227)(76,228,201,105)(77,106,202,229)(78,230,203,107)(79,108,204,231)(80,232,205,109), (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), (1,224,11,234)(2,120,12,110)(3,222,13,232)(4,118,14,108)(5,240,15,230)(6,116,16,106)(7,238,17,228)(8,114,18,104)(9,236,19,226)(10,112,20,102)(21,290,31,300)(22,304,32,314)(23,288,33,298)(24,302,34,312)(25,286,35,296)(26,320,36,310)(27,284,37,294)(28,318,38,308)(29,282,39,292)(30,316,40,306)(41,250,51,260)(42,184,52,194)(43,248,53,258)(44,182,54,192)(45,246,55,256)(46,200,56,190)(47,244,57,254)(48,198,58,188)(49,242,59,252)(50,196,60,186)(61,164,71,174)(62,96,72,86)(63,162,73,172)(64,94,74,84)(65,180,75,170)(66,92,76,82)(67,178,77,168)(68,90,78,100)(69,176,79,166)(70,88,80,98)(81,212,91,202)(83,210,93,220)(85,208,95,218)(87,206,97,216)(89,204,99,214)(101,152,111,142)(103,150,113,160)(105,148,115,158)(107,146,117,156)(109,144,119,154)(121,319,131,309)(122,283,132,293)(123,317,133,307)(124,281,134,291)(125,315,135,305)(126,299,136,289)(127,313,137,303)(128,297,138,287)(129,311,139,301)(130,295,140,285)(141,225,151,235)(143,223,153,233)(145,221,155,231)(147,239,157,229)(149,237,159,227)(161,219,171,209)(163,217,173,207)(165,215,175,205)(167,213,177,203)(169,211,179,201)(181,265,191,275)(183,263,193,273)(185,261,195,271)(187,279,197,269)(189,277,199,267)(241,270,251,280)(243,268,253,278)(245,266,255,276)(247,264,257,274)(249,262,259,272) );
G=PermutationGroup([(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(41,205),(42,206),(43,207),(44,208),(45,209),(46,210),(47,211),(48,212),(49,213),(50,214),(51,215),(52,216),(53,217),(54,218),(55,219),(56,220),(57,201),(58,202),(59,203),(60,204),(61,272),(62,273),(63,274),(64,275),(65,276),(66,277),(67,278),(68,279),(69,280),(70,261),(71,262),(72,263),(73,264),(74,265),(75,266),(76,267),(77,268),(78,269),(79,270),(80,271),(81,188),(82,189),(83,190),(84,191),(85,192),(86,193),(87,194),(88,195),(89,196),(90,197),(91,198),(92,199),(93,200),(94,181),(95,182),(96,183),(97,184),(98,185),(99,186),(100,187),(101,315),(102,316),(103,317),(104,318),(105,319),(106,320),(107,301),(108,302),(109,303),(110,304),(111,305),(112,306),(113,307),(114,308),(115,309),(116,310),(117,311),(118,312),(119,313),(120,314),(121,158),(122,159),(123,160),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157),(161,246),(162,247),(163,248),(164,249),(165,250),(166,251),(167,252),(168,253),(169,254),(170,255),(171,256),(172,257),(173,258),(174,259),(175,260),(176,241),(177,242),(178,243),(179,244),(180,245),(221,297),(222,298),(223,299),(224,300),(225,281),(226,282),(227,283),(228,284),(229,285),(230,286),(231,287),(232,288),(233,289),(234,290),(235,291),(236,292),(237,293),(238,294),(239,295),(240,296)], [(1,163,142,96),(2,97,143,164),(3,165,144,98),(4,99,145,166),(5,167,146,100),(6,81,147,168),(7,169,148,82),(8,83,149,170),(9,171,150,84),(10,85,151,172),(11,173,152,86),(12,87,153,174),(13,175,154,88),(14,89,155,176),(15,177,156,90),(16,91,157,178),(17,179,158,92),(18,93,159,180),(19,161,160,94),(20,95,141,162),(21,258,135,193),(22,194,136,259),(23,260,137,195),(24,196,138,241),(25,242,139,197),(26,198,140,243),(27,244,121,199),(28,200,122,245),(29,246,123,181),(30,182,124,247),(31,248,125,183),(32,184,126,249),(33,250,127,185),(34,186,128,251),(35,252,129,187),(36,188,130,253),(37,254,131,189),(38,190,132,255),(39,256,133,191),(40,192,134,257),(41,303,271,288),(42,289,272,304),(43,305,273,290),(44,291,274,306),(45,307,275,292),(46,293,276,308),(47,309,277,294),(48,295,278,310),(49,311,279,296),(50,297,280,312),(51,313,261,298),(52,299,262,314),(53,315,263,300),(54,281,264,316),(55,317,265,282),(56,283,266,318),(57,319,267,284),(58,285,268,320),(59,301,269,286),(60,287,270,302),(61,110,206,233),(62,234,207,111),(63,112,208,235),(64,236,209,113),(65,114,210,237),(66,238,211,115),(67,116,212,239),(68,240,213,117),(69,118,214,221),(70,222,215,119),(71,120,216,223),(72,224,217,101),(73,102,218,225),(74,226,219,103),(75,104,220,227),(76,228,201,105),(77,106,202,229),(78,230,203,107),(79,108,204,231),(80,232,205,109)], [(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)], [(1,224,11,234),(2,120,12,110),(3,222,13,232),(4,118,14,108),(5,240,15,230),(6,116,16,106),(7,238,17,228),(8,114,18,104),(9,236,19,226),(10,112,20,102),(21,290,31,300),(22,304,32,314),(23,288,33,298),(24,302,34,312),(25,286,35,296),(26,320,36,310),(27,284,37,294),(28,318,38,308),(29,282,39,292),(30,316,40,306),(41,250,51,260),(42,184,52,194),(43,248,53,258),(44,182,54,192),(45,246,55,256),(46,200,56,190),(47,244,57,254),(48,198,58,188),(49,242,59,252),(50,196,60,186),(61,164,71,174),(62,96,72,86),(63,162,73,172),(64,94,74,84),(65,180,75,170),(66,92,76,82),(67,178,77,168),(68,90,78,100),(69,176,79,166),(70,88,80,98),(81,212,91,202),(83,210,93,220),(85,208,95,218),(87,206,97,216),(89,204,99,214),(101,152,111,142),(103,150,113,160),(105,148,115,158),(107,146,117,156),(109,144,119,154),(121,319,131,309),(122,283,132,293),(123,317,133,307),(124,281,134,291),(125,315,135,305),(126,299,136,289),(127,313,137,303),(128,297,138,287),(129,311,139,301),(130,295,140,285),(141,225,151,235),(143,223,153,233),(145,221,155,231),(147,239,157,229),(149,237,159,227),(161,219,171,209),(163,217,173,207),(165,215,175,205),(167,213,177,203),(169,211,179,201),(181,265,191,275),(183,263,193,273),(185,261,195,271),(187,279,197,269),(189,277,199,267),(241,270,251,280),(243,268,253,278),(245,266,255,276),(247,264,257,274),(249,262,259,272)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 |
| 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 | 13 | 37 |
| 0 | 0 | 0 | 0 | 22 | 28 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 27 | 40 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 12 | 0 | 0 |
| 0 | 0 | 23 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 36 |
| 0 | 0 | 0 | 0 | 7 | 35 |
G:=sub<GL(6,GF(41))| [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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,22,0,0,0,0,37,28],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,14,11,0,0,0,0,30,9,0,0,0,0,0,0,1,27,0,0,0,0,0,40],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,16,23,0,0,0,0,12,25,0,0,0,0,0,0,6,7,0,0,0,0,36,35] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | Dic10 | D4⋊2D5 | Q8⋊2D5 |
| kernel | C2×C4.Dic10 | C4.Dic10 | C2×C4×Dic5 | C2×C10.D4 | C2×C4⋊Dic5 | C10×C4⋊C4 | C2×C20 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 3 | 1 | 4 | 2 | 8 | 8 | 6 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_4.Dic_{10} % in TeX
G:=Group("C2xC4.Dic10"); // GroupNames label
G:=SmallGroup(320,1171);
// by ID
G=gap.SmallGroup(320,1171);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1571,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀