metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.77D10, C4⋊Q8.8D5, (C2×C20).294D4, (C2×Q8).43D10, C20.79(C4○D4), C20.6Q8.9C2, C4.25(D4⋊2D5), (C2×C20).400C23, (C4×C20).129C22, Q8⋊Dic5.12C2, (Q8×C10).61C22, C42.D5.8C2, C10.46(C4.4D4), C10.94(C8.C22), C4⋊Dic5.160C22, C2.15(C20.C23), C2.13(C20.17D4), C5⋊4(C42.30C22), (C5×C4⋊Q8).8C2, (C2×C10).531(C2×D4), (C2×C4).72(C5⋊D4), (C2×C4).497(C22×D5), C22.203(C2×C5⋊D4), (C2×C5⋊2C8).134C22, SmallGroup(320,709)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.77D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc9 >
Subgroups: 270 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, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×Q8, C42.30C22, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, Q8×C10, C42.D5, Q8⋊Dic5, C20.6Q8, C5×C4⋊Q8, C42.77D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8.C22, C5⋊D4, C22×D5, C42.30C22, D4⋊2D5, C2×C5⋊D4, C20.17D4, C20.C23, C42.77D10
►Regular action on 320 points
Generators in S320
(1 29 151 129)(2 130 152 30)(3 31 153 131)(4 132 154 32)(5 33 155 133)(6 134 156 34)(7 35 157 135)(8 136 158 36)(9 37 159 137)(10 138 160 38)(11 39 141 139)(12 140 142 40)(13 21 143 121)(14 122 144 22)(15 23 145 123)(16 124 146 24)(17 25 147 125)(18 126 148 26)(19 27 149 127)(20 128 150 28)(41 79 113 233)(42 234 114 80)(43 61 115 235)(44 236 116 62)(45 63 117 237)(46 238 118 64)(47 65 119 239)(48 240 120 66)(49 67 101 221)(50 222 102 68)(51 69 103 223)(52 224 104 70)(53 71 105 225)(54 226 106 72)(55 73 107 227)(56 228 108 74)(57 75 109 229)(58 230 110 76)(59 77 111 231)(60 232 112 78)(81 268 253 293)(82 294 254 269)(83 270 255 295)(84 296 256 271)(85 272 257 297)(86 298 258 273)(87 274 259 299)(88 300 260 275)(89 276 241 281)(90 282 242 277)(91 278 243 283)(92 284 244 279)(93 280 245 285)(94 286 246 261)(95 262 247 287)(96 288 248 263)(97 264 249 289)(98 290 250 265)(99 266 251 291)(100 292 252 267)(161 205 200 302)(162 303 181 206)(163 207 182 304)(164 305 183 208)(165 209 184 306)(166 307 185 210)(167 211 186 308)(168 309 187 212)(169 213 188 310)(170 311 189 214)(171 215 190 312)(172 313 191 216)(173 217 192 314)(174 315 193 218)(175 219 194 316)(176 317 195 220)(177 201 196 318)(178 319 197 202)(179 203 198 320)(180 301 199 204)
(1 114 11 104)(2 105 12 115)(3 116 13 106)(4 107 14 117)(5 118 15 108)(6 109 16 119)(7 120 17 110)(8 111 18 101)(9 102 19 112)(10 113 20 103)(21 72 31 62)(22 63 32 73)(23 74 33 64)(24 65 34 75)(25 76 35 66)(26 67 36 77)(27 78 37 68)(28 69 38 79)(29 80 39 70)(30 71 40 61)(41 150 51 160)(42 141 52 151)(43 152 53 142)(44 143 54 153)(45 154 55 144)(46 145 56 155)(47 156 57 146)(48 147 58 157)(49 158 59 148)(50 149 60 159)(81 173 91 163)(82 164 92 174)(83 175 93 165)(84 166 94 176)(85 177 95 167)(86 168 96 178)(87 179 97 169)(88 170 98 180)(89 161 99 171)(90 172 100 162)(121 226 131 236)(122 237 132 227)(123 228 133 238)(124 239 134 229)(125 230 135 240)(126 221 136 231)(127 232 137 222)(128 223 138 233)(129 234 139 224)(130 225 140 235)(181 242 191 252)(182 253 192 243)(183 244 193 254)(184 255 194 245)(185 246 195 256)(186 257 196 247)(187 248 197 258)(188 259 198 249)(189 250 199 260)(190 241 200 251)(201 262 211 272)(202 273 212 263)(203 264 213 274)(204 275 214 265)(205 266 215 276)(206 277 216 267)(207 268 217 278)(208 279 218 269)(209 270 219 280)(210 261 220 271)(281 302 291 312)(282 313 292 303)(283 304 293 314)(284 315 294 305)(285 306 295 316)(286 317 296 307)(287 308 297 318)(288 319 298 309)(289 310 299 320)(290 301 300 311)
(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)
(1 176 141 185)(2 245 142 83)(3 174 143 183)(4 243 144 81)(5 172 145 181)(6 241 146 99)(7 170 147 199)(8 259 148 97)(9 168 149 197)(10 257 150 95)(11 166 151 195)(12 255 152 93)(13 164 153 193)(14 253 154 91)(15 162 155 191)(16 251 156 89)(17 180 157 189)(18 249 158 87)(19 178 159 187)(20 247 160 85)(21 218 131 305)(22 283 132 268)(23 216 133 303)(24 281 134 266)(25 214 135 301)(26 299 136 264)(27 212 137 319)(28 297 138 262)(29 210 139 317)(30 295 140 280)(31 208 121 315)(32 293 122 278)(33 206 123 313)(34 291 124 276)(35 204 125 311)(36 289 126 274)(37 202 127 309)(38 287 128 272)(39 220 129 307)(40 285 130 270)(41 167 103 196)(42 246 104 84)(43 165 105 194)(44 244 106 82)(45 163 107 192)(46 242 108 100)(47 161 109 190)(48 260 110 98)(49 179 111 188)(50 258 112 96)(51 177 113 186)(52 256 114 94)(53 175 115 184)(54 254 116 92)(55 173 117 182)(56 252 118 90)(57 171 119 200)(58 250 120 88)(59 169 101 198)(60 248 102 86)(61 316 225 209)(62 269 226 284)(63 314 227 207)(64 267 228 282)(65 312 229 205)(66 265 230 300)(67 310 231 203)(68 263 232 298)(69 308 233 201)(70 261 234 296)(71 306 235 219)(72 279 236 294)(73 304 237 217)(74 277 238 292)(75 302 239 215)(76 275 240 290)(77 320 221 213)(78 273 222 288)(79 318 223 211)(80 271 224 286)
G:=sub<Sym(320)| (1,29,151,129)(2,130,152,30)(3,31,153,131)(4,132,154,32)(5,33,155,133)(6,134,156,34)(7,35,157,135)(8,136,158,36)(9,37,159,137)(10,138,160,38)(11,39,141,139)(12,140,142,40)(13,21,143,121)(14,122,144,22)(15,23,145,123)(16,124,146,24)(17,25,147,125)(18,126,148,26)(19,27,149,127)(20,128,150,28)(41,79,113,233)(42,234,114,80)(43,61,115,235)(44,236,116,62)(45,63,117,237)(46,238,118,64)(47,65,119,239)(48,240,120,66)(49,67,101,221)(50,222,102,68)(51,69,103,223)(52,224,104,70)(53,71,105,225)(54,226,106,72)(55,73,107,227)(56,228,108,74)(57,75,109,229)(58,230,110,76)(59,77,111,231)(60,232,112,78)(81,268,253,293)(82,294,254,269)(83,270,255,295)(84,296,256,271)(85,272,257,297)(86,298,258,273)(87,274,259,299)(88,300,260,275)(89,276,241,281)(90,282,242,277)(91,278,243,283)(92,284,244,279)(93,280,245,285)(94,286,246,261)(95,262,247,287)(96,288,248,263)(97,264,249,289)(98,290,250,265)(99,266,251,291)(100,292,252,267)(161,205,200,302)(162,303,181,206)(163,207,182,304)(164,305,183,208)(165,209,184,306)(166,307,185,210)(167,211,186,308)(168,309,187,212)(169,213,188,310)(170,311,189,214)(171,215,190,312)(172,313,191,216)(173,217,192,314)(174,315,193,218)(175,219,194,316)(176,317,195,220)(177,201,196,318)(178,319,197,202)(179,203,198,320)(180,301,199,204), (1,114,11,104)(2,105,12,115)(3,116,13,106)(4,107,14,117)(5,118,15,108)(6,109,16,119)(7,120,17,110)(8,111,18,101)(9,102,19,112)(10,113,20,103)(21,72,31,62)(22,63,32,73)(23,74,33,64)(24,65,34,75)(25,76,35,66)(26,67,36,77)(27,78,37,68)(28,69,38,79)(29,80,39,70)(30,71,40,61)(41,150,51,160)(42,141,52,151)(43,152,53,142)(44,143,54,153)(45,154,55,144)(46,145,56,155)(47,156,57,146)(48,147,58,157)(49,158,59,148)(50,149,60,159)(81,173,91,163)(82,164,92,174)(83,175,93,165)(84,166,94,176)(85,177,95,167)(86,168,96,178)(87,179,97,169)(88,170,98,180)(89,161,99,171)(90,172,100,162)(121,226,131,236)(122,237,132,227)(123,228,133,238)(124,239,134,229)(125,230,135,240)(126,221,136,231)(127,232,137,222)(128,223,138,233)(129,234,139,224)(130,225,140,235)(181,242,191,252)(182,253,192,243)(183,244,193,254)(184,255,194,245)(185,246,195,256)(186,257,196,247)(187,248,197,258)(188,259,198,249)(189,250,199,260)(190,241,200,251)(201,262,211,272)(202,273,212,263)(203,264,213,274)(204,275,214,265)(205,266,215,276)(206,277,216,267)(207,268,217,278)(208,279,218,269)(209,270,219,280)(210,261,220,271)(281,302,291,312)(282,313,292,303)(283,304,293,314)(284,315,294,305)(285,306,295,316)(286,317,296,307)(287,308,297,318)(288,319,298,309)(289,310,299,320)(290,301,300,311), (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), (1,176,141,185)(2,245,142,83)(3,174,143,183)(4,243,144,81)(5,172,145,181)(6,241,146,99)(7,170,147,199)(8,259,148,97)(9,168,149,197)(10,257,150,95)(11,166,151,195)(12,255,152,93)(13,164,153,193)(14,253,154,91)(15,162,155,191)(16,251,156,89)(17,180,157,189)(18,249,158,87)(19,178,159,187)(20,247,160,85)(21,218,131,305)(22,283,132,268)(23,216,133,303)(24,281,134,266)(25,214,135,301)(26,299,136,264)(27,212,137,319)(28,297,138,262)(29,210,139,317)(30,295,140,280)(31,208,121,315)(32,293,122,278)(33,206,123,313)(34,291,124,276)(35,204,125,311)(36,289,126,274)(37,202,127,309)(38,287,128,272)(39,220,129,307)(40,285,130,270)(41,167,103,196)(42,246,104,84)(43,165,105,194)(44,244,106,82)(45,163,107,192)(46,242,108,100)(47,161,109,190)(48,260,110,98)(49,179,111,188)(50,258,112,96)(51,177,113,186)(52,256,114,94)(53,175,115,184)(54,254,116,92)(55,173,117,182)(56,252,118,90)(57,171,119,200)(58,250,120,88)(59,169,101,198)(60,248,102,86)(61,316,225,209)(62,269,226,284)(63,314,227,207)(64,267,228,282)(65,312,229,205)(66,265,230,300)(67,310,231,203)(68,263,232,298)(69,308,233,201)(70,261,234,296)(71,306,235,219)(72,279,236,294)(73,304,237,217)(74,277,238,292)(75,302,239,215)(76,275,240,290)(77,320,221,213)(78,273,222,288)(79,318,223,211)(80,271,224,286)>;
G:=Group( (1,29,151,129)(2,130,152,30)(3,31,153,131)(4,132,154,32)(5,33,155,133)(6,134,156,34)(7,35,157,135)(8,136,158,36)(9,37,159,137)(10,138,160,38)(11,39,141,139)(12,140,142,40)(13,21,143,121)(14,122,144,22)(15,23,145,123)(16,124,146,24)(17,25,147,125)(18,126,148,26)(19,27,149,127)(20,128,150,28)(41,79,113,233)(42,234,114,80)(43,61,115,235)(44,236,116,62)(45,63,117,237)(46,238,118,64)(47,65,119,239)(48,240,120,66)(49,67,101,221)(50,222,102,68)(51,69,103,223)(52,224,104,70)(53,71,105,225)(54,226,106,72)(55,73,107,227)(56,228,108,74)(57,75,109,229)(58,230,110,76)(59,77,111,231)(60,232,112,78)(81,268,253,293)(82,294,254,269)(83,270,255,295)(84,296,256,271)(85,272,257,297)(86,298,258,273)(87,274,259,299)(88,300,260,275)(89,276,241,281)(90,282,242,277)(91,278,243,283)(92,284,244,279)(93,280,245,285)(94,286,246,261)(95,262,247,287)(96,288,248,263)(97,264,249,289)(98,290,250,265)(99,266,251,291)(100,292,252,267)(161,205,200,302)(162,303,181,206)(163,207,182,304)(164,305,183,208)(165,209,184,306)(166,307,185,210)(167,211,186,308)(168,309,187,212)(169,213,188,310)(170,311,189,214)(171,215,190,312)(172,313,191,216)(173,217,192,314)(174,315,193,218)(175,219,194,316)(176,317,195,220)(177,201,196,318)(178,319,197,202)(179,203,198,320)(180,301,199,204), (1,114,11,104)(2,105,12,115)(3,116,13,106)(4,107,14,117)(5,118,15,108)(6,109,16,119)(7,120,17,110)(8,111,18,101)(9,102,19,112)(10,113,20,103)(21,72,31,62)(22,63,32,73)(23,74,33,64)(24,65,34,75)(25,76,35,66)(26,67,36,77)(27,78,37,68)(28,69,38,79)(29,80,39,70)(30,71,40,61)(41,150,51,160)(42,141,52,151)(43,152,53,142)(44,143,54,153)(45,154,55,144)(46,145,56,155)(47,156,57,146)(48,147,58,157)(49,158,59,148)(50,149,60,159)(81,173,91,163)(82,164,92,174)(83,175,93,165)(84,166,94,176)(85,177,95,167)(86,168,96,178)(87,179,97,169)(88,170,98,180)(89,161,99,171)(90,172,100,162)(121,226,131,236)(122,237,132,227)(123,228,133,238)(124,239,134,229)(125,230,135,240)(126,221,136,231)(127,232,137,222)(128,223,138,233)(129,234,139,224)(130,225,140,235)(181,242,191,252)(182,253,192,243)(183,244,193,254)(184,255,194,245)(185,246,195,256)(186,257,196,247)(187,248,197,258)(188,259,198,249)(189,250,199,260)(190,241,200,251)(201,262,211,272)(202,273,212,263)(203,264,213,274)(204,275,214,265)(205,266,215,276)(206,277,216,267)(207,268,217,278)(208,279,218,269)(209,270,219,280)(210,261,220,271)(281,302,291,312)(282,313,292,303)(283,304,293,314)(284,315,294,305)(285,306,295,316)(286,317,296,307)(287,308,297,318)(288,319,298,309)(289,310,299,320)(290,301,300,311), (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), (1,176,141,185)(2,245,142,83)(3,174,143,183)(4,243,144,81)(5,172,145,181)(6,241,146,99)(7,170,147,199)(8,259,148,97)(9,168,149,197)(10,257,150,95)(11,166,151,195)(12,255,152,93)(13,164,153,193)(14,253,154,91)(15,162,155,191)(16,251,156,89)(17,180,157,189)(18,249,158,87)(19,178,159,187)(20,247,160,85)(21,218,131,305)(22,283,132,268)(23,216,133,303)(24,281,134,266)(25,214,135,301)(26,299,136,264)(27,212,137,319)(28,297,138,262)(29,210,139,317)(30,295,140,280)(31,208,121,315)(32,293,122,278)(33,206,123,313)(34,291,124,276)(35,204,125,311)(36,289,126,274)(37,202,127,309)(38,287,128,272)(39,220,129,307)(40,285,130,270)(41,167,103,196)(42,246,104,84)(43,165,105,194)(44,244,106,82)(45,163,107,192)(46,242,108,100)(47,161,109,190)(48,260,110,98)(49,179,111,188)(50,258,112,96)(51,177,113,186)(52,256,114,94)(53,175,115,184)(54,254,116,92)(55,173,117,182)(56,252,118,90)(57,171,119,200)(58,250,120,88)(59,169,101,198)(60,248,102,86)(61,316,225,209)(62,269,226,284)(63,314,227,207)(64,267,228,282)(65,312,229,205)(66,265,230,300)(67,310,231,203)(68,263,232,298)(69,308,233,201)(70,261,234,296)(71,306,235,219)(72,279,236,294)(73,304,237,217)(74,277,238,292)(75,302,239,215)(76,275,240,290)(77,320,221,213)(78,273,222,288)(79,318,223,211)(80,271,224,286) );
G=PermutationGroup([[(1,29,151,129),(2,130,152,30),(3,31,153,131),(4,132,154,32),(5,33,155,133),(6,134,156,34),(7,35,157,135),(8,136,158,36),(9,37,159,137),(10,138,160,38),(11,39,141,139),(12,140,142,40),(13,21,143,121),(14,122,144,22),(15,23,145,123),(16,124,146,24),(17,25,147,125),(18,126,148,26),(19,27,149,127),(20,128,150,28),(41,79,113,233),(42,234,114,80),(43,61,115,235),(44,236,116,62),(45,63,117,237),(46,238,118,64),(47,65,119,239),(48,240,120,66),(49,67,101,221),(50,222,102,68),(51,69,103,223),(52,224,104,70),(53,71,105,225),(54,226,106,72),(55,73,107,227),(56,228,108,74),(57,75,109,229),(58,230,110,76),(59,77,111,231),(60,232,112,78),(81,268,253,293),(82,294,254,269),(83,270,255,295),(84,296,256,271),(85,272,257,297),(86,298,258,273),(87,274,259,299),(88,300,260,275),(89,276,241,281),(90,282,242,277),(91,278,243,283),(92,284,244,279),(93,280,245,285),(94,286,246,261),(95,262,247,287),(96,288,248,263),(97,264,249,289),(98,290,250,265),(99,266,251,291),(100,292,252,267),(161,205,200,302),(162,303,181,206),(163,207,182,304),(164,305,183,208),(165,209,184,306),(166,307,185,210),(167,211,186,308),(168,309,187,212),(169,213,188,310),(170,311,189,214),(171,215,190,312),(172,313,191,216),(173,217,192,314),(174,315,193,218),(175,219,194,316),(176,317,195,220),(177,201,196,318),(178,319,197,202),(179,203,198,320),(180,301,199,204)], [(1,114,11,104),(2,105,12,115),(3,116,13,106),(4,107,14,117),(5,118,15,108),(6,109,16,119),(7,120,17,110),(8,111,18,101),(9,102,19,112),(10,113,20,103),(21,72,31,62),(22,63,32,73),(23,74,33,64),(24,65,34,75),(25,76,35,66),(26,67,36,77),(27,78,37,68),(28,69,38,79),(29,80,39,70),(30,71,40,61),(41,150,51,160),(42,141,52,151),(43,152,53,142),(44,143,54,153),(45,154,55,144),(46,145,56,155),(47,156,57,146),(48,147,58,157),(49,158,59,148),(50,149,60,159),(81,173,91,163),(82,164,92,174),(83,175,93,165),(84,166,94,176),(85,177,95,167),(86,168,96,178),(87,179,97,169),(88,170,98,180),(89,161,99,171),(90,172,100,162),(121,226,131,236),(122,237,132,227),(123,228,133,238),(124,239,134,229),(125,230,135,240),(126,221,136,231),(127,232,137,222),(128,223,138,233),(129,234,139,224),(130,225,140,235),(181,242,191,252),(182,253,192,243),(183,244,193,254),(184,255,194,245),(185,246,195,256),(186,257,196,247),(187,248,197,258),(188,259,198,249),(189,250,199,260),(190,241,200,251),(201,262,211,272),(202,273,212,263),(203,264,213,274),(204,275,214,265),(205,266,215,276),(206,277,216,267),(207,268,217,278),(208,279,218,269),(209,270,219,280),(210,261,220,271),(281,302,291,312),(282,313,292,303),(283,304,293,314),(284,315,294,305),(285,306,295,316),(286,317,296,307),(287,308,297,318),(288,319,298,309),(289,310,299,320),(290,301,300,311)], [(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)], [(1,176,141,185),(2,245,142,83),(3,174,143,183),(4,243,144,81),(5,172,145,181),(6,241,146,99),(7,170,147,199),(8,259,148,97),(9,168,149,197),(10,257,150,95),(11,166,151,195),(12,255,152,93),(13,164,153,193),(14,253,154,91),(15,162,155,191),(16,251,156,89),(17,180,157,189),(18,249,158,87),(19,178,159,187),(20,247,160,85),(21,218,131,305),(22,283,132,268),(23,216,133,303),(24,281,134,266),(25,214,135,301),(26,299,136,264),(27,212,137,319),(28,297,138,262),(29,210,139,317),(30,295,140,280),(31,208,121,315),(32,293,122,278),(33,206,123,313),(34,291,124,276),(35,204,125,311),(36,289,126,274),(37,202,127,309),(38,287,128,272),(39,220,129,307),(40,285,130,270),(41,167,103,196),(42,246,104,84),(43,165,105,194),(44,244,106,82),(45,163,107,192),(46,242,108,100),(47,161,109,190),(48,260,110,98),(49,179,111,188),(50,258,112,96),(51,177,113,186),(52,256,114,94),(53,175,115,184),(54,254,116,92),(55,173,117,182),(56,252,118,90),(57,171,119,200),(58,250,120,88),(59,169,101,198),(60,248,102,86),(61,316,225,209),(62,269,226,284),(63,314,227,207),(64,267,228,282),(65,312,229,205),(66,265,230,300),(67,310,231,203),(68,263,232,298),(69,308,233,201),(70,261,234,296),(71,306,235,219),(72,279,236,294),(73,304,237,217),(74,277,238,292),(75,302,239,215),(76,275,240,290),(77,320,221,213),(78,273,222,288),(79,318,223,211),(80,271,224,286)]])
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 | D4⋊2D5 | C20.C23 |
| kernel | C42.77D10 | C42.D5 | Q8⋊Dic5 | C20.6Q8 | C5×C4⋊Q8 | C2×C20 | C4⋊Q8 | C20 | C42 | C2×Q8 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C42.77D10 ►in GL8(𝔽41)
| 15 | 36 | 37 | 15 | 0 | 0 | 0 | 0 |
| 5 | 26 | 26 | 4 | 0 | 0 | 0 | 0 |
| 4 | 26 | 26 | 5 | 0 | 0 | 0 | 0 |
| 15 | 37 | 36 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 23 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 18 |
| 0 | 0 | 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 23 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 | 0 | 0 |
| 35 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 11 | 3 | 13 |
| 0 | 0 | 0 | 0 | 14 | 6 | 24 | 27 |
| 0 | 0 | 0 | 0 | 3 | 13 | 8 | 30 |
| 0 | 0 | 0 | 0 | 24 | 27 | 27 | 35 |
| 12 | 14 | 5 | 40 | 0 | 0 | 0 | 0 |
| 4 | 29 | 29 | 36 | 0 | 0 | 0 | 0 |
| 36 | 1 | 29 | 27 | 0 | 0 | 0 | 0 |
| 12 | 5 | 37 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 7 | 12 | 30 |
| 0 | 0 | 0 | 0 | 20 | 30 | 33 | 29 |
| 0 | 0 | 0 | 0 | 12 | 30 | 30 | 34 |
| 0 | 0 | 0 | 0 | 33 | 29 | 21 | 11 |
G:=sub<GL(8,GF(41))| [15,5,4,15,0,0,0,0,36,26,26,37,0,0,0,0,37,26,26,36,0,0,0,0,15,4,5,15,0,0,0,0,0,0,0,0,0,0,18,5,0,0,0,0,0,0,1,23,0,0,0,0,23,36,0,0,0,0,0,0,40,18,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,0,33,14,3,24,0,0,0,0,11,6,13,27,0,0,0,0,3,24,8,27,0,0,0,0,13,27,30,35],[12,4,36,12,0,0,0,0,14,29,1,5,0,0,0,0,5,29,29,37,0,0,0,0,40,36,27,12,0,0,0,0,0,0,0,0,11,20,12,33,0,0,0,0,7,30,30,29,0,0,0,0,12,33,30,21,0,0,0,0,30,29,34,11] >;
C42.77D10 in GAP, Magma, Sage, TeX
C_4^2._{77}D_{10} % in TeX
G:=Group("C4^2.77D10"); // GroupNames label
G:=SmallGroup(320,709);
// by ID
G=gap.SmallGroup(320,709);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,590,135,184,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^9>;
// generators/relations