metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.8D10, C10.31C4≀C2, C4⋊C4.1Dic5, (C2×C20).232D4, C42.C2.1D5, (C4×C20).236C22, C42.D5.9C2, C2.7(D4⋊2Dic5), C2.3(C20.10D4), C5⋊3(C42.2C22), C10.13(C4.10D4), C22.40(C23.D5), (C5×C4⋊C4).14C4, (C2×C20).343(C2×C4), (C2×C4).10(C2×Dic5), (C5×C42.C2).7C2, (C2×C4).166(C5⋊D4), (C2×C10).165(C22⋊C4), SmallGroup(320,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.8D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a-1b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc9 >
Subgroups: 158 in 60 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C20, C2×C10, C8⋊C4, C42.C2, C5⋊2C8, C2×C20, C2×C20, C2×C20, C42.2C22, C2×C5⋊2C8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C42.D5, C5×C42.C2, C42.8D10
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C4.10D4, C4≀C2, C2×Dic5, C5⋊D4, C42.2C22, C23.D5, C20.10D4, D4⋊2Dic5, C42.8D10
►Regular action on 320 points
Generators in S320
(1 207 231 251)(2 218 232 242)(3 209 233 253)(4 220 234 244)(5 211 235 255)(6 202 236 246)(7 213 237 257)(8 204 238 248)(9 215 239 259)(10 206 240 250)(11 217 221 241)(12 208 222 252)(13 219 223 243)(14 210 224 254)(15 201 225 245)(16 212 226 256)(17 203 227 247)(18 214 228 258)(19 205 229 249)(20 216 230 260)(21 62 170 95)(22 73 171 86)(23 64 172 97)(24 75 173 88)(25 66 174 99)(26 77 175 90)(27 68 176 81)(28 79 177 92)(29 70 178 83)(30 61 179 94)(31 72 180 85)(32 63 161 96)(33 74 162 87)(34 65 163 98)(35 76 164 89)(36 67 165 100)(37 78 166 91)(38 69 167 82)(39 80 168 93)(40 71 169 84)(41 141 113 185)(42 152 114 196)(43 143 115 187)(44 154 116 198)(45 145 117 189)(46 156 118 200)(47 147 119 191)(48 158 120 182)(49 149 101 193)(50 160 102 184)(51 151 103 195)(52 142 104 186)(53 153 105 197)(54 144 106 188)(55 155 107 199)(56 146 108 190)(57 157 109 181)(58 148 110 192)(59 159 111 183)(60 150 112 194)(121 297 316 269)(122 288 317 280)(123 299 318 271)(124 290 319 262)(125 281 320 273)(126 292 301 264)(127 283 302 275)(128 294 303 266)(129 285 304 277)(130 296 305 268)(131 287 306 279)(132 298 307 270)(133 289 308 261)(134 300 309 272)(135 291 310 263)(136 282 311 274)(137 293 312 265)(138 284 313 276)(139 295 314 267)(140 286 315 278)
(1 63 221 86)(2 87 222 64)(3 65 223 88)(4 89 224 66)(5 67 225 90)(6 91 226 68)(7 69 227 92)(8 93 228 70)(9 71 229 94)(10 95 230 72)(11 73 231 96)(12 97 232 74)(13 75 233 98)(14 99 234 76)(15 77 235 100)(16 81 236 78)(17 79 237 82)(18 83 238 80)(19 61 239 84)(20 85 240 62)(21 260 180 206)(22 207 161 241)(23 242 162 208)(24 209 163 243)(25 244 164 210)(26 211 165 245)(27 246 166 212)(28 213 167 247)(29 248 168 214)(30 215 169 249)(31 250 170 216)(32 217 171 251)(33 252 172 218)(34 219 173 253)(35 254 174 220)(36 201 175 255)(37 256 176 202)(38 203 177 257)(39 258 178 204)(40 205 179 259)(41 129 103 314)(42 315 104 130)(43 131 105 316)(44 317 106 132)(45 133 107 318)(46 319 108 134)(47 135 109 320)(48 301 110 136)(49 137 111 302)(50 303 112 138)(51 139 113 304)(52 305 114 140)(53 121 115 306)(54 307 116 122)(55 123 117 308)(56 309 118 124)(57 125 119 310)(58 311 120 126)(59 127 101 312)(60 313 102 128)(141 285 195 267)(142 268 196 286)(143 287 197 269)(144 270 198 288)(145 289 199 271)(146 272 200 290)(147 291 181 273)(148 274 182 292)(149 293 183 275)(150 276 184 294)(151 295 185 277)(152 278 186 296)(153 297 187 279)(154 280 188 298)(155 299 189 261)(156 262 190 300)(157 281 191 263)(158 264 192 282)(159 283 193 265)(160 266 194 284)
(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)
(1 140 32 186 11 130 22 196)(2 41 162 277 12 51 172 267)(3 138 34 184 13 128 24 194)(4 59 164 275 14 49 174 265)(5 136 36 182 15 126 26 192)(6 57 166 273 16 47 176 263)(7 134 38 200 17 124 28 190)(8 55 168 271 18 45 178 261)(9 132 40 198 19 122 30 188)(10 53 170 269 20 43 180 279)(21 297 230 115 31 287 240 105)(23 295 232 113 33 285 222 103)(25 293 234 111 35 283 224 101)(27 291 236 109 37 281 226 119)(29 289 238 107 39 299 228 117)(42 217 286 73 52 207 296 63)(44 215 288 71 54 205 298 61)(46 213 290 69 56 203 300 79)(48 211 292 67 58 201 282 77)(50 209 294 65 60 219 284 75)(62 306 260 197 72 316 250 187)(64 304 242 195 74 314 252 185)(66 302 244 193 76 312 254 183)(68 320 246 191 78 310 256 181)(70 318 248 189 80 308 258 199)(81 125 202 147 91 135 212 157)(82 108 247 272 92 118 257 262)(83 123 204 145 93 133 214 155)(84 106 249 270 94 116 259 280)(85 121 206 143 95 131 216 153)(86 104 251 268 96 114 241 278)(87 139 208 141 97 129 218 151)(88 102 253 266 98 112 243 276)(89 137 210 159 99 127 220 149)(90 120 255 264 100 110 245 274)(142 221 305 171 152 231 315 161)(144 239 307 169 154 229 317 179)(146 237 309 167 156 227 319 177)(148 235 311 165 158 225 301 175)(150 233 313 163 160 223 303 173)
G:=sub<Sym(320)| (1,207,231,251)(2,218,232,242)(3,209,233,253)(4,220,234,244)(5,211,235,255)(6,202,236,246)(7,213,237,257)(8,204,238,248)(9,215,239,259)(10,206,240,250)(11,217,221,241)(12,208,222,252)(13,219,223,243)(14,210,224,254)(15,201,225,245)(16,212,226,256)(17,203,227,247)(18,214,228,258)(19,205,229,249)(20,216,230,260)(21,62,170,95)(22,73,171,86)(23,64,172,97)(24,75,173,88)(25,66,174,99)(26,77,175,90)(27,68,176,81)(28,79,177,92)(29,70,178,83)(30,61,179,94)(31,72,180,85)(32,63,161,96)(33,74,162,87)(34,65,163,98)(35,76,164,89)(36,67,165,100)(37,78,166,91)(38,69,167,82)(39,80,168,93)(40,71,169,84)(41,141,113,185)(42,152,114,196)(43,143,115,187)(44,154,116,198)(45,145,117,189)(46,156,118,200)(47,147,119,191)(48,158,120,182)(49,149,101,193)(50,160,102,184)(51,151,103,195)(52,142,104,186)(53,153,105,197)(54,144,106,188)(55,155,107,199)(56,146,108,190)(57,157,109,181)(58,148,110,192)(59,159,111,183)(60,150,112,194)(121,297,316,269)(122,288,317,280)(123,299,318,271)(124,290,319,262)(125,281,320,273)(126,292,301,264)(127,283,302,275)(128,294,303,266)(129,285,304,277)(130,296,305,268)(131,287,306,279)(132,298,307,270)(133,289,308,261)(134,300,309,272)(135,291,310,263)(136,282,311,274)(137,293,312,265)(138,284,313,276)(139,295,314,267)(140,286,315,278), (1,63,221,86)(2,87,222,64)(3,65,223,88)(4,89,224,66)(5,67,225,90)(6,91,226,68)(7,69,227,92)(8,93,228,70)(9,71,229,94)(10,95,230,72)(11,73,231,96)(12,97,232,74)(13,75,233,98)(14,99,234,76)(15,77,235,100)(16,81,236,78)(17,79,237,82)(18,83,238,80)(19,61,239,84)(20,85,240,62)(21,260,180,206)(22,207,161,241)(23,242,162,208)(24,209,163,243)(25,244,164,210)(26,211,165,245)(27,246,166,212)(28,213,167,247)(29,248,168,214)(30,215,169,249)(31,250,170,216)(32,217,171,251)(33,252,172,218)(34,219,173,253)(35,254,174,220)(36,201,175,255)(37,256,176,202)(38,203,177,257)(39,258,178,204)(40,205,179,259)(41,129,103,314)(42,315,104,130)(43,131,105,316)(44,317,106,132)(45,133,107,318)(46,319,108,134)(47,135,109,320)(48,301,110,136)(49,137,111,302)(50,303,112,138)(51,139,113,304)(52,305,114,140)(53,121,115,306)(54,307,116,122)(55,123,117,308)(56,309,118,124)(57,125,119,310)(58,311,120,126)(59,127,101,312)(60,313,102,128)(141,285,195,267)(142,268,196,286)(143,287,197,269)(144,270,198,288)(145,289,199,271)(146,272,200,290)(147,291,181,273)(148,274,182,292)(149,293,183,275)(150,276,184,294)(151,295,185,277)(152,278,186,296)(153,297,187,279)(154,280,188,298)(155,299,189,261)(156,262,190,300)(157,281,191,263)(158,264,192,282)(159,283,193,265)(160,266,194,284), (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), (1,140,32,186,11,130,22,196)(2,41,162,277,12,51,172,267)(3,138,34,184,13,128,24,194)(4,59,164,275,14,49,174,265)(5,136,36,182,15,126,26,192)(6,57,166,273,16,47,176,263)(7,134,38,200,17,124,28,190)(8,55,168,271,18,45,178,261)(9,132,40,198,19,122,30,188)(10,53,170,269,20,43,180,279)(21,297,230,115,31,287,240,105)(23,295,232,113,33,285,222,103)(25,293,234,111,35,283,224,101)(27,291,236,109,37,281,226,119)(29,289,238,107,39,299,228,117)(42,217,286,73,52,207,296,63)(44,215,288,71,54,205,298,61)(46,213,290,69,56,203,300,79)(48,211,292,67,58,201,282,77)(50,209,294,65,60,219,284,75)(62,306,260,197,72,316,250,187)(64,304,242,195,74,314,252,185)(66,302,244,193,76,312,254,183)(68,320,246,191,78,310,256,181)(70,318,248,189,80,308,258,199)(81,125,202,147,91,135,212,157)(82,108,247,272,92,118,257,262)(83,123,204,145,93,133,214,155)(84,106,249,270,94,116,259,280)(85,121,206,143,95,131,216,153)(86,104,251,268,96,114,241,278)(87,139,208,141,97,129,218,151)(88,102,253,266,98,112,243,276)(89,137,210,159,99,127,220,149)(90,120,255,264,100,110,245,274)(142,221,305,171,152,231,315,161)(144,239,307,169,154,229,317,179)(146,237,309,167,156,227,319,177)(148,235,311,165,158,225,301,175)(150,233,313,163,160,223,303,173)>;
G:=Group( (1,207,231,251)(2,218,232,242)(3,209,233,253)(4,220,234,244)(5,211,235,255)(6,202,236,246)(7,213,237,257)(8,204,238,248)(9,215,239,259)(10,206,240,250)(11,217,221,241)(12,208,222,252)(13,219,223,243)(14,210,224,254)(15,201,225,245)(16,212,226,256)(17,203,227,247)(18,214,228,258)(19,205,229,249)(20,216,230,260)(21,62,170,95)(22,73,171,86)(23,64,172,97)(24,75,173,88)(25,66,174,99)(26,77,175,90)(27,68,176,81)(28,79,177,92)(29,70,178,83)(30,61,179,94)(31,72,180,85)(32,63,161,96)(33,74,162,87)(34,65,163,98)(35,76,164,89)(36,67,165,100)(37,78,166,91)(38,69,167,82)(39,80,168,93)(40,71,169,84)(41,141,113,185)(42,152,114,196)(43,143,115,187)(44,154,116,198)(45,145,117,189)(46,156,118,200)(47,147,119,191)(48,158,120,182)(49,149,101,193)(50,160,102,184)(51,151,103,195)(52,142,104,186)(53,153,105,197)(54,144,106,188)(55,155,107,199)(56,146,108,190)(57,157,109,181)(58,148,110,192)(59,159,111,183)(60,150,112,194)(121,297,316,269)(122,288,317,280)(123,299,318,271)(124,290,319,262)(125,281,320,273)(126,292,301,264)(127,283,302,275)(128,294,303,266)(129,285,304,277)(130,296,305,268)(131,287,306,279)(132,298,307,270)(133,289,308,261)(134,300,309,272)(135,291,310,263)(136,282,311,274)(137,293,312,265)(138,284,313,276)(139,295,314,267)(140,286,315,278), (1,63,221,86)(2,87,222,64)(3,65,223,88)(4,89,224,66)(5,67,225,90)(6,91,226,68)(7,69,227,92)(8,93,228,70)(9,71,229,94)(10,95,230,72)(11,73,231,96)(12,97,232,74)(13,75,233,98)(14,99,234,76)(15,77,235,100)(16,81,236,78)(17,79,237,82)(18,83,238,80)(19,61,239,84)(20,85,240,62)(21,260,180,206)(22,207,161,241)(23,242,162,208)(24,209,163,243)(25,244,164,210)(26,211,165,245)(27,246,166,212)(28,213,167,247)(29,248,168,214)(30,215,169,249)(31,250,170,216)(32,217,171,251)(33,252,172,218)(34,219,173,253)(35,254,174,220)(36,201,175,255)(37,256,176,202)(38,203,177,257)(39,258,178,204)(40,205,179,259)(41,129,103,314)(42,315,104,130)(43,131,105,316)(44,317,106,132)(45,133,107,318)(46,319,108,134)(47,135,109,320)(48,301,110,136)(49,137,111,302)(50,303,112,138)(51,139,113,304)(52,305,114,140)(53,121,115,306)(54,307,116,122)(55,123,117,308)(56,309,118,124)(57,125,119,310)(58,311,120,126)(59,127,101,312)(60,313,102,128)(141,285,195,267)(142,268,196,286)(143,287,197,269)(144,270,198,288)(145,289,199,271)(146,272,200,290)(147,291,181,273)(148,274,182,292)(149,293,183,275)(150,276,184,294)(151,295,185,277)(152,278,186,296)(153,297,187,279)(154,280,188,298)(155,299,189,261)(156,262,190,300)(157,281,191,263)(158,264,192,282)(159,283,193,265)(160,266,194,284), (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), (1,140,32,186,11,130,22,196)(2,41,162,277,12,51,172,267)(3,138,34,184,13,128,24,194)(4,59,164,275,14,49,174,265)(5,136,36,182,15,126,26,192)(6,57,166,273,16,47,176,263)(7,134,38,200,17,124,28,190)(8,55,168,271,18,45,178,261)(9,132,40,198,19,122,30,188)(10,53,170,269,20,43,180,279)(21,297,230,115,31,287,240,105)(23,295,232,113,33,285,222,103)(25,293,234,111,35,283,224,101)(27,291,236,109,37,281,226,119)(29,289,238,107,39,299,228,117)(42,217,286,73,52,207,296,63)(44,215,288,71,54,205,298,61)(46,213,290,69,56,203,300,79)(48,211,292,67,58,201,282,77)(50,209,294,65,60,219,284,75)(62,306,260,197,72,316,250,187)(64,304,242,195,74,314,252,185)(66,302,244,193,76,312,254,183)(68,320,246,191,78,310,256,181)(70,318,248,189,80,308,258,199)(81,125,202,147,91,135,212,157)(82,108,247,272,92,118,257,262)(83,123,204,145,93,133,214,155)(84,106,249,270,94,116,259,280)(85,121,206,143,95,131,216,153)(86,104,251,268,96,114,241,278)(87,139,208,141,97,129,218,151)(88,102,253,266,98,112,243,276)(89,137,210,159,99,127,220,149)(90,120,255,264,100,110,245,274)(142,221,305,171,152,231,315,161)(144,239,307,169,154,229,317,179)(146,237,309,167,156,227,319,177)(148,235,311,165,158,225,301,175)(150,233,313,163,160,223,303,173) );
G=PermutationGroup([[(1,207,231,251),(2,218,232,242),(3,209,233,253),(4,220,234,244),(5,211,235,255),(6,202,236,246),(7,213,237,257),(8,204,238,248),(9,215,239,259),(10,206,240,250),(11,217,221,241),(12,208,222,252),(13,219,223,243),(14,210,224,254),(15,201,225,245),(16,212,226,256),(17,203,227,247),(18,214,228,258),(19,205,229,249),(20,216,230,260),(21,62,170,95),(22,73,171,86),(23,64,172,97),(24,75,173,88),(25,66,174,99),(26,77,175,90),(27,68,176,81),(28,79,177,92),(29,70,178,83),(30,61,179,94),(31,72,180,85),(32,63,161,96),(33,74,162,87),(34,65,163,98),(35,76,164,89),(36,67,165,100),(37,78,166,91),(38,69,167,82),(39,80,168,93),(40,71,169,84),(41,141,113,185),(42,152,114,196),(43,143,115,187),(44,154,116,198),(45,145,117,189),(46,156,118,200),(47,147,119,191),(48,158,120,182),(49,149,101,193),(50,160,102,184),(51,151,103,195),(52,142,104,186),(53,153,105,197),(54,144,106,188),(55,155,107,199),(56,146,108,190),(57,157,109,181),(58,148,110,192),(59,159,111,183),(60,150,112,194),(121,297,316,269),(122,288,317,280),(123,299,318,271),(124,290,319,262),(125,281,320,273),(126,292,301,264),(127,283,302,275),(128,294,303,266),(129,285,304,277),(130,296,305,268),(131,287,306,279),(132,298,307,270),(133,289,308,261),(134,300,309,272),(135,291,310,263),(136,282,311,274),(137,293,312,265),(138,284,313,276),(139,295,314,267),(140,286,315,278)], [(1,63,221,86),(2,87,222,64),(3,65,223,88),(4,89,224,66),(5,67,225,90),(6,91,226,68),(7,69,227,92),(8,93,228,70),(9,71,229,94),(10,95,230,72),(11,73,231,96),(12,97,232,74),(13,75,233,98),(14,99,234,76),(15,77,235,100),(16,81,236,78),(17,79,237,82),(18,83,238,80),(19,61,239,84),(20,85,240,62),(21,260,180,206),(22,207,161,241),(23,242,162,208),(24,209,163,243),(25,244,164,210),(26,211,165,245),(27,246,166,212),(28,213,167,247),(29,248,168,214),(30,215,169,249),(31,250,170,216),(32,217,171,251),(33,252,172,218),(34,219,173,253),(35,254,174,220),(36,201,175,255),(37,256,176,202),(38,203,177,257),(39,258,178,204),(40,205,179,259),(41,129,103,314),(42,315,104,130),(43,131,105,316),(44,317,106,132),(45,133,107,318),(46,319,108,134),(47,135,109,320),(48,301,110,136),(49,137,111,302),(50,303,112,138),(51,139,113,304),(52,305,114,140),(53,121,115,306),(54,307,116,122),(55,123,117,308),(56,309,118,124),(57,125,119,310),(58,311,120,126),(59,127,101,312),(60,313,102,128),(141,285,195,267),(142,268,196,286),(143,287,197,269),(144,270,198,288),(145,289,199,271),(146,272,200,290),(147,291,181,273),(148,274,182,292),(149,293,183,275),(150,276,184,294),(151,295,185,277),(152,278,186,296),(153,297,187,279),(154,280,188,298),(155,299,189,261),(156,262,190,300),(157,281,191,263),(158,264,192,282),(159,283,193,265),(160,266,194,284)], [(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)], [(1,140,32,186,11,130,22,196),(2,41,162,277,12,51,172,267),(3,138,34,184,13,128,24,194),(4,59,164,275,14,49,174,265),(5,136,36,182,15,126,26,192),(6,57,166,273,16,47,176,263),(7,134,38,200,17,124,28,190),(8,55,168,271,18,45,178,261),(9,132,40,198,19,122,30,188),(10,53,170,269,20,43,180,279),(21,297,230,115,31,287,240,105),(23,295,232,113,33,285,222,103),(25,293,234,111,35,283,224,101),(27,291,236,109,37,281,226,119),(29,289,238,107,39,299,228,117),(42,217,286,73,52,207,296,63),(44,215,288,71,54,205,298,61),(46,213,290,69,56,203,300,79),(48,211,292,67,58,201,282,77),(50,209,294,65,60,219,284,75),(62,306,260,197,72,316,250,187),(64,304,242,195,74,314,252,185),(66,302,244,193,76,312,254,183),(68,320,246,191,78,310,256,181),(70,318,248,189,80,308,258,199),(81,125,202,147,91,135,212,157),(82,108,247,272,92,118,257,262),(83,123,204,145,93,133,214,155),(84,106,249,270,94,116,259,280),(85,121,206,143,95,131,216,153),(86,104,251,268,96,114,241,278),(87,139,208,141,97,129,218,151),(88,102,253,266,98,112,243,276),(89,137,210,159,99,127,220,149),(90,120,255,264,100,110,245,274),(142,221,305,171,152,231,315,161),(144,239,307,169,154,229,317,179),(146,237,309,167,156,227,319,177),(148,235,311,165,158,225,301,175),(150,233,313,163,160,223,303,173)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C4 | D4 | D5 | D10 | Dic5 | C4≀C2 | C5⋊D4 | C4.10D4 | C20.10D4 | D4⋊2Dic5 |
| kernel | C42.8D10 | C42.D5 | C5×C42.C2 | C5×C4⋊C4 | C2×C20 | C42.C2 | C42 | C4⋊C4 | C10 | C2×C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 8 | 1 | 4 | 8 |
Matrix representation of C42.8D10 ►in GL6(𝔽41)
| 9 | 39 | 0 | 0 | 0 | 0 |
| 40 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 18 | 28 | 0 | 0 | 0 | 0 |
| 25 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 5 | 21 | 0 | 0 | 0 | 0 |
| 14 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 25 | 0 | 0 |
| 0 | 0 | 28 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 5 | 9 |
G:=sub<GL(6,GF(41))| [9,40,0,0,0,0,39,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,9,0,0,0,0,18,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[18,25,0,0,0,0,28,23,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[5,14,0,0,0,0,21,36,0,0,0,0,0,0,23,28,0,0,0,0,25,18,0,0,0,0,0,0,40,5,0,0,0,0,0,9] >;
C42.8D10 in GAP, Magma, Sage, TeX
C_4^2._8D_{10} % in TeX
G:=Group("C4^2.8D10"); // GroupNames label
G:=SmallGroup(320,101);
// by ID
G=gap.SmallGroup(320,101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,268,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^-1*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^9>;
// generators/relations