metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.2D10, C10.18C4≀C2, C8⋊C4.3D5, C4⋊Dic5.1C4, (C2×C20).224D4, (C2×C4).106D20, (C4×C20).11C22, C20.6Q8.4C2, C2.7(D20⋊4C4), C2.5(D20⋊7C4), C42.D5.1C2, C2.3(C4.12D20), C10.6(C4.10D4), C5⋊2(C42.2C22), C22.57(D10⋊C4), (C2×C4).11(C4×D5), (C5×C8⋊C4).7C2, (C2×C20).196(C2×C4), (C2×C4).207(C5⋊D4), (C2×C10).102(C22⋊C4), SmallGroup(320,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.2D10
G = < a,b,c,d | a4=b4=1, c10=a, d2=ba=ab, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a2bc9 >
Subgroups: 206 in 60 conjugacy classes, 25 normal (all characteristic)
C1, C2, C4, C22, C5, C8, C2×C4, C2×C4, C10, C42, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C8⋊C4, C8⋊C4, C42.C2, C5⋊2C8, C40, C2×Dic5, C2×C20, C42.2C22, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C42.D5, C5×C8⋊C4, C20.6Q8, C42.2D10
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C4.10D4, C4≀C2, C4×D5, D20, C5⋊D4, C42.2C22, D10⋊C4, D20⋊4C4, C4.12D20, D20⋊7C4, C42.2D10
►Regular action on 320 points
Generators in S320
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)(161 171 181 191)(162 172 182 192)(163 173 183 193)(164 174 184 194)(165 175 185 195)(166 176 186 196)(167 177 187 197)(168 178 188 198)(169 179 189 199)(170 180 190 200)(201 211 221 231)(202 212 222 232)(203 213 223 233)(204 214 224 234)(205 215 225 235)(206 216 226 236)(207 217 227 237)(208 218 228 238)(209 219 229 239)(210 220 230 240)(241 251 261 271)(242 252 262 272)(243 253 263 273)(244 254 264 274)(245 255 265 275)(246 256 266 276)(247 257 267 277)(248 258 268 278)(249 259 269 279)(250 260 270 280)(281 291 301 311)(282 292 302 312)(283 293 303 313)(284 294 304 314)(285 295 305 315)(286 296 306 316)(287 297 307 317)(288 298 308 318)(289 299 309 319)(290 300 310 320)
(1 105 289 231)(2 86 290 212)(3 107 291 233)(4 88 292 214)(5 109 293 235)(6 90 294 216)(7 111 295 237)(8 92 296 218)(9 113 297 239)(10 94 298 220)(11 115 299 201)(12 96 300 222)(13 117 301 203)(14 98 302 224)(15 119 303 205)(16 100 304 226)(17 81 305 207)(18 102 306 228)(19 83 307 209)(20 104 308 230)(21 85 309 211)(22 106 310 232)(23 87 311 213)(24 108 312 234)(25 89 313 215)(26 110 314 236)(27 91 315 217)(28 112 316 238)(29 93 317 219)(30 114 318 240)(31 95 319 221)(32 116 320 202)(33 97 281 223)(34 118 282 204)(35 99 283 225)(36 120 284 206)(37 101 285 227)(38 82 286 208)(39 103 287 229)(40 84 288 210)(41 145 247 192)(42 126 248 173)(43 147 249 194)(44 128 250 175)(45 149 251 196)(46 130 252 177)(47 151 253 198)(48 132 254 179)(49 153 255 200)(50 134 256 181)(51 155 257 162)(52 136 258 183)(53 157 259 164)(54 138 260 185)(55 159 261 166)(56 140 262 187)(57 121 263 168)(58 142 264 189)(59 123 265 170)(60 144 266 191)(61 125 267 172)(62 146 268 193)(63 127 269 174)(64 148 270 195)(65 129 271 176)(66 150 272 197)(67 131 273 178)(68 152 274 199)(69 133 275 180)(70 154 276 161)(71 135 277 182)(72 156 278 163)(73 137 279 184)(74 158 280 165)(75 139 241 186)(76 160 242 167)(77 141 243 188)(78 122 244 169)(79 143 245 190)(80 124 246 171)
(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 47 115 121 309 273 221 188)(2 160 96 252 310 187 202 66)(3 251 117 166 311 65 223 139)(4 165 98 44 312 138 204 270)(5 43 119 157 313 269 225 184)(6 156 100 248 314 183 206 62)(7 247 81 162 315 61 227 135)(8 161 102 80 316 134 208 266)(9 79 83 153 317 265 229 180)(10 152 104 244 318 179 210 58)(11 243 85 198 319 57 231 131)(12 197 106 76 320 130 212 262)(13 75 87 149 281 261 233 176)(14 148 108 280 282 175 214 54)(15 279 89 194 283 53 235 127)(16 193 110 72 284 126 216 258)(17 71 91 145 285 257 237 172)(18 144 112 276 286 171 218 50)(19 275 93 190 287 49 239 123)(20 189 114 68 288 122 220 254)(21 67 95 141 289 253 201 168)(22 140 116 272 290 167 222 46)(23 271 97 186 291 45 203 159)(24 185 118 64 292 158 224 250)(25 63 99 137 293 249 205 164)(26 136 120 268 294 163 226 42)(27 267 101 182 295 41 207 155)(28 181 82 60 296 154 228 246)(29 59 103 133 297 245 209 200)(30 132 84 264 298 199 230 78)(31 263 105 178 299 77 211 151)(32 177 86 56 300 150 232 242)(33 55 107 129 301 241 213 196)(34 128 88 260 302 195 234 74)(35 259 109 174 303 73 215 147)(36 173 90 52 304 146 236 278)(37 51 111 125 305 277 217 192)(38 124 92 256 306 191 238 70)(39 255 113 170 307 69 219 143)(40 169 94 48 308 142 240 274)
G:=sub<Sym(320)| (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160)(161,171,181,191)(162,172,182,192)(163,173,183,193)(164,174,184,194)(165,175,185,195)(166,176,186,196)(167,177,187,197)(168,178,188,198)(169,179,189,199)(170,180,190,200)(201,211,221,231)(202,212,222,232)(203,213,223,233)(204,214,224,234)(205,215,225,235)(206,216,226,236)(207,217,227,237)(208,218,228,238)(209,219,229,239)(210,220,230,240)(241,251,261,271)(242,252,262,272)(243,253,263,273)(244,254,264,274)(245,255,265,275)(246,256,266,276)(247,257,267,277)(248,258,268,278)(249,259,269,279)(250,260,270,280)(281,291,301,311)(282,292,302,312)(283,293,303,313)(284,294,304,314)(285,295,305,315)(286,296,306,316)(287,297,307,317)(288,298,308,318)(289,299,309,319)(290,300,310,320), (1,105,289,231)(2,86,290,212)(3,107,291,233)(4,88,292,214)(5,109,293,235)(6,90,294,216)(7,111,295,237)(8,92,296,218)(9,113,297,239)(10,94,298,220)(11,115,299,201)(12,96,300,222)(13,117,301,203)(14,98,302,224)(15,119,303,205)(16,100,304,226)(17,81,305,207)(18,102,306,228)(19,83,307,209)(20,104,308,230)(21,85,309,211)(22,106,310,232)(23,87,311,213)(24,108,312,234)(25,89,313,215)(26,110,314,236)(27,91,315,217)(28,112,316,238)(29,93,317,219)(30,114,318,240)(31,95,319,221)(32,116,320,202)(33,97,281,223)(34,118,282,204)(35,99,283,225)(36,120,284,206)(37,101,285,227)(38,82,286,208)(39,103,287,229)(40,84,288,210)(41,145,247,192)(42,126,248,173)(43,147,249,194)(44,128,250,175)(45,149,251,196)(46,130,252,177)(47,151,253,198)(48,132,254,179)(49,153,255,200)(50,134,256,181)(51,155,257,162)(52,136,258,183)(53,157,259,164)(54,138,260,185)(55,159,261,166)(56,140,262,187)(57,121,263,168)(58,142,264,189)(59,123,265,170)(60,144,266,191)(61,125,267,172)(62,146,268,193)(63,127,269,174)(64,148,270,195)(65,129,271,176)(66,150,272,197)(67,131,273,178)(68,152,274,199)(69,133,275,180)(70,154,276,161)(71,135,277,182)(72,156,278,163)(73,137,279,184)(74,158,280,165)(75,139,241,186)(76,160,242,167)(77,141,243,188)(78,122,244,169)(79,143,245,190)(80,124,246,171), (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), (1,47,115,121,309,273,221,188)(2,160,96,252,310,187,202,66)(3,251,117,166,311,65,223,139)(4,165,98,44,312,138,204,270)(5,43,119,157,313,269,225,184)(6,156,100,248,314,183,206,62)(7,247,81,162,315,61,227,135)(8,161,102,80,316,134,208,266)(9,79,83,153,317,265,229,180)(10,152,104,244,318,179,210,58)(11,243,85,198,319,57,231,131)(12,197,106,76,320,130,212,262)(13,75,87,149,281,261,233,176)(14,148,108,280,282,175,214,54)(15,279,89,194,283,53,235,127)(16,193,110,72,284,126,216,258)(17,71,91,145,285,257,237,172)(18,144,112,276,286,171,218,50)(19,275,93,190,287,49,239,123)(20,189,114,68,288,122,220,254)(21,67,95,141,289,253,201,168)(22,140,116,272,290,167,222,46)(23,271,97,186,291,45,203,159)(24,185,118,64,292,158,224,250)(25,63,99,137,293,249,205,164)(26,136,120,268,294,163,226,42)(27,267,101,182,295,41,207,155)(28,181,82,60,296,154,228,246)(29,59,103,133,297,245,209,200)(30,132,84,264,298,199,230,78)(31,263,105,178,299,77,211,151)(32,177,86,56,300,150,232,242)(33,55,107,129,301,241,213,196)(34,128,88,260,302,195,234,74)(35,259,109,174,303,73,215,147)(36,173,90,52,304,146,236,278)(37,51,111,125,305,277,217,192)(38,124,92,256,306,191,238,70)(39,255,113,170,307,69,219,143)(40,169,94,48,308,142,240,274)>;
G:=Group( (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160)(161,171,181,191)(162,172,182,192)(163,173,183,193)(164,174,184,194)(165,175,185,195)(166,176,186,196)(167,177,187,197)(168,178,188,198)(169,179,189,199)(170,180,190,200)(201,211,221,231)(202,212,222,232)(203,213,223,233)(204,214,224,234)(205,215,225,235)(206,216,226,236)(207,217,227,237)(208,218,228,238)(209,219,229,239)(210,220,230,240)(241,251,261,271)(242,252,262,272)(243,253,263,273)(244,254,264,274)(245,255,265,275)(246,256,266,276)(247,257,267,277)(248,258,268,278)(249,259,269,279)(250,260,270,280)(281,291,301,311)(282,292,302,312)(283,293,303,313)(284,294,304,314)(285,295,305,315)(286,296,306,316)(287,297,307,317)(288,298,308,318)(289,299,309,319)(290,300,310,320), (1,105,289,231)(2,86,290,212)(3,107,291,233)(4,88,292,214)(5,109,293,235)(6,90,294,216)(7,111,295,237)(8,92,296,218)(9,113,297,239)(10,94,298,220)(11,115,299,201)(12,96,300,222)(13,117,301,203)(14,98,302,224)(15,119,303,205)(16,100,304,226)(17,81,305,207)(18,102,306,228)(19,83,307,209)(20,104,308,230)(21,85,309,211)(22,106,310,232)(23,87,311,213)(24,108,312,234)(25,89,313,215)(26,110,314,236)(27,91,315,217)(28,112,316,238)(29,93,317,219)(30,114,318,240)(31,95,319,221)(32,116,320,202)(33,97,281,223)(34,118,282,204)(35,99,283,225)(36,120,284,206)(37,101,285,227)(38,82,286,208)(39,103,287,229)(40,84,288,210)(41,145,247,192)(42,126,248,173)(43,147,249,194)(44,128,250,175)(45,149,251,196)(46,130,252,177)(47,151,253,198)(48,132,254,179)(49,153,255,200)(50,134,256,181)(51,155,257,162)(52,136,258,183)(53,157,259,164)(54,138,260,185)(55,159,261,166)(56,140,262,187)(57,121,263,168)(58,142,264,189)(59,123,265,170)(60,144,266,191)(61,125,267,172)(62,146,268,193)(63,127,269,174)(64,148,270,195)(65,129,271,176)(66,150,272,197)(67,131,273,178)(68,152,274,199)(69,133,275,180)(70,154,276,161)(71,135,277,182)(72,156,278,163)(73,137,279,184)(74,158,280,165)(75,139,241,186)(76,160,242,167)(77,141,243,188)(78,122,244,169)(79,143,245,190)(80,124,246,171), (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), (1,47,115,121,309,273,221,188)(2,160,96,252,310,187,202,66)(3,251,117,166,311,65,223,139)(4,165,98,44,312,138,204,270)(5,43,119,157,313,269,225,184)(6,156,100,248,314,183,206,62)(7,247,81,162,315,61,227,135)(8,161,102,80,316,134,208,266)(9,79,83,153,317,265,229,180)(10,152,104,244,318,179,210,58)(11,243,85,198,319,57,231,131)(12,197,106,76,320,130,212,262)(13,75,87,149,281,261,233,176)(14,148,108,280,282,175,214,54)(15,279,89,194,283,53,235,127)(16,193,110,72,284,126,216,258)(17,71,91,145,285,257,237,172)(18,144,112,276,286,171,218,50)(19,275,93,190,287,49,239,123)(20,189,114,68,288,122,220,254)(21,67,95,141,289,253,201,168)(22,140,116,272,290,167,222,46)(23,271,97,186,291,45,203,159)(24,185,118,64,292,158,224,250)(25,63,99,137,293,249,205,164)(26,136,120,268,294,163,226,42)(27,267,101,182,295,41,207,155)(28,181,82,60,296,154,228,246)(29,59,103,133,297,245,209,200)(30,132,84,264,298,199,230,78)(31,263,105,178,299,77,211,151)(32,177,86,56,300,150,232,242)(33,55,107,129,301,241,213,196)(34,128,88,260,302,195,234,74)(35,259,109,174,303,73,215,147)(36,173,90,52,304,146,236,278)(37,51,111,125,305,277,217,192)(38,124,92,256,306,191,238,70)(39,255,113,170,307,69,219,143)(40,169,94,48,308,142,240,274) );
G=PermutationGroup([[(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160),(161,171,181,191),(162,172,182,192),(163,173,183,193),(164,174,184,194),(165,175,185,195),(166,176,186,196),(167,177,187,197),(168,178,188,198),(169,179,189,199),(170,180,190,200),(201,211,221,231),(202,212,222,232),(203,213,223,233),(204,214,224,234),(205,215,225,235),(206,216,226,236),(207,217,227,237),(208,218,228,238),(209,219,229,239),(210,220,230,240),(241,251,261,271),(242,252,262,272),(243,253,263,273),(244,254,264,274),(245,255,265,275),(246,256,266,276),(247,257,267,277),(248,258,268,278),(249,259,269,279),(250,260,270,280),(281,291,301,311),(282,292,302,312),(283,293,303,313),(284,294,304,314),(285,295,305,315),(286,296,306,316),(287,297,307,317),(288,298,308,318),(289,299,309,319),(290,300,310,320)], [(1,105,289,231),(2,86,290,212),(3,107,291,233),(4,88,292,214),(5,109,293,235),(6,90,294,216),(7,111,295,237),(8,92,296,218),(9,113,297,239),(10,94,298,220),(11,115,299,201),(12,96,300,222),(13,117,301,203),(14,98,302,224),(15,119,303,205),(16,100,304,226),(17,81,305,207),(18,102,306,228),(19,83,307,209),(20,104,308,230),(21,85,309,211),(22,106,310,232),(23,87,311,213),(24,108,312,234),(25,89,313,215),(26,110,314,236),(27,91,315,217),(28,112,316,238),(29,93,317,219),(30,114,318,240),(31,95,319,221),(32,116,320,202),(33,97,281,223),(34,118,282,204),(35,99,283,225),(36,120,284,206),(37,101,285,227),(38,82,286,208),(39,103,287,229),(40,84,288,210),(41,145,247,192),(42,126,248,173),(43,147,249,194),(44,128,250,175),(45,149,251,196),(46,130,252,177),(47,151,253,198),(48,132,254,179),(49,153,255,200),(50,134,256,181),(51,155,257,162),(52,136,258,183),(53,157,259,164),(54,138,260,185),(55,159,261,166),(56,140,262,187),(57,121,263,168),(58,142,264,189),(59,123,265,170),(60,144,266,191),(61,125,267,172),(62,146,268,193),(63,127,269,174),(64,148,270,195),(65,129,271,176),(66,150,272,197),(67,131,273,178),(68,152,274,199),(69,133,275,180),(70,154,276,161),(71,135,277,182),(72,156,278,163),(73,137,279,184),(74,158,280,165),(75,139,241,186),(76,160,242,167),(77,141,243,188),(78,122,244,169),(79,143,245,190),(80,124,246,171)], [(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)], [(1,47,115,121,309,273,221,188),(2,160,96,252,310,187,202,66),(3,251,117,166,311,65,223,139),(4,165,98,44,312,138,204,270),(5,43,119,157,313,269,225,184),(6,156,100,248,314,183,206,62),(7,247,81,162,315,61,227,135),(8,161,102,80,316,134,208,266),(9,79,83,153,317,265,229,180),(10,152,104,244,318,179,210,58),(11,243,85,198,319,57,231,131),(12,197,106,76,320,130,212,262),(13,75,87,149,281,261,233,176),(14,148,108,280,282,175,214,54),(15,279,89,194,283,53,235,127),(16,193,110,72,284,126,216,258),(17,71,91,145,285,257,237,172),(18,144,112,276,286,171,218,50),(19,275,93,190,287,49,239,123),(20,189,114,68,288,122,220,254),(21,67,95,141,289,253,201,168),(22,140,116,272,290,167,222,46),(23,271,97,186,291,45,203,159),(24,185,118,64,292,158,224,250),(25,63,99,137,293,249,205,164),(26,136,120,268,294,163,226,42),(27,267,101,182,295,41,207,155),(28,181,82,60,296,154,228,246),(29,59,103,133,297,245,209,200),(30,132,84,264,298,199,230,78),(31,263,105,178,299,77,211,151),(32,177,86,56,300,150,232,242),(33,55,107,129,301,241,213,196),(34,128,88,260,302,195,234,74),(35,259,109,174,303,73,215,147),(36,173,90,52,304,146,236,278),(37,51,111,125,305,277,217,192),(38,124,92,256,306,191,238,70),(39,255,113,170,307,69,219,143),(40,169,94,48,308,142,240,274)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D5 | D10 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | D20⋊4C4 | C4.10D4 | C4.12D20 | D20⋊7C4 |
| kernel | C42.2D10 | C42.D5 | C5×C8⋊C4 | C20.6Q8 | C4⋊Dic5 | C2×C20 | C8⋊C4 | C42 | C10 | C2×C4 | C2×C4 | C2×C4 | C2 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C42.2D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
| 1 | 39 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 32 | 30 |
| 24 | 13 | 0 | 0 |
| 10 | 17 | 0 | 0 |
| 0 | 0 | 6 | 23 |
| 0 | 0 | 18 | 9 |
| 0 | 33 | 0 | 0 |
| 4 | 33 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,24,40,0,0,1,17],[1,1,0,0,39,40,0,0,0,0,11,32,0,0,9,30],[24,10,0,0,13,17,0,0,0,0,6,18,0,0,23,9],[0,4,0,0,33,33,0,0,0,0,0,3,0,0,3,0] >;
C42.2D10 in GAP, Magma, Sage, TeX
C_4^2._2D_{10} % in TeX
G:=Group("C4^2.2D10"); // GroupNames label
G:=SmallGroup(320,23);
// by ID
G=gap.SmallGroup(320,23);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,1571,570,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a,d^2=b*a=a*b,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^9>;
// generators/relations