metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.71D10, C4⋊C4.78D10, (C2×C20).86D4, C42.C2.5D5, C20.73(C4○D4), C20⋊2Q8.18C2, (C4×C20).117C22, (C2×C20).387C23, C4.15(Q8⋊2D5), C10.Q16.13C2, C42.D5.6C2, C10.57(C4.4D4), C2.22(D4.9D10), C2.10(C20.23D4), C10.123(C8.C22), C5⋊3(C42.30C22), (C2×Dic10).114C22, (C2×C10).518(C2×D4), (C2×C4).68(C5⋊D4), (C5×C42.C2).4C2, (C5×C4⋊C4).125C22, (C2×C4).485(C22×D5), C22.191(C2×C5⋊D4), (C2×C5⋊2C8).128C22, SmallGroup(320,696)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.71D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >
Subgroups: 302 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C42.30C22, C2×C5⋊2C8, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C42.D5, C10.Q16, C20⋊2Q8, C5×C42.C2, C42.71D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8.C22, C5⋊D4, C22×D5, C42.30C22, Q8⋊2D5, C2×C5⋊D4, C20.23D4, D4.9D10, C42.71D10
►Regular action on 320 points
Generators in S320
(1 249 11 259)(2 289 12 299)(3 251 13 241)(4 291 14 281)(5 253 15 243)(6 293 16 283)(7 255 17 245)(8 295 18 285)(9 257 19 247)(10 297 20 287)(21 312 31 302)(22 105 32 115)(23 314 33 304)(24 107 34 117)(25 316 35 306)(26 109 36 119)(27 318 37 308)(28 111 38 101)(29 320 39 310)(30 113 40 103)(41 198 51 188)(42 134 52 124)(43 200 53 190)(44 136 54 126)(45 182 55 192)(46 138 56 128)(47 184 57 194)(48 140 58 130)(49 186 59 196)(50 122 60 132)(61 165 71 175)(62 146 72 156)(63 167 73 177)(64 148 74 158)(65 169 75 179)(66 150 76 160)(67 171 77 161)(68 152 78 142)(69 173 79 163)(70 154 80 144)(81 121 91 131)(82 187 92 197)(83 123 93 133)(84 189 94 199)(85 125 95 135)(86 191 96 181)(87 127 97 137)(88 193 98 183)(89 129 99 139)(90 195 100 185)(102 264 112 274)(104 266 114 276)(106 268 116 278)(108 270 118 280)(110 272 120 262)(141 216 151 206)(143 218 153 208)(145 220 155 210)(147 202 157 212)(149 204 159 214)(162 217 172 207)(164 219 174 209)(166 201 176 211)(168 203 178 213)(170 205 180 215)(221 242 231 252)(222 282 232 292)(223 244 233 254)(224 284 234 294)(225 246 235 256)(226 286 236 296)(227 248 237 258)(228 288 238 298)(229 250 239 260)(230 290 240 300)(261 317 271 307)(263 319 273 309)(265 301 275 311)(267 303 277 313)(269 305 279 315)
(1 90 238 48)(2 49 239 91)(3 92 240 50)(4 51 221 93)(5 94 222 52)(6 53 223 95)(7 96 224 54)(8 55 225 97)(9 98 226 56)(10 57 227 99)(11 100 228 58)(12 59 229 81)(13 82 230 60)(14 41 231 83)(15 84 232 42)(16 43 233 85)(17 86 234 44)(18 45 235 87)(19 88 236 46)(20 47 237 89)(21 158 266 168)(22 169 267 159)(23 160 268 170)(24 171 269 141)(25 142 270 172)(26 173 271 143)(27 144 272 174)(28 175 273 145)(29 146 274 176)(30 177 275 147)(31 148 276 178)(32 179 277 149)(33 150 278 180)(34 161 279 151)(35 152 280 162)(36 163 261 153)(37 154 262 164)(38 165 263 155)(39 156 264 166)(40 167 265 157)(61 309 220 111)(62 112 201 310)(63 311 202 113)(64 114 203 312)(65 313 204 115)(66 116 205 314)(67 315 206 117)(68 118 207 316)(69 317 208 119)(70 120 209 318)(71 319 210 101)(72 102 211 320)(73 301 212 103)(74 104 213 302)(75 303 214 105)(76 106 215 304)(77 305 216 107)(78 108 217 306)(79 307 218 109)(80 110 219 308)(121 299 196 250)(122 251 197 300)(123 281 198 252)(124 253 199 282)(125 283 200 254)(126 255 181 284)(127 285 182 256)(128 257 183 286)(129 287 184 258)(130 259 185 288)(131 289 186 260)(132 241 187 290)(133 291 188 242)(134 243 189 292)(135 293 190 244)(136 245 191 294)(137 295 192 246)(138 247 193 296)(139 297 194 248)(140 249 195 298)
(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)
(1 102 100 201 238 320 58 62)(2 220 59 101 239 61 81 319)(3 120 82 219 240 318 60 80)(4 218 41 119 221 79 83 317)(5 118 84 217 222 316 42 78)(6 216 43 117 223 77 85 315)(7 116 86 215 224 314 44 76)(8 214 45 115 225 75 87 313)(9 114 88 213 226 312 46 74)(10 212 47 113 227 73 89 311)(11 112 90 211 228 310 48 72)(12 210 49 111 229 71 91 309)(13 110 92 209 230 308 50 70)(14 208 51 109 231 69 93 307)(15 108 94 207 232 306 52 68)(16 206 53 107 233 67 95 305)(17 106 96 205 234 304 54 66)(18 204 55 105 235 65 97 303)(19 104 98 203 236 302 56 64)(20 202 57 103 237 63 99 301)(21 183 148 296 266 128 178 247)(22 295 179 182 267 246 149 127)(23 181 150 294 268 126 180 245)(24 293 161 200 269 244 151 125)(25 199 152 292 270 124 162 243)(26 291 163 198 271 242 153 123)(27 197 154 290 272 122 164 241)(28 289 165 196 273 260 155 121)(29 195 156 288 274 140 166 259)(30 287 167 194 275 258 157 139)(31 193 158 286 276 138 168 257)(32 285 169 192 277 256 159 137)(33 191 160 284 278 136 170 255)(34 283 171 190 279 254 141 135)(35 189 142 282 280 134 172 253)(36 281 173 188 261 252 143 133)(37 187 144 300 262 132 174 251)(38 299 175 186 263 250 145 131)(39 185 146 298 264 130 176 249)(40 297 177 184 265 248 147 129)
G:=sub<Sym(320)| (1,249,11,259)(2,289,12,299)(3,251,13,241)(4,291,14,281)(5,253,15,243)(6,293,16,283)(7,255,17,245)(8,295,18,285)(9,257,19,247)(10,297,20,287)(21,312,31,302)(22,105,32,115)(23,314,33,304)(24,107,34,117)(25,316,35,306)(26,109,36,119)(27,318,37,308)(28,111,38,101)(29,320,39,310)(30,113,40,103)(41,198,51,188)(42,134,52,124)(43,200,53,190)(44,136,54,126)(45,182,55,192)(46,138,56,128)(47,184,57,194)(48,140,58,130)(49,186,59,196)(50,122,60,132)(61,165,71,175)(62,146,72,156)(63,167,73,177)(64,148,74,158)(65,169,75,179)(66,150,76,160)(67,171,77,161)(68,152,78,142)(69,173,79,163)(70,154,80,144)(81,121,91,131)(82,187,92,197)(83,123,93,133)(84,189,94,199)(85,125,95,135)(86,191,96,181)(87,127,97,137)(88,193,98,183)(89,129,99,139)(90,195,100,185)(102,264,112,274)(104,266,114,276)(106,268,116,278)(108,270,118,280)(110,272,120,262)(141,216,151,206)(143,218,153,208)(145,220,155,210)(147,202,157,212)(149,204,159,214)(162,217,172,207)(164,219,174,209)(166,201,176,211)(168,203,178,213)(170,205,180,215)(221,242,231,252)(222,282,232,292)(223,244,233,254)(224,284,234,294)(225,246,235,256)(226,286,236,296)(227,248,237,258)(228,288,238,298)(229,250,239,260)(230,290,240,300)(261,317,271,307)(263,319,273,309)(265,301,275,311)(267,303,277,313)(269,305,279,315), (1,90,238,48)(2,49,239,91)(3,92,240,50)(4,51,221,93)(5,94,222,52)(6,53,223,95)(7,96,224,54)(8,55,225,97)(9,98,226,56)(10,57,227,99)(11,100,228,58)(12,59,229,81)(13,82,230,60)(14,41,231,83)(15,84,232,42)(16,43,233,85)(17,86,234,44)(18,45,235,87)(19,88,236,46)(20,47,237,89)(21,158,266,168)(22,169,267,159)(23,160,268,170)(24,171,269,141)(25,142,270,172)(26,173,271,143)(27,144,272,174)(28,175,273,145)(29,146,274,176)(30,177,275,147)(31,148,276,178)(32,179,277,149)(33,150,278,180)(34,161,279,151)(35,152,280,162)(36,163,261,153)(37,154,262,164)(38,165,263,155)(39,156,264,166)(40,167,265,157)(61,309,220,111)(62,112,201,310)(63,311,202,113)(64,114,203,312)(65,313,204,115)(66,116,205,314)(67,315,206,117)(68,118,207,316)(69,317,208,119)(70,120,209,318)(71,319,210,101)(72,102,211,320)(73,301,212,103)(74,104,213,302)(75,303,214,105)(76,106,215,304)(77,305,216,107)(78,108,217,306)(79,307,218,109)(80,110,219,308)(121,299,196,250)(122,251,197,300)(123,281,198,252)(124,253,199,282)(125,283,200,254)(126,255,181,284)(127,285,182,256)(128,257,183,286)(129,287,184,258)(130,259,185,288)(131,289,186,260)(132,241,187,290)(133,291,188,242)(134,243,189,292)(135,293,190,244)(136,245,191,294)(137,295,192,246)(138,247,193,296)(139,297,194,248)(140,249,195,298), (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), (1,102,100,201,238,320,58,62)(2,220,59,101,239,61,81,319)(3,120,82,219,240,318,60,80)(4,218,41,119,221,79,83,317)(5,118,84,217,222,316,42,78)(6,216,43,117,223,77,85,315)(7,116,86,215,224,314,44,76)(8,214,45,115,225,75,87,313)(9,114,88,213,226,312,46,74)(10,212,47,113,227,73,89,311)(11,112,90,211,228,310,48,72)(12,210,49,111,229,71,91,309)(13,110,92,209,230,308,50,70)(14,208,51,109,231,69,93,307)(15,108,94,207,232,306,52,68)(16,206,53,107,233,67,95,305)(17,106,96,205,234,304,54,66)(18,204,55,105,235,65,97,303)(19,104,98,203,236,302,56,64)(20,202,57,103,237,63,99,301)(21,183,148,296,266,128,178,247)(22,295,179,182,267,246,149,127)(23,181,150,294,268,126,180,245)(24,293,161,200,269,244,151,125)(25,199,152,292,270,124,162,243)(26,291,163,198,271,242,153,123)(27,197,154,290,272,122,164,241)(28,289,165,196,273,260,155,121)(29,195,156,288,274,140,166,259)(30,287,167,194,275,258,157,139)(31,193,158,286,276,138,168,257)(32,285,169,192,277,256,159,137)(33,191,160,284,278,136,170,255)(34,283,171,190,279,254,141,135)(35,189,142,282,280,134,172,253)(36,281,173,188,261,252,143,133)(37,187,144,300,262,132,174,251)(38,299,175,186,263,250,145,131)(39,185,146,298,264,130,176,249)(40,297,177,184,265,248,147,129)>;
G:=Group( (1,249,11,259)(2,289,12,299)(3,251,13,241)(4,291,14,281)(5,253,15,243)(6,293,16,283)(7,255,17,245)(8,295,18,285)(9,257,19,247)(10,297,20,287)(21,312,31,302)(22,105,32,115)(23,314,33,304)(24,107,34,117)(25,316,35,306)(26,109,36,119)(27,318,37,308)(28,111,38,101)(29,320,39,310)(30,113,40,103)(41,198,51,188)(42,134,52,124)(43,200,53,190)(44,136,54,126)(45,182,55,192)(46,138,56,128)(47,184,57,194)(48,140,58,130)(49,186,59,196)(50,122,60,132)(61,165,71,175)(62,146,72,156)(63,167,73,177)(64,148,74,158)(65,169,75,179)(66,150,76,160)(67,171,77,161)(68,152,78,142)(69,173,79,163)(70,154,80,144)(81,121,91,131)(82,187,92,197)(83,123,93,133)(84,189,94,199)(85,125,95,135)(86,191,96,181)(87,127,97,137)(88,193,98,183)(89,129,99,139)(90,195,100,185)(102,264,112,274)(104,266,114,276)(106,268,116,278)(108,270,118,280)(110,272,120,262)(141,216,151,206)(143,218,153,208)(145,220,155,210)(147,202,157,212)(149,204,159,214)(162,217,172,207)(164,219,174,209)(166,201,176,211)(168,203,178,213)(170,205,180,215)(221,242,231,252)(222,282,232,292)(223,244,233,254)(224,284,234,294)(225,246,235,256)(226,286,236,296)(227,248,237,258)(228,288,238,298)(229,250,239,260)(230,290,240,300)(261,317,271,307)(263,319,273,309)(265,301,275,311)(267,303,277,313)(269,305,279,315), (1,90,238,48)(2,49,239,91)(3,92,240,50)(4,51,221,93)(5,94,222,52)(6,53,223,95)(7,96,224,54)(8,55,225,97)(9,98,226,56)(10,57,227,99)(11,100,228,58)(12,59,229,81)(13,82,230,60)(14,41,231,83)(15,84,232,42)(16,43,233,85)(17,86,234,44)(18,45,235,87)(19,88,236,46)(20,47,237,89)(21,158,266,168)(22,169,267,159)(23,160,268,170)(24,171,269,141)(25,142,270,172)(26,173,271,143)(27,144,272,174)(28,175,273,145)(29,146,274,176)(30,177,275,147)(31,148,276,178)(32,179,277,149)(33,150,278,180)(34,161,279,151)(35,152,280,162)(36,163,261,153)(37,154,262,164)(38,165,263,155)(39,156,264,166)(40,167,265,157)(61,309,220,111)(62,112,201,310)(63,311,202,113)(64,114,203,312)(65,313,204,115)(66,116,205,314)(67,315,206,117)(68,118,207,316)(69,317,208,119)(70,120,209,318)(71,319,210,101)(72,102,211,320)(73,301,212,103)(74,104,213,302)(75,303,214,105)(76,106,215,304)(77,305,216,107)(78,108,217,306)(79,307,218,109)(80,110,219,308)(121,299,196,250)(122,251,197,300)(123,281,198,252)(124,253,199,282)(125,283,200,254)(126,255,181,284)(127,285,182,256)(128,257,183,286)(129,287,184,258)(130,259,185,288)(131,289,186,260)(132,241,187,290)(133,291,188,242)(134,243,189,292)(135,293,190,244)(136,245,191,294)(137,295,192,246)(138,247,193,296)(139,297,194,248)(140,249,195,298), (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), (1,102,100,201,238,320,58,62)(2,220,59,101,239,61,81,319)(3,120,82,219,240,318,60,80)(4,218,41,119,221,79,83,317)(5,118,84,217,222,316,42,78)(6,216,43,117,223,77,85,315)(7,116,86,215,224,314,44,76)(8,214,45,115,225,75,87,313)(9,114,88,213,226,312,46,74)(10,212,47,113,227,73,89,311)(11,112,90,211,228,310,48,72)(12,210,49,111,229,71,91,309)(13,110,92,209,230,308,50,70)(14,208,51,109,231,69,93,307)(15,108,94,207,232,306,52,68)(16,206,53,107,233,67,95,305)(17,106,96,205,234,304,54,66)(18,204,55,105,235,65,97,303)(19,104,98,203,236,302,56,64)(20,202,57,103,237,63,99,301)(21,183,148,296,266,128,178,247)(22,295,179,182,267,246,149,127)(23,181,150,294,268,126,180,245)(24,293,161,200,269,244,151,125)(25,199,152,292,270,124,162,243)(26,291,163,198,271,242,153,123)(27,197,154,290,272,122,164,241)(28,289,165,196,273,260,155,121)(29,195,156,288,274,140,166,259)(30,287,167,194,275,258,157,139)(31,193,158,286,276,138,168,257)(32,285,169,192,277,256,159,137)(33,191,160,284,278,136,170,255)(34,283,171,190,279,254,141,135)(35,189,142,282,280,134,172,253)(36,281,173,188,261,252,143,133)(37,187,144,300,262,132,174,251)(38,299,175,186,263,250,145,131)(39,185,146,298,264,130,176,249)(40,297,177,184,265,248,147,129) );
G=PermutationGroup([[(1,249,11,259),(2,289,12,299),(3,251,13,241),(4,291,14,281),(5,253,15,243),(6,293,16,283),(7,255,17,245),(8,295,18,285),(9,257,19,247),(10,297,20,287),(21,312,31,302),(22,105,32,115),(23,314,33,304),(24,107,34,117),(25,316,35,306),(26,109,36,119),(27,318,37,308),(28,111,38,101),(29,320,39,310),(30,113,40,103),(41,198,51,188),(42,134,52,124),(43,200,53,190),(44,136,54,126),(45,182,55,192),(46,138,56,128),(47,184,57,194),(48,140,58,130),(49,186,59,196),(50,122,60,132),(61,165,71,175),(62,146,72,156),(63,167,73,177),(64,148,74,158),(65,169,75,179),(66,150,76,160),(67,171,77,161),(68,152,78,142),(69,173,79,163),(70,154,80,144),(81,121,91,131),(82,187,92,197),(83,123,93,133),(84,189,94,199),(85,125,95,135),(86,191,96,181),(87,127,97,137),(88,193,98,183),(89,129,99,139),(90,195,100,185),(102,264,112,274),(104,266,114,276),(106,268,116,278),(108,270,118,280),(110,272,120,262),(141,216,151,206),(143,218,153,208),(145,220,155,210),(147,202,157,212),(149,204,159,214),(162,217,172,207),(164,219,174,209),(166,201,176,211),(168,203,178,213),(170,205,180,215),(221,242,231,252),(222,282,232,292),(223,244,233,254),(224,284,234,294),(225,246,235,256),(226,286,236,296),(227,248,237,258),(228,288,238,298),(229,250,239,260),(230,290,240,300),(261,317,271,307),(263,319,273,309),(265,301,275,311),(267,303,277,313),(269,305,279,315)], [(1,90,238,48),(2,49,239,91),(3,92,240,50),(4,51,221,93),(5,94,222,52),(6,53,223,95),(7,96,224,54),(8,55,225,97),(9,98,226,56),(10,57,227,99),(11,100,228,58),(12,59,229,81),(13,82,230,60),(14,41,231,83),(15,84,232,42),(16,43,233,85),(17,86,234,44),(18,45,235,87),(19,88,236,46),(20,47,237,89),(21,158,266,168),(22,169,267,159),(23,160,268,170),(24,171,269,141),(25,142,270,172),(26,173,271,143),(27,144,272,174),(28,175,273,145),(29,146,274,176),(30,177,275,147),(31,148,276,178),(32,179,277,149),(33,150,278,180),(34,161,279,151),(35,152,280,162),(36,163,261,153),(37,154,262,164),(38,165,263,155),(39,156,264,166),(40,167,265,157),(61,309,220,111),(62,112,201,310),(63,311,202,113),(64,114,203,312),(65,313,204,115),(66,116,205,314),(67,315,206,117),(68,118,207,316),(69,317,208,119),(70,120,209,318),(71,319,210,101),(72,102,211,320),(73,301,212,103),(74,104,213,302),(75,303,214,105),(76,106,215,304),(77,305,216,107),(78,108,217,306),(79,307,218,109),(80,110,219,308),(121,299,196,250),(122,251,197,300),(123,281,198,252),(124,253,199,282),(125,283,200,254),(126,255,181,284),(127,285,182,256),(128,257,183,286),(129,287,184,258),(130,259,185,288),(131,289,186,260),(132,241,187,290),(133,291,188,242),(134,243,189,292),(135,293,190,244),(136,245,191,294),(137,295,192,246),(138,247,193,296),(139,297,194,248),(140,249,195,298)], [(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)], [(1,102,100,201,238,320,58,62),(2,220,59,101,239,61,81,319),(3,120,82,219,240,318,60,80),(4,218,41,119,221,79,83,317),(5,118,84,217,222,316,42,78),(6,216,43,117,223,77,85,315),(7,116,86,215,224,314,44,76),(8,214,45,115,225,75,87,313),(9,114,88,213,226,312,46,74),(10,212,47,113,227,73,89,311),(11,112,90,211,228,310,48,72),(12,210,49,111,229,71,91,309),(13,110,92,209,230,308,50,70),(14,208,51,109,231,69,93,307),(15,108,94,207,232,306,52,68),(16,206,53,107,233,67,95,305),(17,106,96,205,234,304,54,66),(18,204,55,105,235,65,97,303),(19,104,98,203,236,302,56,64),(20,202,57,103,237,63,99,301),(21,183,148,296,266,128,178,247),(22,295,179,182,267,246,149,127),(23,181,150,294,268,126,180,245),(24,293,161,200,269,244,151,125),(25,199,152,292,270,124,162,243),(26,291,163,198,271,242,153,123),(27,197,154,290,272,122,164,241),(28,289,165,196,273,260,155,121),(29,195,156,288,274,140,166,259),(30,287,167,194,275,258,157,139),(31,193,158,286,276,138,168,257),(32,285,169,192,277,256,159,137),(33,191,160,284,278,136,170,255),(34,283,171,190,279,254,141,135),(35,189,142,282,280,134,172,253),(36,281,173,188,261,252,143,133),(37,187,144,300,262,132,174,251),(38,299,175,186,263,250,145,131),(39,185,146,298,264,130,176,249),(40,297,177,184,265,248,147,129)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C8.C22 | Q8⋊2D5 | D4.9D10 |
| kernel | C42.71D10 | C42.D5 | C10.Q16 | C20⋊2Q8 | C5×C42.C2 | C2×C20 | C42.C2 | C20 | C42 | C4⋊C4 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C42.71D10 ►in GL6(𝔽41)
| 26 | 23 | 0 | 0 | 0 | 0 |
| 8 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 25 | 9 | 0 | 0 | 0 | 0 |
| 17 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 22 | 40 | 2 |
| 0 | 0 | 19 | 16 | 39 | 26 |
| 0 | 0 | 40 | 2 | 35 | 19 |
| 0 | 0 | 39 | 26 | 22 | 25 |
| 29 | 2 | 0 | 0 | 0 | 0 |
| 31 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 25 | 23 | 21 |
| 0 | 0 | 36 | 25 | 24 | 18 |
| 0 | 0 | 18 | 20 | 16 | 25 |
| 0 | 0 | 17 | 23 | 36 | 25 |
G:=sub<GL(6,GF(41))| [26,8,0,0,0,0,23,15,0,0,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,11,32,0,0,0,0,9,30,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[25,17,0,0,0,0,9,16,0,0,0,0,0,0,6,19,40,39,0,0,22,16,2,26,0,0,40,39,35,22,0,0,2,26,19,25],[29,31,0,0,0,0,2,12,0,0,0,0,0,0,16,36,18,17,0,0,25,25,20,23,0,0,23,24,16,36,0,0,21,18,25,25] >;
C42.71D10 in GAP, Magma, Sage, TeX
C_4^2._{71}D_{10} % in TeX
G:=Group("C4^2.71D10"); // GroupNames label
G:=SmallGroup(320,696);
// by ID
G=gap.SmallGroup(320,696);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,555,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations