metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.210D10, C20.25M4(2), C5⋊2C8⋊11Q8, C5⋊8(C8⋊4Q8), (C4×Q8).5D5, C4.58(Q8×D5), C4⋊C4.9Dic5, (Q8×C20).6C2, C10.36(C4×Q8), C2.5(Q8×Dic5), (Q8×C10).19C4, C20.116(C2×Q8), (C2×Q8).5Dic5, C20⋊3C8.18C2, C10.66(C8○D4), (C4×C20).95C22, C4.3(C4.Dic5), C20.339(C4○D4), (C2×C20).852C23, C4.59(Q8⋊2D5), C10.76(C2×M4(2)), C2.8(D4.Dic5), C42.D5.3C2, C22.47(C22×Dic5), (C5×C4⋊C4).27C4, (C4×C5⋊2C8).8C2, (C2×C20).339(C2×C4), (C2×C4).45(C2×Dic5), C2.10(C2×C4.Dic5), (C2×C4).794(C22×D5), (C2×C10).290(C22×C4), (C2×C5⋊2C8).206C22, SmallGroup(320,651)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.210D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >
Subgroups: 190 in 94 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C20, C2×C10, C4×C8, C8⋊C4, C4⋊C8, C4×Q8, C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C5×Q8, C8⋊4Q8, C2×C5⋊2C8, C2×C5⋊2C8, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Q8×C10, C4×C5⋊2C8, C42.D5, C20⋊3C8, C20⋊3C8, Q8×C20, C42.210D10
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, M4(2), C22×C4, C2×Q8, C4○D4, Dic5, D10, C4×Q8, C2×M4(2), C8○D4, C2×Dic5, C22×D5, C8⋊4Q8, C4.Dic5, Q8×D5, Q8⋊2D5, C22×Dic5, C2×C4.Dic5, Q8×Dic5, D4.Dic5, C42.210D10
►Regular action on 320 points
Generators in S320
(1 187 11 197)(2 198 12 188)(3 189 13 199)(4 200 14 190)(5 191 15 181)(6 182 16 192)(7 193 17 183)(8 184 18 194)(9 195 19 185)(10 186 20 196)(21 204 31 214)(22 215 32 205)(23 206 33 216)(24 217 34 207)(25 208 35 218)(26 219 36 209)(27 210 37 220)(28 201 38 211)(29 212 39 202)(30 203 40 213)(41 176 51 166)(42 167 52 177)(43 178 53 168)(44 169 54 179)(45 180 55 170)(46 171 56 161)(47 162 57 172)(48 173 58 163)(49 164 59 174)(50 175 60 165)(61 105 71 115)(62 116 72 106)(63 107 73 117)(64 118 74 108)(65 109 75 119)(66 120 76 110)(67 111 77 101)(68 102 78 112)(69 113 79 103)(70 104 80 114)(81 159 91 149)(82 150 92 160)(83 141 93 151)(84 152 94 142)(85 143 95 153)(86 154 96 144)(87 145 97 155)(88 156 98 146)(89 147 99 157)(90 158 100 148)(121 319 131 309)(122 310 132 320)(123 301 133 311)(124 312 134 302)(125 303 135 313)(126 314 136 304)(127 305 137 315)(128 316 138 306)(129 307 139 317)(130 318 140 308)(221 275 231 265)(222 266 232 276)(223 277 233 267)(224 268 234 278)(225 279 235 269)(226 270 236 280)(227 261 237 271)(228 272 238 262)(229 263 239 273)(230 274 240 264)(241 288 251 298)(242 299 252 289)(243 290 253 300)(244 281 254 291)(245 292 255 282)(246 283 256 293)(247 294 257 284)(248 285 258 295)(249 296 259 286)(250 287 260 297)
(1 130 164 65)(2 131 165 66)(3 132 166 67)(4 133 167 68)(5 134 168 69)(6 135 169 70)(7 136 170 71)(8 137 171 72)(9 138 172 73)(10 139 173 74)(11 140 174 75)(12 121 175 76)(13 122 176 77)(14 123 177 78)(15 124 178 79)(16 125 179 80)(17 126 180 61)(18 127 161 62)(19 128 162 63)(20 129 163 64)(21 255 153 231)(22 256 154 232)(23 257 155 233)(24 258 156 234)(25 259 157 235)(26 260 158 236)(27 241 159 237)(28 242 160 238)(29 243 141 239)(30 244 142 240)(31 245 143 221)(32 246 144 222)(33 247 145 223)(34 248 146 224)(35 249 147 225)(36 250 148 226)(37 251 149 227)(38 252 150 228)(39 253 151 229)(40 254 152 230)(41 111 189 320)(42 112 190 301)(43 113 191 302)(44 114 192 303)(45 115 193 304)(46 116 194 305)(47 117 195 306)(48 118 196 307)(49 119 197 308)(50 120 198 309)(51 101 199 310)(52 102 200 311)(53 103 181 312)(54 104 182 313)(55 105 183 314)(56 106 184 315)(57 107 185 316)(58 108 186 317)(59 109 187 318)(60 110 188 319)(81 261 220 298)(82 262 201 299)(83 263 202 300)(84 264 203 281)(85 265 204 282)(86 266 205 283)(87 267 206 284)(88 268 207 285)(89 269 208 286)(90 270 209 287)(91 271 210 288)(92 272 211 289)(93 273 212 290)(94 274 213 291)(95 275 214 292)(96 276 215 293)(97 277 216 294)(98 278 217 295)(99 279 218 296)(100 280 219 297)
(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 98 75 285 164 217 140 268)(2 206 76 277 165 87 121 294)(3 96 77 283 166 215 122 266)(4 204 78 275 167 85 123 292)(5 94 79 281 168 213 124 264)(6 202 80 273 169 83 125 290)(7 92 61 299 170 211 126 262)(8 220 62 271 171 81 127 288)(9 90 63 297 172 209 128 280)(10 218 64 269 173 99 129 286)(11 88 65 295 174 207 130 278)(12 216 66 267 175 97 131 284)(13 86 67 293 176 205 132 276)(14 214 68 265 177 95 133 282)(15 84 69 291 178 203 134 274)(16 212 70 263 179 93 135 300)(17 82 71 289 180 201 136 272)(18 210 72 261 161 91 137 298)(19 100 73 287 162 219 138 270)(20 208 74 279 163 89 139 296)(21 112 221 52 153 301 245 200)(22 310 222 189 154 101 246 41)(23 110 223 50 155 319 247 198)(24 308 224 187 156 119 248 59)(25 108 225 48 157 317 249 196)(26 306 226 185 158 117 250 57)(27 106 227 46 159 315 251 194)(28 304 228 183 160 115 252 55)(29 104 229 44 141 313 253 192)(30 302 230 181 142 113 254 53)(31 102 231 42 143 311 255 190)(32 320 232 199 144 111 256 51)(33 120 233 60 145 309 257 188)(34 318 234 197 146 109 258 49)(35 118 235 58 147 307 259 186)(36 316 236 195 148 107 260 47)(37 116 237 56 149 305 241 184)(38 314 238 193 150 105 242 45)(39 114 239 54 151 303 243 182)(40 312 240 191 152 103 244 43)
G:=sub<Sym(320)| (1,187,11,197)(2,198,12,188)(3,189,13,199)(4,200,14,190)(5,191,15,181)(6,182,16,192)(7,193,17,183)(8,184,18,194)(9,195,19,185)(10,186,20,196)(21,204,31,214)(22,215,32,205)(23,206,33,216)(24,217,34,207)(25,208,35,218)(26,219,36,209)(27,210,37,220)(28,201,38,211)(29,212,39,202)(30,203,40,213)(41,176,51,166)(42,167,52,177)(43,178,53,168)(44,169,54,179)(45,180,55,170)(46,171,56,161)(47,162,57,172)(48,173,58,163)(49,164,59,174)(50,175,60,165)(61,105,71,115)(62,116,72,106)(63,107,73,117)(64,118,74,108)(65,109,75,119)(66,120,76,110)(67,111,77,101)(68,102,78,112)(69,113,79,103)(70,104,80,114)(81,159,91,149)(82,150,92,160)(83,141,93,151)(84,152,94,142)(85,143,95,153)(86,154,96,144)(87,145,97,155)(88,156,98,146)(89,147,99,157)(90,158,100,148)(121,319,131,309)(122,310,132,320)(123,301,133,311)(124,312,134,302)(125,303,135,313)(126,314,136,304)(127,305,137,315)(128,316,138,306)(129,307,139,317)(130,318,140,308)(221,275,231,265)(222,266,232,276)(223,277,233,267)(224,268,234,278)(225,279,235,269)(226,270,236,280)(227,261,237,271)(228,272,238,262)(229,263,239,273)(230,274,240,264)(241,288,251,298)(242,299,252,289)(243,290,253,300)(244,281,254,291)(245,292,255,282)(246,283,256,293)(247,294,257,284)(248,285,258,295)(249,296,259,286)(250,287,260,297), (1,130,164,65)(2,131,165,66)(3,132,166,67)(4,133,167,68)(5,134,168,69)(6,135,169,70)(7,136,170,71)(8,137,171,72)(9,138,172,73)(10,139,173,74)(11,140,174,75)(12,121,175,76)(13,122,176,77)(14,123,177,78)(15,124,178,79)(16,125,179,80)(17,126,180,61)(18,127,161,62)(19,128,162,63)(20,129,163,64)(21,255,153,231)(22,256,154,232)(23,257,155,233)(24,258,156,234)(25,259,157,235)(26,260,158,236)(27,241,159,237)(28,242,160,238)(29,243,141,239)(30,244,142,240)(31,245,143,221)(32,246,144,222)(33,247,145,223)(34,248,146,224)(35,249,147,225)(36,250,148,226)(37,251,149,227)(38,252,150,228)(39,253,151,229)(40,254,152,230)(41,111,189,320)(42,112,190,301)(43,113,191,302)(44,114,192,303)(45,115,193,304)(46,116,194,305)(47,117,195,306)(48,118,196,307)(49,119,197,308)(50,120,198,309)(51,101,199,310)(52,102,200,311)(53,103,181,312)(54,104,182,313)(55,105,183,314)(56,106,184,315)(57,107,185,316)(58,108,186,317)(59,109,187,318)(60,110,188,319)(81,261,220,298)(82,262,201,299)(83,263,202,300)(84,264,203,281)(85,265,204,282)(86,266,205,283)(87,267,206,284)(88,268,207,285)(89,269,208,286)(90,270,209,287)(91,271,210,288)(92,272,211,289)(93,273,212,290)(94,274,213,291)(95,275,214,292)(96,276,215,293)(97,277,216,294)(98,278,217,295)(99,279,218,296)(100,280,219,297), (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), (1,98,75,285,164,217,140,268)(2,206,76,277,165,87,121,294)(3,96,77,283,166,215,122,266)(4,204,78,275,167,85,123,292)(5,94,79,281,168,213,124,264)(6,202,80,273,169,83,125,290)(7,92,61,299,170,211,126,262)(8,220,62,271,171,81,127,288)(9,90,63,297,172,209,128,280)(10,218,64,269,173,99,129,286)(11,88,65,295,174,207,130,278)(12,216,66,267,175,97,131,284)(13,86,67,293,176,205,132,276)(14,214,68,265,177,95,133,282)(15,84,69,291,178,203,134,274)(16,212,70,263,179,93,135,300)(17,82,71,289,180,201,136,272)(18,210,72,261,161,91,137,298)(19,100,73,287,162,219,138,270)(20,208,74,279,163,89,139,296)(21,112,221,52,153,301,245,200)(22,310,222,189,154,101,246,41)(23,110,223,50,155,319,247,198)(24,308,224,187,156,119,248,59)(25,108,225,48,157,317,249,196)(26,306,226,185,158,117,250,57)(27,106,227,46,159,315,251,194)(28,304,228,183,160,115,252,55)(29,104,229,44,141,313,253,192)(30,302,230,181,142,113,254,53)(31,102,231,42,143,311,255,190)(32,320,232,199,144,111,256,51)(33,120,233,60,145,309,257,188)(34,318,234,197,146,109,258,49)(35,118,235,58,147,307,259,186)(36,316,236,195,148,107,260,47)(37,116,237,56,149,305,241,184)(38,314,238,193,150,105,242,45)(39,114,239,54,151,303,243,182)(40,312,240,191,152,103,244,43)>;
G:=Group( (1,187,11,197)(2,198,12,188)(3,189,13,199)(4,200,14,190)(5,191,15,181)(6,182,16,192)(7,193,17,183)(8,184,18,194)(9,195,19,185)(10,186,20,196)(21,204,31,214)(22,215,32,205)(23,206,33,216)(24,217,34,207)(25,208,35,218)(26,219,36,209)(27,210,37,220)(28,201,38,211)(29,212,39,202)(30,203,40,213)(41,176,51,166)(42,167,52,177)(43,178,53,168)(44,169,54,179)(45,180,55,170)(46,171,56,161)(47,162,57,172)(48,173,58,163)(49,164,59,174)(50,175,60,165)(61,105,71,115)(62,116,72,106)(63,107,73,117)(64,118,74,108)(65,109,75,119)(66,120,76,110)(67,111,77,101)(68,102,78,112)(69,113,79,103)(70,104,80,114)(81,159,91,149)(82,150,92,160)(83,141,93,151)(84,152,94,142)(85,143,95,153)(86,154,96,144)(87,145,97,155)(88,156,98,146)(89,147,99,157)(90,158,100,148)(121,319,131,309)(122,310,132,320)(123,301,133,311)(124,312,134,302)(125,303,135,313)(126,314,136,304)(127,305,137,315)(128,316,138,306)(129,307,139,317)(130,318,140,308)(221,275,231,265)(222,266,232,276)(223,277,233,267)(224,268,234,278)(225,279,235,269)(226,270,236,280)(227,261,237,271)(228,272,238,262)(229,263,239,273)(230,274,240,264)(241,288,251,298)(242,299,252,289)(243,290,253,300)(244,281,254,291)(245,292,255,282)(246,283,256,293)(247,294,257,284)(248,285,258,295)(249,296,259,286)(250,287,260,297), (1,130,164,65)(2,131,165,66)(3,132,166,67)(4,133,167,68)(5,134,168,69)(6,135,169,70)(7,136,170,71)(8,137,171,72)(9,138,172,73)(10,139,173,74)(11,140,174,75)(12,121,175,76)(13,122,176,77)(14,123,177,78)(15,124,178,79)(16,125,179,80)(17,126,180,61)(18,127,161,62)(19,128,162,63)(20,129,163,64)(21,255,153,231)(22,256,154,232)(23,257,155,233)(24,258,156,234)(25,259,157,235)(26,260,158,236)(27,241,159,237)(28,242,160,238)(29,243,141,239)(30,244,142,240)(31,245,143,221)(32,246,144,222)(33,247,145,223)(34,248,146,224)(35,249,147,225)(36,250,148,226)(37,251,149,227)(38,252,150,228)(39,253,151,229)(40,254,152,230)(41,111,189,320)(42,112,190,301)(43,113,191,302)(44,114,192,303)(45,115,193,304)(46,116,194,305)(47,117,195,306)(48,118,196,307)(49,119,197,308)(50,120,198,309)(51,101,199,310)(52,102,200,311)(53,103,181,312)(54,104,182,313)(55,105,183,314)(56,106,184,315)(57,107,185,316)(58,108,186,317)(59,109,187,318)(60,110,188,319)(81,261,220,298)(82,262,201,299)(83,263,202,300)(84,264,203,281)(85,265,204,282)(86,266,205,283)(87,267,206,284)(88,268,207,285)(89,269,208,286)(90,270,209,287)(91,271,210,288)(92,272,211,289)(93,273,212,290)(94,274,213,291)(95,275,214,292)(96,276,215,293)(97,277,216,294)(98,278,217,295)(99,279,218,296)(100,280,219,297), (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), (1,98,75,285,164,217,140,268)(2,206,76,277,165,87,121,294)(3,96,77,283,166,215,122,266)(4,204,78,275,167,85,123,292)(5,94,79,281,168,213,124,264)(6,202,80,273,169,83,125,290)(7,92,61,299,170,211,126,262)(8,220,62,271,171,81,127,288)(9,90,63,297,172,209,128,280)(10,218,64,269,173,99,129,286)(11,88,65,295,174,207,130,278)(12,216,66,267,175,97,131,284)(13,86,67,293,176,205,132,276)(14,214,68,265,177,95,133,282)(15,84,69,291,178,203,134,274)(16,212,70,263,179,93,135,300)(17,82,71,289,180,201,136,272)(18,210,72,261,161,91,137,298)(19,100,73,287,162,219,138,270)(20,208,74,279,163,89,139,296)(21,112,221,52,153,301,245,200)(22,310,222,189,154,101,246,41)(23,110,223,50,155,319,247,198)(24,308,224,187,156,119,248,59)(25,108,225,48,157,317,249,196)(26,306,226,185,158,117,250,57)(27,106,227,46,159,315,251,194)(28,304,228,183,160,115,252,55)(29,104,229,44,141,313,253,192)(30,302,230,181,142,113,254,53)(31,102,231,42,143,311,255,190)(32,320,232,199,144,111,256,51)(33,120,233,60,145,309,257,188)(34,318,234,197,146,109,258,49)(35,118,235,58,147,307,259,186)(36,316,236,195,148,107,260,47)(37,116,237,56,149,305,241,184)(38,314,238,193,150,105,242,45)(39,114,239,54,151,303,243,182)(40,312,240,191,152,103,244,43) );
G=PermutationGroup([[(1,187,11,197),(2,198,12,188),(3,189,13,199),(4,200,14,190),(5,191,15,181),(6,182,16,192),(7,193,17,183),(8,184,18,194),(9,195,19,185),(10,186,20,196),(21,204,31,214),(22,215,32,205),(23,206,33,216),(24,217,34,207),(25,208,35,218),(26,219,36,209),(27,210,37,220),(28,201,38,211),(29,212,39,202),(30,203,40,213),(41,176,51,166),(42,167,52,177),(43,178,53,168),(44,169,54,179),(45,180,55,170),(46,171,56,161),(47,162,57,172),(48,173,58,163),(49,164,59,174),(50,175,60,165),(61,105,71,115),(62,116,72,106),(63,107,73,117),(64,118,74,108),(65,109,75,119),(66,120,76,110),(67,111,77,101),(68,102,78,112),(69,113,79,103),(70,104,80,114),(81,159,91,149),(82,150,92,160),(83,141,93,151),(84,152,94,142),(85,143,95,153),(86,154,96,144),(87,145,97,155),(88,156,98,146),(89,147,99,157),(90,158,100,148),(121,319,131,309),(122,310,132,320),(123,301,133,311),(124,312,134,302),(125,303,135,313),(126,314,136,304),(127,305,137,315),(128,316,138,306),(129,307,139,317),(130,318,140,308),(221,275,231,265),(222,266,232,276),(223,277,233,267),(224,268,234,278),(225,279,235,269),(226,270,236,280),(227,261,237,271),(228,272,238,262),(229,263,239,273),(230,274,240,264),(241,288,251,298),(242,299,252,289),(243,290,253,300),(244,281,254,291),(245,292,255,282),(246,283,256,293),(247,294,257,284),(248,285,258,295),(249,296,259,286),(250,287,260,297)], [(1,130,164,65),(2,131,165,66),(3,132,166,67),(4,133,167,68),(5,134,168,69),(6,135,169,70),(7,136,170,71),(8,137,171,72),(9,138,172,73),(10,139,173,74),(11,140,174,75),(12,121,175,76),(13,122,176,77),(14,123,177,78),(15,124,178,79),(16,125,179,80),(17,126,180,61),(18,127,161,62),(19,128,162,63),(20,129,163,64),(21,255,153,231),(22,256,154,232),(23,257,155,233),(24,258,156,234),(25,259,157,235),(26,260,158,236),(27,241,159,237),(28,242,160,238),(29,243,141,239),(30,244,142,240),(31,245,143,221),(32,246,144,222),(33,247,145,223),(34,248,146,224),(35,249,147,225),(36,250,148,226),(37,251,149,227),(38,252,150,228),(39,253,151,229),(40,254,152,230),(41,111,189,320),(42,112,190,301),(43,113,191,302),(44,114,192,303),(45,115,193,304),(46,116,194,305),(47,117,195,306),(48,118,196,307),(49,119,197,308),(50,120,198,309),(51,101,199,310),(52,102,200,311),(53,103,181,312),(54,104,182,313),(55,105,183,314),(56,106,184,315),(57,107,185,316),(58,108,186,317),(59,109,187,318),(60,110,188,319),(81,261,220,298),(82,262,201,299),(83,263,202,300),(84,264,203,281),(85,265,204,282),(86,266,205,283),(87,267,206,284),(88,268,207,285),(89,269,208,286),(90,270,209,287),(91,271,210,288),(92,272,211,289),(93,273,212,290),(94,274,213,291),(95,275,214,292),(96,276,215,293),(97,277,216,294),(98,278,217,295),(99,279,218,296),(100,280,219,297)], [(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)], [(1,98,75,285,164,217,140,268),(2,206,76,277,165,87,121,294),(3,96,77,283,166,215,122,266),(4,204,78,275,167,85,123,292),(5,94,79,281,168,213,124,264),(6,202,80,273,169,83,125,290),(7,92,61,299,170,211,126,262),(8,220,62,271,171,81,127,288),(9,90,63,297,172,209,128,280),(10,218,64,269,173,99,129,286),(11,88,65,295,174,207,130,278),(12,216,66,267,175,97,131,284),(13,86,67,293,176,205,132,276),(14,214,68,265,177,95,133,282),(15,84,69,291,178,203,134,274),(16,212,70,263,179,93,135,300),(17,82,71,289,180,201,136,272),(18,210,72,261,161,91,137,298),(19,100,73,287,162,219,138,270),(20,208,74,279,163,89,139,296),(21,112,221,52,153,301,245,200),(22,310,222,189,154,101,246,41),(23,110,223,50,155,319,247,198),(24,308,224,187,156,119,248,59),(25,108,225,48,157,317,249,196),(26,306,226,185,158,117,250,57),(27,106,227,46,159,315,251,194),(28,304,228,183,160,115,252,55),(29,104,229,44,141,313,253,192),(30,302,230,181,142,113,254,53),(31,102,231,42,143,311,255,190),(32,320,232,199,144,111,256,51),(33,120,233,60,145,309,257,188),(34,318,234,197,146,109,258,49),(35,118,235,58,147,307,259,186),(36,316,236,195,148,107,260,47),(37,116,237,56,149,305,241,184),(38,314,238,193,150,105,242,45),(39,114,239,54,151,303,243,182),(40,312,240,191,152,103,244,43)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | Q8 | D5 | M4(2) | C4○D4 | D10 | Dic5 | Dic5 | C8○D4 | C4.Dic5 | Q8×D5 | Q8⋊2D5 | D4.Dic5 |
| kernel | C42.210D10 | C4×C5⋊2C8 | C42.D5 | C20⋊3C8 | Q8×C20 | C5×C4⋊C4 | Q8×C10 | C5⋊2C8 | C4×Q8 | C20 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 6 | 2 | 2 | 2 | 4 | 2 | 6 | 6 | 2 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C42.210D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 31 | 19 |
| 0 | 0 | 40 | 10 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 21 | 20 | 0 | 0 |
| 21 | 18 | 0 | 0 |
| 0 | 0 | 29 | 37 |
| 0 | 0 | 26 | 12 |
| 37 | 30 | 0 | 0 |
| 6 | 4 | 0 | 0 |
| 0 | 0 | 14 | 20 |
| 0 | 0 | 5 | 27 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,31,40,0,0,19,10],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[21,21,0,0,20,18,0,0,0,0,29,26,0,0,37,12],[37,6,0,0,30,4,0,0,0,0,14,5,0,0,20,27] >;
C42.210D10 in GAP, Magma, Sage, TeX
C_4^2._{210}D_{10} % in TeX
G:=Group("C4^2.210D10"); // GroupNames label
G:=SmallGroup(320,651);
// by ID
G=gap.SmallGroup(320,651);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,100,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^9>;
// generators/relations