metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.59D10, C5⋊Q16⋊9C4, (C4×Q8).7D5, Q8.6(C4×D5), (Q8×C20).8C2, C4⋊C4.256D10, C5⋊5(Q16⋊C4), (C2×C20).260D4, C10.106(C4×D4), C4.43(C4○D20), C20.63(C4○D4), C20.63(C22×C4), (C2×Q8).163D10, (C2×C20).350C23, (C4×C20).101C22, (C4×Dic10).15C2, Dic10.32(C2×C4), Q8⋊Dic5.10C2, C10.Q16.10C2, C20.Q8.12C2, C42.D5.4C2, C2.4(D4.9D10), C2.4(C20.C23), C4⋊Dic5.333C22, (Q8×C10).198C22, C10.111(C8.C22), (C2×Dic10).275C22, C4.28(C2×C4×D5), C5⋊2C8.4(C2×C4), C2.22(C4×C5⋊D4), (C5×Q8).28(C2×C4), (C2×C5⋊Q16).4C2, (C2×C10).481(C2×D4), C22.82(C2×C5⋊D4), (C2×C4).223(C5⋊D4), (C5×C4⋊C4).287C22, (C2×C4).450(C22×D5), (C2×C5⋊2C8).103C22, SmallGroup(320,657)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.59D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c9 >
Subgroups: 310 in 108 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C4×Q8, C2×Q16, C5⋊2C8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q16⋊C4, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C5⋊Q16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C42.D5, C20.Q8, C10.Q16, Q8⋊Dic5, C4×Dic10, C2×C5⋊Q16, Q8×C20, C42.59D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8.C22, C4×D5, C5⋊D4, C22×D5, Q16⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C20.C23, D4.9D10, C42.59D10
►Regular action on 320 points
Generators in S320
(1 225 84 252)(2 236 85 243)(3 227 86 254)(4 238 87 245)(5 229 88 256)(6 240 89 247)(7 231 90 258)(8 222 91 249)(9 233 92 260)(10 224 93 251)(11 235 94 242)(12 226 95 253)(13 237 96 244)(14 228 97 255)(15 239 98 246)(16 230 99 257)(17 221 100 248)(18 232 81 259)(19 223 82 250)(20 234 83 241)(21 52 213 155)(22 43 214 146)(23 54 215 157)(24 45 216 148)(25 56 217 159)(26 47 218 150)(27 58 219 141)(28 49 220 152)(29 60 201 143)(30 51 202 154)(31 42 203 145)(32 53 204 156)(33 44 205 147)(34 55 206 158)(35 46 207 149)(36 57 208 160)(37 48 209 151)(38 59 210 142)(39 50 211 153)(40 41 212 144)(61 263 128 182)(62 274 129 193)(63 265 130 184)(64 276 131 195)(65 267 132 186)(66 278 133 197)(67 269 134 188)(68 280 135 199)(69 271 136 190)(70 262 137 181)(71 273 138 192)(72 264 139 183)(73 275 140 194)(74 266 121 185)(75 277 122 196)(76 268 123 187)(77 279 124 198)(78 270 125 189)(79 261 126 200)(80 272 127 191)(101 162 290 311)(102 173 291 302)(103 164 292 313)(104 175 293 304)(105 166 294 315)(106 177 295 306)(107 168 296 317)(108 179 297 308)(109 170 298 319)(110 161 299 310)(111 172 300 301)(112 163 281 312)(113 174 282 303)(114 165 283 314)(115 176 284 305)(116 167 285 316)(117 178 286 307)(118 169 287 318)(119 180 288 309)(120 171 289 320)
(1 273 11 263)(2 264 12 274)(3 275 13 265)(4 266 14 276)(5 277 15 267)(6 268 16 278)(7 279 17 269)(8 270 18 280)(9 261 19 271)(10 272 20 262)(21 177 31 167)(22 168 32 178)(23 179 33 169)(24 170 34 180)(25 161 35 171)(26 172 36 162)(27 163 37 173)(28 174 38 164)(29 165 39 175)(30 176 40 166)(41 294 51 284)(42 285 52 295)(43 296 53 286)(44 287 54 297)(45 298 55 288)(46 289 56 299)(47 300 57 290)(48 291 58 281)(49 282 59 292)(50 293 60 283)(61 252 71 242)(62 243 72 253)(63 254 73 244)(64 245 74 255)(65 256 75 246)(66 247 76 257)(67 258 77 248)(68 249 78 259)(69 260 79 250)(70 251 80 241)(81 199 91 189)(82 190 92 200)(83 181 93 191)(84 192 94 182)(85 183 95 193)(86 194 96 184)(87 185 97 195)(88 196 98 186)(89 187 99 197)(90 198 100 188)(101 150 111 160)(102 141 112 151)(103 152 113 142)(104 143 114 153)(105 154 115 144)(106 145 116 155)(107 156 117 146)(108 147 118 157)(109 158 119 148)(110 149 120 159)(121 228 131 238)(122 239 132 229)(123 230 133 240)(124 221 134 231)(125 232 135 222)(126 223 136 233)(127 234 137 224)(128 225 138 235)(129 236 139 226)(130 227 140 237)(201 314 211 304)(202 305 212 315)(203 316 213 306)(204 307 214 317)(205 318 215 308)(206 309 216 319)(207 320 217 310)(208 311 218 301)(209 302 219 312)(210 313 220 303)
(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)
(1 201 263 304 11 211 273 314)(2 303 274 220 12 313 264 210)(3 219 265 302 13 209 275 312)(4 301 276 218 14 311 266 208)(5 217 267 320 15 207 277 310)(6 319 278 216 16 309 268 206)(7 215 269 318 17 205 279 308)(8 317 280 214 18 307 270 204)(9 213 271 316 19 203 261 306)(10 315 262 212 20 305 272 202)(21 190 167 82 31 200 177 92)(22 81 178 189 32 91 168 199)(23 188 169 100 33 198 179 90)(24 99 180 187 34 89 170 197)(25 186 171 98 35 196 161 88)(26 97 162 185 36 87 172 195)(27 184 173 96 37 194 163 86)(28 95 164 183 38 85 174 193)(29 182 175 94 39 192 165 84)(30 93 166 181 40 83 176 191)(41 251 284 70 51 241 294 80)(42 69 295 250 52 79 285 260)(43 249 286 68 53 259 296 78)(44 67 297 248 54 77 287 258)(45 247 288 66 55 257 298 76)(46 65 299 246 56 75 289 256)(47 245 290 64 57 255 300 74)(48 63 281 244 58 73 291 254)(49 243 292 62 59 253 282 72)(50 61 283 242 60 71 293 252)(101 131 160 228 111 121 150 238)(102 227 151 130 112 237 141 140)(103 129 142 226 113 139 152 236)(104 225 153 128 114 235 143 138)(105 127 144 224 115 137 154 234)(106 223 155 126 116 233 145 136)(107 125 146 222 117 135 156 232)(108 221 157 124 118 231 147 134)(109 123 148 240 119 133 158 230)(110 239 159 122 120 229 149 132)
G:=sub<Sym(320)| (1,225,84,252)(2,236,85,243)(3,227,86,254)(4,238,87,245)(5,229,88,256)(6,240,89,247)(7,231,90,258)(8,222,91,249)(9,233,92,260)(10,224,93,251)(11,235,94,242)(12,226,95,253)(13,237,96,244)(14,228,97,255)(15,239,98,246)(16,230,99,257)(17,221,100,248)(18,232,81,259)(19,223,82,250)(20,234,83,241)(21,52,213,155)(22,43,214,146)(23,54,215,157)(24,45,216,148)(25,56,217,159)(26,47,218,150)(27,58,219,141)(28,49,220,152)(29,60,201,143)(30,51,202,154)(31,42,203,145)(32,53,204,156)(33,44,205,147)(34,55,206,158)(35,46,207,149)(36,57,208,160)(37,48,209,151)(38,59,210,142)(39,50,211,153)(40,41,212,144)(61,263,128,182)(62,274,129,193)(63,265,130,184)(64,276,131,195)(65,267,132,186)(66,278,133,197)(67,269,134,188)(68,280,135,199)(69,271,136,190)(70,262,137,181)(71,273,138,192)(72,264,139,183)(73,275,140,194)(74,266,121,185)(75,277,122,196)(76,268,123,187)(77,279,124,198)(78,270,125,189)(79,261,126,200)(80,272,127,191)(101,162,290,311)(102,173,291,302)(103,164,292,313)(104,175,293,304)(105,166,294,315)(106,177,295,306)(107,168,296,317)(108,179,297,308)(109,170,298,319)(110,161,299,310)(111,172,300,301)(112,163,281,312)(113,174,282,303)(114,165,283,314)(115,176,284,305)(116,167,285,316)(117,178,286,307)(118,169,287,318)(119,180,288,309)(120,171,289,320), (1,273,11,263)(2,264,12,274)(3,275,13,265)(4,266,14,276)(5,277,15,267)(6,268,16,278)(7,279,17,269)(8,270,18,280)(9,261,19,271)(10,272,20,262)(21,177,31,167)(22,168,32,178)(23,179,33,169)(24,170,34,180)(25,161,35,171)(26,172,36,162)(27,163,37,173)(28,174,38,164)(29,165,39,175)(30,176,40,166)(41,294,51,284)(42,285,52,295)(43,296,53,286)(44,287,54,297)(45,298,55,288)(46,289,56,299)(47,300,57,290)(48,291,58,281)(49,282,59,292)(50,293,60,283)(61,252,71,242)(62,243,72,253)(63,254,73,244)(64,245,74,255)(65,256,75,246)(66,247,76,257)(67,258,77,248)(68,249,78,259)(69,260,79,250)(70,251,80,241)(81,199,91,189)(82,190,92,200)(83,181,93,191)(84,192,94,182)(85,183,95,193)(86,194,96,184)(87,185,97,195)(88,196,98,186)(89,187,99,197)(90,198,100,188)(101,150,111,160)(102,141,112,151)(103,152,113,142)(104,143,114,153)(105,154,115,144)(106,145,116,155)(107,156,117,146)(108,147,118,157)(109,158,119,148)(110,149,120,159)(121,228,131,238)(122,239,132,229)(123,230,133,240)(124,221,134,231)(125,232,135,222)(126,223,136,233)(127,234,137,224)(128,225,138,235)(129,236,139,226)(130,227,140,237)(201,314,211,304)(202,305,212,315)(203,316,213,306)(204,307,214,317)(205,318,215,308)(206,309,216,319)(207,320,217,310)(208,311,218,301)(209,302,219,312)(210,313,220,303), (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), (1,201,263,304,11,211,273,314)(2,303,274,220,12,313,264,210)(3,219,265,302,13,209,275,312)(4,301,276,218,14,311,266,208)(5,217,267,320,15,207,277,310)(6,319,278,216,16,309,268,206)(7,215,269,318,17,205,279,308)(8,317,280,214,18,307,270,204)(9,213,271,316,19,203,261,306)(10,315,262,212,20,305,272,202)(21,190,167,82,31,200,177,92)(22,81,178,189,32,91,168,199)(23,188,169,100,33,198,179,90)(24,99,180,187,34,89,170,197)(25,186,171,98,35,196,161,88)(26,97,162,185,36,87,172,195)(27,184,173,96,37,194,163,86)(28,95,164,183,38,85,174,193)(29,182,175,94,39,192,165,84)(30,93,166,181,40,83,176,191)(41,251,284,70,51,241,294,80)(42,69,295,250,52,79,285,260)(43,249,286,68,53,259,296,78)(44,67,297,248,54,77,287,258)(45,247,288,66,55,257,298,76)(46,65,299,246,56,75,289,256)(47,245,290,64,57,255,300,74)(48,63,281,244,58,73,291,254)(49,243,292,62,59,253,282,72)(50,61,283,242,60,71,293,252)(101,131,160,228,111,121,150,238)(102,227,151,130,112,237,141,140)(103,129,142,226,113,139,152,236)(104,225,153,128,114,235,143,138)(105,127,144,224,115,137,154,234)(106,223,155,126,116,233,145,136)(107,125,146,222,117,135,156,232)(108,221,157,124,118,231,147,134)(109,123,148,240,119,133,158,230)(110,239,159,122,120,229,149,132)>;
G:=Group( (1,225,84,252)(2,236,85,243)(3,227,86,254)(4,238,87,245)(5,229,88,256)(6,240,89,247)(7,231,90,258)(8,222,91,249)(9,233,92,260)(10,224,93,251)(11,235,94,242)(12,226,95,253)(13,237,96,244)(14,228,97,255)(15,239,98,246)(16,230,99,257)(17,221,100,248)(18,232,81,259)(19,223,82,250)(20,234,83,241)(21,52,213,155)(22,43,214,146)(23,54,215,157)(24,45,216,148)(25,56,217,159)(26,47,218,150)(27,58,219,141)(28,49,220,152)(29,60,201,143)(30,51,202,154)(31,42,203,145)(32,53,204,156)(33,44,205,147)(34,55,206,158)(35,46,207,149)(36,57,208,160)(37,48,209,151)(38,59,210,142)(39,50,211,153)(40,41,212,144)(61,263,128,182)(62,274,129,193)(63,265,130,184)(64,276,131,195)(65,267,132,186)(66,278,133,197)(67,269,134,188)(68,280,135,199)(69,271,136,190)(70,262,137,181)(71,273,138,192)(72,264,139,183)(73,275,140,194)(74,266,121,185)(75,277,122,196)(76,268,123,187)(77,279,124,198)(78,270,125,189)(79,261,126,200)(80,272,127,191)(101,162,290,311)(102,173,291,302)(103,164,292,313)(104,175,293,304)(105,166,294,315)(106,177,295,306)(107,168,296,317)(108,179,297,308)(109,170,298,319)(110,161,299,310)(111,172,300,301)(112,163,281,312)(113,174,282,303)(114,165,283,314)(115,176,284,305)(116,167,285,316)(117,178,286,307)(118,169,287,318)(119,180,288,309)(120,171,289,320), (1,273,11,263)(2,264,12,274)(3,275,13,265)(4,266,14,276)(5,277,15,267)(6,268,16,278)(7,279,17,269)(8,270,18,280)(9,261,19,271)(10,272,20,262)(21,177,31,167)(22,168,32,178)(23,179,33,169)(24,170,34,180)(25,161,35,171)(26,172,36,162)(27,163,37,173)(28,174,38,164)(29,165,39,175)(30,176,40,166)(41,294,51,284)(42,285,52,295)(43,296,53,286)(44,287,54,297)(45,298,55,288)(46,289,56,299)(47,300,57,290)(48,291,58,281)(49,282,59,292)(50,293,60,283)(61,252,71,242)(62,243,72,253)(63,254,73,244)(64,245,74,255)(65,256,75,246)(66,247,76,257)(67,258,77,248)(68,249,78,259)(69,260,79,250)(70,251,80,241)(81,199,91,189)(82,190,92,200)(83,181,93,191)(84,192,94,182)(85,183,95,193)(86,194,96,184)(87,185,97,195)(88,196,98,186)(89,187,99,197)(90,198,100,188)(101,150,111,160)(102,141,112,151)(103,152,113,142)(104,143,114,153)(105,154,115,144)(106,145,116,155)(107,156,117,146)(108,147,118,157)(109,158,119,148)(110,149,120,159)(121,228,131,238)(122,239,132,229)(123,230,133,240)(124,221,134,231)(125,232,135,222)(126,223,136,233)(127,234,137,224)(128,225,138,235)(129,236,139,226)(130,227,140,237)(201,314,211,304)(202,305,212,315)(203,316,213,306)(204,307,214,317)(205,318,215,308)(206,309,216,319)(207,320,217,310)(208,311,218,301)(209,302,219,312)(210,313,220,303), (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), (1,201,263,304,11,211,273,314)(2,303,274,220,12,313,264,210)(3,219,265,302,13,209,275,312)(4,301,276,218,14,311,266,208)(5,217,267,320,15,207,277,310)(6,319,278,216,16,309,268,206)(7,215,269,318,17,205,279,308)(8,317,280,214,18,307,270,204)(9,213,271,316,19,203,261,306)(10,315,262,212,20,305,272,202)(21,190,167,82,31,200,177,92)(22,81,178,189,32,91,168,199)(23,188,169,100,33,198,179,90)(24,99,180,187,34,89,170,197)(25,186,171,98,35,196,161,88)(26,97,162,185,36,87,172,195)(27,184,173,96,37,194,163,86)(28,95,164,183,38,85,174,193)(29,182,175,94,39,192,165,84)(30,93,166,181,40,83,176,191)(41,251,284,70,51,241,294,80)(42,69,295,250,52,79,285,260)(43,249,286,68,53,259,296,78)(44,67,297,248,54,77,287,258)(45,247,288,66,55,257,298,76)(46,65,299,246,56,75,289,256)(47,245,290,64,57,255,300,74)(48,63,281,244,58,73,291,254)(49,243,292,62,59,253,282,72)(50,61,283,242,60,71,293,252)(101,131,160,228,111,121,150,238)(102,227,151,130,112,237,141,140)(103,129,142,226,113,139,152,236)(104,225,153,128,114,235,143,138)(105,127,144,224,115,137,154,234)(106,223,155,126,116,233,145,136)(107,125,146,222,117,135,156,232)(108,221,157,124,118,231,147,134)(109,123,148,240,119,133,158,230)(110,239,159,122,120,229,149,132) );
G=PermutationGroup([[(1,225,84,252),(2,236,85,243),(3,227,86,254),(4,238,87,245),(5,229,88,256),(6,240,89,247),(7,231,90,258),(8,222,91,249),(9,233,92,260),(10,224,93,251),(11,235,94,242),(12,226,95,253),(13,237,96,244),(14,228,97,255),(15,239,98,246),(16,230,99,257),(17,221,100,248),(18,232,81,259),(19,223,82,250),(20,234,83,241),(21,52,213,155),(22,43,214,146),(23,54,215,157),(24,45,216,148),(25,56,217,159),(26,47,218,150),(27,58,219,141),(28,49,220,152),(29,60,201,143),(30,51,202,154),(31,42,203,145),(32,53,204,156),(33,44,205,147),(34,55,206,158),(35,46,207,149),(36,57,208,160),(37,48,209,151),(38,59,210,142),(39,50,211,153),(40,41,212,144),(61,263,128,182),(62,274,129,193),(63,265,130,184),(64,276,131,195),(65,267,132,186),(66,278,133,197),(67,269,134,188),(68,280,135,199),(69,271,136,190),(70,262,137,181),(71,273,138,192),(72,264,139,183),(73,275,140,194),(74,266,121,185),(75,277,122,196),(76,268,123,187),(77,279,124,198),(78,270,125,189),(79,261,126,200),(80,272,127,191),(101,162,290,311),(102,173,291,302),(103,164,292,313),(104,175,293,304),(105,166,294,315),(106,177,295,306),(107,168,296,317),(108,179,297,308),(109,170,298,319),(110,161,299,310),(111,172,300,301),(112,163,281,312),(113,174,282,303),(114,165,283,314),(115,176,284,305),(116,167,285,316),(117,178,286,307),(118,169,287,318),(119,180,288,309),(120,171,289,320)], [(1,273,11,263),(2,264,12,274),(3,275,13,265),(4,266,14,276),(5,277,15,267),(6,268,16,278),(7,279,17,269),(8,270,18,280),(9,261,19,271),(10,272,20,262),(21,177,31,167),(22,168,32,178),(23,179,33,169),(24,170,34,180),(25,161,35,171),(26,172,36,162),(27,163,37,173),(28,174,38,164),(29,165,39,175),(30,176,40,166),(41,294,51,284),(42,285,52,295),(43,296,53,286),(44,287,54,297),(45,298,55,288),(46,289,56,299),(47,300,57,290),(48,291,58,281),(49,282,59,292),(50,293,60,283),(61,252,71,242),(62,243,72,253),(63,254,73,244),(64,245,74,255),(65,256,75,246),(66,247,76,257),(67,258,77,248),(68,249,78,259),(69,260,79,250),(70,251,80,241),(81,199,91,189),(82,190,92,200),(83,181,93,191),(84,192,94,182),(85,183,95,193),(86,194,96,184),(87,185,97,195),(88,196,98,186),(89,187,99,197),(90,198,100,188),(101,150,111,160),(102,141,112,151),(103,152,113,142),(104,143,114,153),(105,154,115,144),(106,145,116,155),(107,156,117,146),(108,147,118,157),(109,158,119,148),(110,149,120,159),(121,228,131,238),(122,239,132,229),(123,230,133,240),(124,221,134,231),(125,232,135,222),(126,223,136,233),(127,234,137,224),(128,225,138,235),(129,236,139,226),(130,227,140,237),(201,314,211,304),(202,305,212,315),(203,316,213,306),(204,307,214,317),(205,318,215,308),(206,309,216,319),(207,320,217,310),(208,311,218,301),(209,302,219,312),(210,313,220,303)], [(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)], [(1,201,263,304,11,211,273,314),(2,303,274,220,12,313,264,210),(3,219,265,302,13,209,275,312),(4,301,276,218,14,311,266,208),(5,217,267,320,15,207,277,310),(6,319,278,216,16,309,268,206),(7,215,269,318,17,205,279,308),(8,317,280,214,18,307,270,204),(9,213,271,316,19,203,261,306),(10,315,262,212,20,305,272,202),(21,190,167,82,31,200,177,92),(22,81,178,189,32,91,168,199),(23,188,169,100,33,198,179,90),(24,99,180,187,34,89,170,197),(25,186,171,98,35,196,161,88),(26,97,162,185,36,87,172,195),(27,184,173,96,37,194,163,86),(28,95,164,183,38,85,174,193),(29,182,175,94,39,192,165,84),(30,93,166,181,40,83,176,191),(41,251,284,70,51,241,294,80),(42,69,295,250,52,79,285,260),(43,249,286,68,53,259,296,78),(44,67,297,248,54,77,287,258),(45,247,288,66,55,257,298,76),(46,65,299,246,56,75,289,256),(47,245,290,64,57,255,300,74),(48,63,281,244,58,73,291,254),(49,243,292,62,59,253,282,72),(50,61,283,242,60,71,293,252),(101,131,160,228,111,121,150,238),(102,227,151,130,112,237,141,140),(103,129,142,226,113,139,152,236),(104,225,153,128,114,235,143,138),(105,127,144,224,115,137,154,234),(106,223,155,126,116,233,145,136),(107,125,146,222,117,135,156,232),(108,221,157,124,118,231,147,134),(109,123,148,240,119,133,158,230),(110,239,159,122,120,229,149,132)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 | C8.C22 | C20.C23 | D4.9D10 |
| kernel | C42.59D10 | C42.D5 | C20.Q8 | C10.Q16 | Q8⋊Dic5 | C4×Dic10 | C2×C5⋊Q16 | Q8×C20 | C5⋊Q16 | C2×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 2 | 4 | 4 |
Matrix representation of C42.59D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 6 |
| 0 | 0 | 0 | 0 | 35 | 18 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 6 | 23 | 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 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 20 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 10 | 29 | 11 |
| 0 | 0 | 31 | 13 | 30 | 13 |
| 0 | 0 | 29 | 11 | 6 | 31 |
| 0 | 0 | 30 | 13 | 10 | 28 |
| 35 | 31 | 0 | 0 | 0 | 0 |
| 16 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 4 | 18 | 34 |
| 0 | 0 | 12 | 26 | 19 | 23 |
| 0 | 0 | 23 | 7 | 15 | 4 |
| 0 | 0 | 22 | 18 | 12 | 26 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,23,35,0,0,0,0,6,18,0,0],[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],[31,20,0,0,0,0,0,37,0,0,0,0,0,0,35,31,29,30,0,0,10,13,11,13,0,0,29,30,6,10,0,0,11,13,31,28],[35,16,0,0,0,0,31,6,0,0,0,0,0,0,15,12,23,22,0,0,4,26,7,18,0,0,18,19,15,12,0,0,34,23,4,26] >;
C42.59D10 in GAP, Magma, Sage, TeX
C_4^2._{59}D_{10} % in TeX
G:=Group("C4^2.59D10"); // GroupNames label
G:=SmallGroup(320,657);
// by ID
G=gap.SmallGroup(320,657);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,387,58,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations