direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.44D4, C23.55D20, C22.5Dic20, (C2×C4).92D20, C10.9(C2×Q16), (C2×C10).9Q16, (C22×C8).7D5, (C2×C8).292D10, C20.409(C2×D4), (C2×C20).385D4, C2.2(C2×Dic20), C10⋊3(Q8⋊C4), Dic10⋊25(C2×C4), (C2×Dic10)⋊18C4, (C22×C40).10C2, C10.14(C2×SD16), (C2×C10).20SD16, C22.49(C2×D20), C20.97(C22⋊C4), C20.170(C22×C4), (C2×C40).352C22, (C2×C20).762C23, (C22×C10).134D4, (C22×C4).422D10, C4.25(D10⋊C4), C22.10(C40⋊C2), C4⋊Dic5.278C22, (C22×Dic10).6C2, (C22×C20).514C22, C22.47(D10⋊C4), (C2×Dic10).224C22, C4.69(C2×C4×D5), C5⋊4(C2×Q8⋊C4), C2.2(C2×C40⋊C2), (C2×C4).114(C4×D5), C4.102(C2×C5⋊D4), (C2×C20).400(C2×C4), (C2×C10).152(C2×D4), C10.90(C2×C22⋊C4), (C2×C4⋊Dic5).22C2, C2.22(C2×D10⋊C4), (C2×C4).252(C5⋊D4), (C2×C4).709(C22×D5), (C2×C10).124(C22⋊C4), SmallGroup(320,730)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.44D4
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c-1 >
Subgroups: 574 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×Q8⋊C4, C4⋊Dic5, C4⋊Dic5, C2×C40, C2×C40, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C20.44D4, C2×C4⋊Dic5, C22×C40, C22×Dic10, C2×C20.44D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, SD16, Q16, C22×C4, C2×D4, D10, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4×D5, D20, C5⋊D4, C22×D5, C2×Q8⋊C4, C40⋊C2, Dic20, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C20.44D4, C2×C40⋊C2, C2×Dic20, C2×D10⋊C4, C2×C20.44D4
►Regular action on 320 points
Generators in S320
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 81)(39 82)(40 83)(41 220)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 209)(51 210)(52 211)(53 212)(54 213)(55 214)(56 215)(57 216)(58 217)(59 218)(60 219)(101 312)(102 313)(103 314)(104 315)(105 316)(106 317)(107 318)(108 319)(109 320)(110 301)(111 302)(112 303)(113 304)(114 305)(115 306)(116 307)(117 308)(118 309)(119 310)(120 311)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 141)(132 142)(133 143)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)(161 284)(162 285)(163 286)(164 287)(165 288)(166 289)(167 290)(168 291)(169 292)(170 293)(171 294)(172 295)(173 296)(174 297)(175 298)(176 299)(177 300)(178 281)(179 282)(180 283)(181 228)(182 229)(183 230)(184 231)(185 232)(186 233)(187 234)(188 235)(189 236)(190 237)(191 238)(192 239)(193 240)(194 221)(195 222)(196 223)(197 224)(198 225)(199 226)(200 227)(241 272)(242 273)(243 274)(244 275)(245 276)(246 277)(247 278)(248 279)(249 280)(250 261)(251 262)(252 263)(253 264)(254 265)(255 266)(256 267)(257 268)(258 269)(259 270)(260 271)
(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 22 60 228)(2 21 41 227)(3 40 42 226)(4 39 43 225)(5 38 44 224)(6 37 45 223)(7 36 46 222)(8 35 47 221)(9 34 48 240)(10 33 49 239)(11 32 50 238)(12 31 51 237)(13 30 52 236)(14 29 53 235)(15 28 54 234)(16 27 55 233)(17 26 56 232)(18 25 57 231)(19 24 58 230)(20 23 59 229)(61 91 213 187)(62 90 214 186)(63 89 215 185)(64 88 216 184)(65 87 217 183)(66 86 218 182)(67 85 219 181)(68 84 220 200)(69 83 201 199)(70 82 202 198)(71 81 203 197)(72 100 204 196)(73 99 205 195)(74 98 206 194)(75 97 207 193)(76 96 208 192)(77 95 209 191)(78 94 210 190)(79 93 211 189)(80 92 212 188)(101 296 247 153)(102 295 248 152)(103 294 249 151)(104 293 250 150)(105 292 251 149)(106 291 252 148)(107 290 253 147)(108 289 254 146)(109 288 255 145)(110 287 256 144)(111 286 257 143)(112 285 258 142)(113 284 259 141)(114 283 260 160)(115 282 241 159)(116 281 242 158)(117 300 243 157)(118 299 244 156)(119 298 245 155)(120 297 246 154)(121 314 171 280)(122 313 172 279)(123 312 173 278)(124 311 174 277)(125 310 175 276)(126 309 176 275)(127 308 177 274)(128 307 178 273)(129 306 179 272)(130 305 180 271)(131 304 161 270)(132 303 162 269)(133 302 163 268)(134 301 164 267)(135 320 165 266)(136 319 166 265)(137 318 167 264)(138 317 168 263)(139 316 169 262)(140 315 170 261)
(1 286 11 296)(2 285 12 295)(3 284 13 294)(4 283 14 293)(5 282 15 292)(6 281 16 291)(7 300 17 290)(8 299 18 289)(9 298 19 288)(10 297 20 287)(21 107 31 117)(22 106 32 116)(23 105 33 115)(24 104 34 114)(25 103 35 113)(26 102 36 112)(27 101 37 111)(28 120 38 110)(29 119 39 109)(30 118 40 108)(41 142 51 152)(42 141 52 151)(43 160 53 150)(44 159 54 149)(45 158 55 148)(46 157 56 147)(47 156 57 146)(48 155 58 145)(49 154 59 144)(50 153 60 143)(61 169 71 179)(62 168 72 178)(63 167 73 177)(64 166 74 176)(65 165 75 175)(66 164 76 174)(67 163 77 173)(68 162 78 172)(69 161 79 171)(70 180 80 170)(81 301 91 311)(82 320 92 310)(83 319 93 309)(84 318 94 308)(85 317 95 307)(86 316 96 306)(87 315 97 305)(88 314 98 304)(89 313 99 303)(90 312 100 302)(121 201 131 211)(122 220 132 210)(123 219 133 209)(124 218 134 208)(125 217 135 207)(126 216 136 206)(127 215 137 205)(128 214 138 204)(129 213 139 203)(130 212 140 202)(181 263 191 273)(182 262 192 272)(183 261 193 271)(184 280 194 270)(185 279 195 269)(186 278 196 268)(187 277 197 267)(188 276 198 266)(189 275 199 265)(190 274 200 264)(221 259 231 249)(222 258 232 248)(223 257 233 247)(224 256 234 246)(225 255 235 245)(226 254 236 244)(227 253 237 243)(228 252 238 242)(229 251 239 241)(230 250 240 260)
G:=sub<Sym(320)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,81)(39,82)(40,83)(41,220)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(57,216)(58,217)(59,218)(60,219)(101,312)(102,313)(103,314)(104,315)(105,316)(106,317)(107,318)(108,319)(109,320)(110,301)(111,302)(112,303)(113,304)(114,305)(115,306)(116,307)(117,308)(118,309)(119,310)(120,311)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(161,284)(162,285)(163,286)(164,287)(165,288)(166,289)(167,290)(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)(177,300)(178,281)(179,282)(180,283)(181,228)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239)(193,240)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)(200,227)(241,272)(242,273)(243,274)(244,275)(245,276)(246,277)(247,278)(248,279)(249,280)(250,261)(251,262)(252,263)(253,264)(254,265)(255,266)(256,267)(257,268)(258,269)(259,270)(260,271), (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), (1,22,60,228)(2,21,41,227)(3,40,42,226)(4,39,43,225)(5,38,44,224)(6,37,45,223)(7,36,46,222)(8,35,47,221)(9,34,48,240)(10,33,49,239)(11,32,50,238)(12,31,51,237)(13,30,52,236)(14,29,53,235)(15,28,54,234)(16,27,55,233)(17,26,56,232)(18,25,57,231)(19,24,58,230)(20,23,59,229)(61,91,213,187)(62,90,214,186)(63,89,215,185)(64,88,216,184)(65,87,217,183)(66,86,218,182)(67,85,219,181)(68,84,220,200)(69,83,201,199)(70,82,202,198)(71,81,203,197)(72,100,204,196)(73,99,205,195)(74,98,206,194)(75,97,207,193)(76,96,208,192)(77,95,209,191)(78,94,210,190)(79,93,211,189)(80,92,212,188)(101,296,247,153)(102,295,248,152)(103,294,249,151)(104,293,250,150)(105,292,251,149)(106,291,252,148)(107,290,253,147)(108,289,254,146)(109,288,255,145)(110,287,256,144)(111,286,257,143)(112,285,258,142)(113,284,259,141)(114,283,260,160)(115,282,241,159)(116,281,242,158)(117,300,243,157)(118,299,244,156)(119,298,245,155)(120,297,246,154)(121,314,171,280)(122,313,172,279)(123,312,173,278)(124,311,174,277)(125,310,175,276)(126,309,176,275)(127,308,177,274)(128,307,178,273)(129,306,179,272)(130,305,180,271)(131,304,161,270)(132,303,162,269)(133,302,163,268)(134,301,164,267)(135,320,165,266)(136,319,166,265)(137,318,167,264)(138,317,168,263)(139,316,169,262)(140,315,170,261), (1,286,11,296)(2,285,12,295)(3,284,13,294)(4,283,14,293)(5,282,15,292)(6,281,16,291)(7,300,17,290)(8,299,18,289)(9,298,19,288)(10,297,20,287)(21,107,31,117)(22,106,32,116)(23,105,33,115)(24,104,34,114)(25,103,35,113)(26,102,36,112)(27,101,37,111)(28,120,38,110)(29,119,39,109)(30,118,40,108)(41,142,51,152)(42,141,52,151)(43,160,53,150)(44,159,54,149)(45,158,55,148)(46,157,56,147)(47,156,57,146)(48,155,58,145)(49,154,59,144)(50,153,60,143)(61,169,71,179)(62,168,72,178)(63,167,73,177)(64,166,74,176)(65,165,75,175)(66,164,76,174)(67,163,77,173)(68,162,78,172)(69,161,79,171)(70,180,80,170)(81,301,91,311)(82,320,92,310)(83,319,93,309)(84,318,94,308)(85,317,95,307)(86,316,96,306)(87,315,97,305)(88,314,98,304)(89,313,99,303)(90,312,100,302)(121,201,131,211)(122,220,132,210)(123,219,133,209)(124,218,134,208)(125,217,135,207)(126,216,136,206)(127,215,137,205)(128,214,138,204)(129,213,139,203)(130,212,140,202)(181,263,191,273)(182,262,192,272)(183,261,193,271)(184,280,194,270)(185,279,195,269)(186,278,196,268)(187,277,197,267)(188,276,198,266)(189,275,199,265)(190,274,200,264)(221,259,231,249)(222,258,232,248)(223,257,233,247)(224,256,234,246)(225,255,235,245)(226,254,236,244)(227,253,237,243)(228,252,238,242)(229,251,239,241)(230,250,240,260)>;
G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,81)(39,82)(40,83)(41,220)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(57,216)(58,217)(59,218)(60,219)(101,312)(102,313)(103,314)(104,315)(105,316)(106,317)(107,318)(108,319)(109,320)(110,301)(111,302)(112,303)(113,304)(114,305)(115,306)(116,307)(117,308)(118,309)(119,310)(120,311)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(161,284)(162,285)(163,286)(164,287)(165,288)(166,289)(167,290)(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)(177,300)(178,281)(179,282)(180,283)(181,228)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239)(193,240)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)(200,227)(241,272)(242,273)(243,274)(244,275)(245,276)(246,277)(247,278)(248,279)(249,280)(250,261)(251,262)(252,263)(253,264)(254,265)(255,266)(256,267)(257,268)(258,269)(259,270)(260,271), (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), (1,22,60,228)(2,21,41,227)(3,40,42,226)(4,39,43,225)(5,38,44,224)(6,37,45,223)(7,36,46,222)(8,35,47,221)(9,34,48,240)(10,33,49,239)(11,32,50,238)(12,31,51,237)(13,30,52,236)(14,29,53,235)(15,28,54,234)(16,27,55,233)(17,26,56,232)(18,25,57,231)(19,24,58,230)(20,23,59,229)(61,91,213,187)(62,90,214,186)(63,89,215,185)(64,88,216,184)(65,87,217,183)(66,86,218,182)(67,85,219,181)(68,84,220,200)(69,83,201,199)(70,82,202,198)(71,81,203,197)(72,100,204,196)(73,99,205,195)(74,98,206,194)(75,97,207,193)(76,96,208,192)(77,95,209,191)(78,94,210,190)(79,93,211,189)(80,92,212,188)(101,296,247,153)(102,295,248,152)(103,294,249,151)(104,293,250,150)(105,292,251,149)(106,291,252,148)(107,290,253,147)(108,289,254,146)(109,288,255,145)(110,287,256,144)(111,286,257,143)(112,285,258,142)(113,284,259,141)(114,283,260,160)(115,282,241,159)(116,281,242,158)(117,300,243,157)(118,299,244,156)(119,298,245,155)(120,297,246,154)(121,314,171,280)(122,313,172,279)(123,312,173,278)(124,311,174,277)(125,310,175,276)(126,309,176,275)(127,308,177,274)(128,307,178,273)(129,306,179,272)(130,305,180,271)(131,304,161,270)(132,303,162,269)(133,302,163,268)(134,301,164,267)(135,320,165,266)(136,319,166,265)(137,318,167,264)(138,317,168,263)(139,316,169,262)(140,315,170,261), (1,286,11,296)(2,285,12,295)(3,284,13,294)(4,283,14,293)(5,282,15,292)(6,281,16,291)(7,300,17,290)(8,299,18,289)(9,298,19,288)(10,297,20,287)(21,107,31,117)(22,106,32,116)(23,105,33,115)(24,104,34,114)(25,103,35,113)(26,102,36,112)(27,101,37,111)(28,120,38,110)(29,119,39,109)(30,118,40,108)(41,142,51,152)(42,141,52,151)(43,160,53,150)(44,159,54,149)(45,158,55,148)(46,157,56,147)(47,156,57,146)(48,155,58,145)(49,154,59,144)(50,153,60,143)(61,169,71,179)(62,168,72,178)(63,167,73,177)(64,166,74,176)(65,165,75,175)(66,164,76,174)(67,163,77,173)(68,162,78,172)(69,161,79,171)(70,180,80,170)(81,301,91,311)(82,320,92,310)(83,319,93,309)(84,318,94,308)(85,317,95,307)(86,316,96,306)(87,315,97,305)(88,314,98,304)(89,313,99,303)(90,312,100,302)(121,201,131,211)(122,220,132,210)(123,219,133,209)(124,218,134,208)(125,217,135,207)(126,216,136,206)(127,215,137,205)(128,214,138,204)(129,213,139,203)(130,212,140,202)(181,263,191,273)(182,262,192,272)(183,261,193,271)(184,280,194,270)(185,279,195,269)(186,278,196,268)(187,277,197,267)(188,276,198,266)(189,275,199,265)(190,274,200,264)(221,259,231,249)(222,258,232,248)(223,257,233,247)(224,256,234,246)(225,255,235,245)(226,254,236,244)(227,253,237,243)(228,252,238,242)(229,251,239,241)(230,250,240,260) );
G=PermutationGroup([[(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,81),(39,82),(40,83),(41,220),(42,201),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,208),(50,209),(51,210),(52,211),(53,212),(54,213),(55,214),(56,215),(57,216),(58,217),(59,218),(60,219),(101,312),(102,313),(103,314),(104,315),(105,316),(106,317),(107,318),(108,319),(109,320),(110,301),(111,302),(112,303),(113,304),(114,305),(115,306),(116,307),(117,308),(118,309),(119,310),(120,311),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,141),(132,142),(133,143),(134,144),(135,145),(136,146),(137,147),(138,148),(139,149),(140,150),(161,284),(162,285),(163,286),(164,287),(165,288),(166,289),(167,290),(168,291),(169,292),(170,293),(171,294),(172,295),(173,296),(174,297),(175,298),(176,299),(177,300),(178,281),(179,282),(180,283),(181,228),(182,229),(183,230),(184,231),(185,232),(186,233),(187,234),(188,235),(189,236),(190,237),(191,238),(192,239),(193,240),(194,221),(195,222),(196,223),(197,224),(198,225),(199,226),(200,227),(241,272),(242,273),(243,274),(244,275),(245,276),(246,277),(247,278),(248,279),(249,280),(250,261),(251,262),(252,263),(253,264),(254,265),(255,266),(256,267),(257,268),(258,269),(259,270),(260,271)], [(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)], [(1,22,60,228),(2,21,41,227),(3,40,42,226),(4,39,43,225),(5,38,44,224),(6,37,45,223),(7,36,46,222),(8,35,47,221),(9,34,48,240),(10,33,49,239),(11,32,50,238),(12,31,51,237),(13,30,52,236),(14,29,53,235),(15,28,54,234),(16,27,55,233),(17,26,56,232),(18,25,57,231),(19,24,58,230),(20,23,59,229),(61,91,213,187),(62,90,214,186),(63,89,215,185),(64,88,216,184),(65,87,217,183),(66,86,218,182),(67,85,219,181),(68,84,220,200),(69,83,201,199),(70,82,202,198),(71,81,203,197),(72,100,204,196),(73,99,205,195),(74,98,206,194),(75,97,207,193),(76,96,208,192),(77,95,209,191),(78,94,210,190),(79,93,211,189),(80,92,212,188),(101,296,247,153),(102,295,248,152),(103,294,249,151),(104,293,250,150),(105,292,251,149),(106,291,252,148),(107,290,253,147),(108,289,254,146),(109,288,255,145),(110,287,256,144),(111,286,257,143),(112,285,258,142),(113,284,259,141),(114,283,260,160),(115,282,241,159),(116,281,242,158),(117,300,243,157),(118,299,244,156),(119,298,245,155),(120,297,246,154),(121,314,171,280),(122,313,172,279),(123,312,173,278),(124,311,174,277),(125,310,175,276),(126,309,176,275),(127,308,177,274),(128,307,178,273),(129,306,179,272),(130,305,180,271),(131,304,161,270),(132,303,162,269),(133,302,163,268),(134,301,164,267),(135,320,165,266),(136,319,166,265),(137,318,167,264),(138,317,168,263),(139,316,169,262),(140,315,170,261)], [(1,286,11,296),(2,285,12,295),(3,284,13,294),(4,283,14,293),(5,282,15,292),(6,281,16,291),(7,300,17,290),(8,299,18,289),(9,298,19,288),(10,297,20,287),(21,107,31,117),(22,106,32,116),(23,105,33,115),(24,104,34,114),(25,103,35,113),(26,102,36,112),(27,101,37,111),(28,120,38,110),(29,119,39,109),(30,118,40,108),(41,142,51,152),(42,141,52,151),(43,160,53,150),(44,159,54,149),(45,158,55,148),(46,157,56,147),(47,156,57,146),(48,155,58,145),(49,154,59,144),(50,153,60,143),(61,169,71,179),(62,168,72,178),(63,167,73,177),(64,166,74,176),(65,165,75,175),(66,164,76,174),(67,163,77,173),(68,162,78,172),(69,161,79,171),(70,180,80,170),(81,301,91,311),(82,320,92,310),(83,319,93,309),(84,318,94,308),(85,317,95,307),(86,316,96,306),(87,315,97,305),(88,314,98,304),(89,313,99,303),(90,312,100,302),(121,201,131,211),(122,220,132,210),(123,219,133,209),(124,218,134,208),(125,217,135,207),(126,216,136,206),(127,215,137,205),(128,214,138,204),(129,213,139,203),(130,212,140,202),(181,263,191,273),(182,262,192,272),(183,261,193,271),(184,280,194,270),(185,279,195,269),(186,278,196,268),(187,277,197,267),(188,276,198,266),(189,275,199,265),(190,274,200,264),(221,259,231,249),(222,258,232,248),(223,257,233,247),(224,256,234,246),(225,255,235,245),(226,254,236,244),(227,253,237,243),(228,252,238,242),(229,251,239,241),(230,250,240,260)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | SD16 | Q16 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D20 | C40⋊C2 | Dic20 |
| kernel | C2×C20.44D4 | C20.44D4 | C2×C4⋊Dic5 | C22×C40 | C22×Dic10 | C2×Dic10 | C2×C20 | C22×C10 | C22×C8 | C2×C10 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 4 | 16 | 16 |
Matrix representation of C2×C20.44D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 |
| 0 | 38 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 32 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,25,0,0,0,0,0,23],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,1,0],[1,0,0,0,0,0,0,38,0,0,0,14,0,0,0,0,0,0,0,32,0,0,0,9,0] >;
C2×C20.44D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{44}D_4 % in TeX
G:=Group("C2xC20.44D4"); // GroupNames label
G:=SmallGroup(320,730);
// by ID
G=gap.SmallGroup(320,730);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^-1>;
// generators/relations