metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.68D10, C5⋊4(C8⋊Q8), C5⋊2C8⋊3Q8, C4.33(Q8×D5), C4⋊C4.74D10, (C2×C20).84D4, C20.33(C2×Q8), C10.29(C4⋊Q8), C42.C2.4D5, C20⋊2Q8.17C2, (C4×C20).113C22, (C2×C20).383C23, C10.D8.14C2, C20.Q8.15C2, C2.20(D4⋊D10), C42.D5.5C2, C10.121(C8⋊C22), C2.9(Dic5⋊Q8), C4⋊Dic5.153C22, C2.21(D4.9D10), C10.122(C8.C22), (C2×C10).514(C2×D4), (C2×C4).65(C5⋊D4), (C5×C42.C2).3C2, (C5×C4⋊C4).121C22, (C2×C4).481(C22×D5), C22.187(C2×C5⋊D4), (C2×C5⋊2C8).125C22, SmallGroup(320,692)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.68D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc9 >
Subgroups: 302 in 90 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C8⋊Q8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C42.D5, C10.D8, C20.Q8, C20⋊2Q8, C5×C42.C2, C42.68D10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C8⋊Q8, Q8×D5, C2×C5⋊D4, Dic5⋊Q8, D4⋊D10, D4.9D10, C42.68D10
►Regular action on 320 points
Generators in S320
(1 25 11 35)(2 72 12 62)(3 27 13 37)(4 74 14 64)(5 29 15 39)(6 76 16 66)(7 31 17 21)(8 78 18 68)(9 33 19 23)(10 80 20 70)(22 273 32 263)(24 275 34 265)(26 277 36 267)(28 279 38 269)(30 261 40 271)(41 129 51 139)(42 146 52 156)(43 131 53 121)(44 148 54 158)(45 133 55 123)(46 150 56 160)(47 135 57 125)(48 152 58 142)(49 137 59 127)(50 154 60 144)(61 266 71 276)(63 268 73 278)(65 270 75 280)(67 272 77 262)(69 274 79 264)(81 303 91 313)(82 297 92 287)(83 305 93 315)(84 299 94 289)(85 307 95 317)(86 281 96 291)(87 309 97 319)(88 283 98 293)(89 311 99 301)(90 285 100 295)(101 282 111 292)(102 310 112 320)(103 284 113 294)(104 312 114 302)(105 286 115 296)(106 314 116 304)(107 288 117 298)(108 316 118 306)(109 290 119 300)(110 318 120 308)(122 195 132 185)(124 197 134 187)(126 199 136 189)(128 181 138 191)(130 183 140 193)(141 198 151 188)(143 200 153 190)(145 182 155 192)(147 184 157 194)(149 186 159 196)(161 222 171 232)(162 248 172 258)(163 224 173 234)(164 250 174 260)(165 226 175 236)(166 252 176 242)(167 228 177 238)(168 254 178 244)(169 230 179 240)(170 256 180 246)(201 239 211 229)(202 245 212 255)(203 221 213 231)(204 247 214 257)(205 223 215 233)(206 249 216 259)(207 225 217 235)(208 251 218 241)(209 227 219 237)(210 253 220 243)
(1 53 276 194)(2 195 277 54)(3 55 278 196)(4 197 279 56)(5 57 280 198)(6 199 261 58)(7 59 262 200)(8 181 263 60)(9 41 264 182)(10 183 265 42)(11 43 266 184)(12 185 267 44)(13 45 268 186)(14 187 269 46)(15 47 270 188)(16 189 271 48)(17 49 272 190)(18 191 273 50)(19 51 274 192)(20 193 275 52)(21 137 77 143)(22 144 78 138)(23 139 79 145)(24 146 80 140)(25 121 61 147)(26 148 62 122)(27 123 63 149)(28 150 64 124)(29 125 65 151)(30 152 66 126)(31 127 67 153)(32 154 68 128)(33 129 69 155)(34 156 70 130)(35 131 71 157)(36 158 72 132)(37 133 73 159)(38 160 74 134)(39 135 75 141)(40 142 76 136)(81 240 105 255)(82 256 106 221)(83 222 107 257)(84 258 108 223)(85 224 109 259)(86 260 110 225)(87 226 111 241)(88 242 112 227)(89 228 113 243)(90 244 114 229)(91 230 115 245)(92 246 116 231)(93 232 117 247)(94 248 118 233)(95 234 119 249)(96 250 120 235)(97 236 101 251)(98 252 102 237)(99 238 103 253)(100 254 104 239)(161 298 214 315)(162 316 215 299)(163 300 216 317)(164 318 217 281)(165 282 218 319)(166 320 219 283)(167 284 220 301)(168 302 201 285)(169 286 202 303)(170 304 203 287)(171 288 204 305)(172 306 205 289)(173 290 206 307)(174 308 207 291)(175 292 208 309)(176 310 209 293)(177 294 210 311)(178 312 211 295)(179 296 212 313)(180 314 213 297)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 119 11 109)(2 258 12 248)(3 117 13 107)(4 256 14 246)(5 115 15 105)(6 254 16 244)(7 113 17 103)(8 252 18 242)(9 111 19 101)(10 250 20 260)(21 294 31 284)(22 219 32 209)(23 292 33 282)(24 217 34 207)(25 290 35 300)(26 215 36 205)(27 288 37 298)(28 213 38 203)(29 286 39 296)(30 211 40 201)(41 226 51 236)(42 120 52 110)(43 224 53 234)(44 118 54 108)(45 222 55 232)(46 116 56 106)(47 240 57 230)(48 114 58 104)(49 238 59 228)(50 112 60 102)(61 307 71 317)(62 162 72 172)(63 305 73 315)(64 180 74 170)(65 303 75 313)(66 178 76 168)(67 301 77 311)(68 176 78 166)(69 319 79 309)(70 174 80 164)(81 280 91 270)(82 187 92 197)(83 278 93 268)(84 185 94 195)(85 276 95 266)(86 183 96 193)(87 274 97 264)(88 181 98 191)(89 272 99 262)(90 199 100 189)(121 173 131 163)(122 299 132 289)(123 171 133 161)(124 297 134 287)(125 169 135 179)(126 295 136 285)(127 167 137 177)(128 293 138 283)(129 165 139 175)(130 291 140 281)(141 212 151 202)(142 302 152 312)(143 210 153 220)(144 320 154 310)(145 208 155 218)(146 318 156 308)(147 206 157 216)(148 316 158 306)(149 204 159 214)(150 314 160 304)(182 241 192 251)(184 259 194 249)(186 257 196 247)(188 255 198 245)(190 253 200 243)(221 269 231 279)(223 267 233 277)(225 265 235 275)(227 263 237 273)(229 261 239 271)
G:=sub<Sym(320)| (1,25,11,35)(2,72,12,62)(3,27,13,37)(4,74,14,64)(5,29,15,39)(6,76,16,66)(7,31,17,21)(8,78,18,68)(9,33,19,23)(10,80,20,70)(22,273,32,263)(24,275,34,265)(26,277,36,267)(28,279,38,269)(30,261,40,271)(41,129,51,139)(42,146,52,156)(43,131,53,121)(44,148,54,158)(45,133,55,123)(46,150,56,160)(47,135,57,125)(48,152,58,142)(49,137,59,127)(50,154,60,144)(61,266,71,276)(63,268,73,278)(65,270,75,280)(67,272,77,262)(69,274,79,264)(81,303,91,313)(82,297,92,287)(83,305,93,315)(84,299,94,289)(85,307,95,317)(86,281,96,291)(87,309,97,319)(88,283,98,293)(89,311,99,301)(90,285,100,295)(101,282,111,292)(102,310,112,320)(103,284,113,294)(104,312,114,302)(105,286,115,296)(106,314,116,304)(107,288,117,298)(108,316,118,306)(109,290,119,300)(110,318,120,308)(122,195,132,185)(124,197,134,187)(126,199,136,189)(128,181,138,191)(130,183,140,193)(141,198,151,188)(143,200,153,190)(145,182,155,192)(147,184,157,194)(149,186,159,196)(161,222,171,232)(162,248,172,258)(163,224,173,234)(164,250,174,260)(165,226,175,236)(166,252,176,242)(167,228,177,238)(168,254,178,244)(169,230,179,240)(170,256,180,246)(201,239,211,229)(202,245,212,255)(203,221,213,231)(204,247,214,257)(205,223,215,233)(206,249,216,259)(207,225,217,235)(208,251,218,241)(209,227,219,237)(210,253,220,243), (1,53,276,194)(2,195,277,54)(3,55,278,196)(4,197,279,56)(5,57,280,198)(6,199,261,58)(7,59,262,200)(8,181,263,60)(9,41,264,182)(10,183,265,42)(11,43,266,184)(12,185,267,44)(13,45,268,186)(14,187,269,46)(15,47,270,188)(16,189,271,48)(17,49,272,190)(18,191,273,50)(19,51,274,192)(20,193,275,52)(21,137,77,143)(22,144,78,138)(23,139,79,145)(24,146,80,140)(25,121,61,147)(26,148,62,122)(27,123,63,149)(28,150,64,124)(29,125,65,151)(30,152,66,126)(31,127,67,153)(32,154,68,128)(33,129,69,155)(34,156,70,130)(35,131,71,157)(36,158,72,132)(37,133,73,159)(38,160,74,134)(39,135,75,141)(40,142,76,136)(81,240,105,255)(82,256,106,221)(83,222,107,257)(84,258,108,223)(85,224,109,259)(86,260,110,225)(87,226,111,241)(88,242,112,227)(89,228,113,243)(90,244,114,229)(91,230,115,245)(92,246,116,231)(93,232,117,247)(94,248,118,233)(95,234,119,249)(96,250,120,235)(97,236,101,251)(98,252,102,237)(99,238,103,253)(100,254,104,239)(161,298,214,315)(162,316,215,299)(163,300,216,317)(164,318,217,281)(165,282,218,319)(166,320,219,283)(167,284,220,301)(168,302,201,285)(169,286,202,303)(170,304,203,287)(171,288,204,305)(172,306,205,289)(173,290,206,307)(174,308,207,291)(175,292,208,309)(176,310,209,293)(177,294,210,311)(178,312,211,295)(179,296,212,313)(180,314,213,297), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,119,11,109)(2,258,12,248)(3,117,13,107)(4,256,14,246)(5,115,15,105)(6,254,16,244)(7,113,17,103)(8,252,18,242)(9,111,19,101)(10,250,20,260)(21,294,31,284)(22,219,32,209)(23,292,33,282)(24,217,34,207)(25,290,35,300)(26,215,36,205)(27,288,37,298)(28,213,38,203)(29,286,39,296)(30,211,40,201)(41,226,51,236)(42,120,52,110)(43,224,53,234)(44,118,54,108)(45,222,55,232)(46,116,56,106)(47,240,57,230)(48,114,58,104)(49,238,59,228)(50,112,60,102)(61,307,71,317)(62,162,72,172)(63,305,73,315)(64,180,74,170)(65,303,75,313)(66,178,76,168)(67,301,77,311)(68,176,78,166)(69,319,79,309)(70,174,80,164)(81,280,91,270)(82,187,92,197)(83,278,93,268)(84,185,94,195)(85,276,95,266)(86,183,96,193)(87,274,97,264)(88,181,98,191)(89,272,99,262)(90,199,100,189)(121,173,131,163)(122,299,132,289)(123,171,133,161)(124,297,134,287)(125,169,135,179)(126,295,136,285)(127,167,137,177)(128,293,138,283)(129,165,139,175)(130,291,140,281)(141,212,151,202)(142,302,152,312)(143,210,153,220)(144,320,154,310)(145,208,155,218)(146,318,156,308)(147,206,157,216)(148,316,158,306)(149,204,159,214)(150,314,160,304)(182,241,192,251)(184,259,194,249)(186,257,196,247)(188,255,198,245)(190,253,200,243)(221,269,231,279)(223,267,233,277)(225,265,235,275)(227,263,237,273)(229,261,239,271)>;
G:=Group( (1,25,11,35)(2,72,12,62)(3,27,13,37)(4,74,14,64)(5,29,15,39)(6,76,16,66)(7,31,17,21)(8,78,18,68)(9,33,19,23)(10,80,20,70)(22,273,32,263)(24,275,34,265)(26,277,36,267)(28,279,38,269)(30,261,40,271)(41,129,51,139)(42,146,52,156)(43,131,53,121)(44,148,54,158)(45,133,55,123)(46,150,56,160)(47,135,57,125)(48,152,58,142)(49,137,59,127)(50,154,60,144)(61,266,71,276)(63,268,73,278)(65,270,75,280)(67,272,77,262)(69,274,79,264)(81,303,91,313)(82,297,92,287)(83,305,93,315)(84,299,94,289)(85,307,95,317)(86,281,96,291)(87,309,97,319)(88,283,98,293)(89,311,99,301)(90,285,100,295)(101,282,111,292)(102,310,112,320)(103,284,113,294)(104,312,114,302)(105,286,115,296)(106,314,116,304)(107,288,117,298)(108,316,118,306)(109,290,119,300)(110,318,120,308)(122,195,132,185)(124,197,134,187)(126,199,136,189)(128,181,138,191)(130,183,140,193)(141,198,151,188)(143,200,153,190)(145,182,155,192)(147,184,157,194)(149,186,159,196)(161,222,171,232)(162,248,172,258)(163,224,173,234)(164,250,174,260)(165,226,175,236)(166,252,176,242)(167,228,177,238)(168,254,178,244)(169,230,179,240)(170,256,180,246)(201,239,211,229)(202,245,212,255)(203,221,213,231)(204,247,214,257)(205,223,215,233)(206,249,216,259)(207,225,217,235)(208,251,218,241)(209,227,219,237)(210,253,220,243), (1,53,276,194)(2,195,277,54)(3,55,278,196)(4,197,279,56)(5,57,280,198)(6,199,261,58)(7,59,262,200)(8,181,263,60)(9,41,264,182)(10,183,265,42)(11,43,266,184)(12,185,267,44)(13,45,268,186)(14,187,269,46)(15,47,270,188)(16,189,271,48)(17,49,272,190)(18,191,273,50)(19,51,274,192)(20,193,275,52)(21,137,77,143)(22,144,78,138)(23,139,79,145)(24,146,80,140)(25,121,61,147)(26,148,62,122)(27,123,63,149)(28,150,64,124)(29,125,65,151)(30,152,66,126)(31,127,67,153)(32,154,68,128)(33,129,69,155)(34,156,70,130)(35,131,71,157)(36,158,72,132)(37,133,73,159)(38,160,74,134)(39,135,75,141)(40,142,76,136)(81,240,105,255)(82,256,106,221)(83,222,107,257)(84,258,108,223)(85,224,109,259)(86,260,110,225)(87,226,111,241)(88,242,112,227)(89,228,113,243)(90,244,114,229)(91,230,115,245)(92,246,116,231)(93,232,117,247)(94,248,118,233)(95,234,119,249)(96,250,120,235)(97,236,101,251)(98,252,102,237)(99,238,103,253)(100,254,104,239)(161,298,214,315)(162,316,215,299)(163,300,216,317)(164,318,217,281)(165,282,218,319)(166,320,219,283)(167,284,220,301)(168,302,201,285)(169,286,202,303)(170,304,203,287)(171,288,204,305)(172,306,205,289)(173,290,206,307)(174,308,207,291)(175,292,208,309)(176,310,209,293)(177,294,210,311)(178,312,211,295)(179,296,212,313)(180,314,213,297), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,119,11,109)(2,258,12,248)(3,117,13,107)(4,256,14,246)(5,115,15,105)(6,254,16,244)(7,113,17,103)(8,252,18,242)(9,111,19,101)(10,250,20,260)(21,294,31,284)(22,219,32,209)(23,292,33,282)(24,217,34,207)(25,290,35,300)(26,215,36,205)(27,288,37,298)(28,213,38,203)(29,286,39,296)(30,211,40,201)(41,226,51,236)(42,120,52,110)(43,224,53,234)(44,118,54,108)(45,222,55,232)(46,116,56,106)(47,240,57,230)(48,114,58,104)(49,238,59,228)(50,112,60,102)(61,307,71,317)(62,162,72,172)(63,305,73,315)(64,180,74,170)(65,303,75,313)(66,178,76,168)(67,301,77,311)(68,176,78,166)(69,319,79,309)(70,174,80,164)(81,280,91,270)(82,187,92,197)(83,278,93,268)(84,185,94,195)(85,276,95,266)(86,183,96,193)(87,274,97,264)(88,181,98,191)(89,272,99,262)(90,199,100,189)(121,173,131,163)(122,299,132,289)(123,171,133,161)(124,297,134,287)(125,169,135,179)(126,295,136,285)(127,167,137,177)(128,293,138,283)(129,165,139,175)(130,291,140,281)(141,212,151,202)(142,302,152,312)(143,210,153,220)(144,320,154,310)(145,208,155,218)(146,318,156,308)(147,206,157,216)(148,316,158,306)(149,204,159,214)(150,314,160,304)(182,241,192,251)(184,259,194,249)(186,257,196,247)(188,255,198,245)(190,253,200,243)(221,269,231,279)(223,267,233,277)(225,265,235,275)(227,263,237,273)(229,261,239,271) );
G=PermutationGroup([[(1,25,11,35),(2,72,12,62),(3,27,13,37),(4,74,14,64),(5,29,15,39),(6,76,16,66),(7,31,17,21),(8,78,18,68),(9,33,19,23),(10,80,20,70),(22,273,32,263),(24,275,34,265),(26,277,36,267),(28,279,38,269),(30,261,40,271),(41,129,51,139),(42,146,52,156),(43,131,53,121),(44,148,54,158),(45,133,55,123),(46,150,56,160),(47,135,57,125),(48,152,58,142),(49,137,59,127),(50,154,60,144),(61,266,71,276),(63,268,73,278),(65,270,75,280),(67,272,77,262),(69,274,79,264),(81,303,91,313),(82,297,92,287),(83,305,93,315),(84,299,94,289),(85,307,95,317),(86,281,96,291),(87,309,97,319),(88,283,98,293),(89,311,99,301),(90,285,100,295),(101,282,111,292),(102,310,112,320),(103,284,113,294),(104,312,114,302),(105,286,115,296),(106,314,116,304),(107,288,117,298),(108,316,118,306),(109,290,119,300),(110,318,120,308),(122,195,132,185),(124,197,134,187),(126,199,136,189),(128,181,138,191),(130,183,140,193),(141,198,151,188),(143,200,153,190),(145,182,155,192),(147,184,157,194),(149,186,159,196),(161,222,171,232),(162,248,172,258),(163,224,173,234),(164,250,174,260),(165,226,175,236),(166,252,176,242),(167,228,177,238),(168,254,178,244),(169,230,179,240),(170,256,180,246),(201,239,211,229),(202,245,212,255),(203,221,213,231),(204,247,214,257),(205,223,215,233),(206,249,216,259),(207,225,217,235),(208,251,218,241),(209,227,219,237),(210,253,220,243)], [(1,53,276,194),(2,195,277,54),(3,55,278,196),(4,197,279,56),(5,57,280,198),(6,199,261,58),(7,59,262,200),(8,181,263,60),(9,41,264,182),(10,183,265,42),(11,43,266,184),(12,185,267,44),(13,45,268,186),(14,187,269,46),(15,47,270,188),(16,189,271,48),(17,49,272,190),(18,191,273,50),(19,51,274,192),(20,193,275,52),(21,137,77,143),(22,144,78,138),(23,139,79,145),(24,146,80,140),(25,121,61,147),(26,148,62,122),(27,123,63,149),(28,150,64,124),(29,125,65,151),(30,152,66,126),(31,127,67,153),(32,154,68,128),(33,129,69,155),(34,156,70,130),(35,131,71,157),(36,158,72,132),(37,133,73,159),(38,160,74,134),(39,135,75,141),(40,142,76,136),(81,240,105,255),(82,256,106,221),(83,222,107,257),(84,258,108,223),(85,224,109,259),(86,260,110,225),(87,226,111,241),(88,242,112,227),(89,228,113,243),(90,244,114,229),(91,230,115,245),(92,246,116,231),(93,232,117,247),(94,248,118,233),(95,234,119,249),(96,250,120,235),(97,236,101,251),(98,252,102,237),(99,238,103,253),(100,254,104,239),(161,298,214,315),(162,316,215,299),(163,300,216,317),(164,318,217,281),(165,282,218,319),(166,320,219,283),(167,284,220,301),(168,302,201,285),(169,286,202,303),(170,304,203,287),(171,288,204,305),(172,306,205,289),(173,290,206,307),(174,308,207,291),(175,292,208,309),(176,310,209,293),(177,294,210,311),(178,312,211,295),(179,296,212,313),(180,314,213,297)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,119,11,109),(2,258,12,248),(3,117,13,107),(4,256,14,246),(5,115,15,105),(6,254,16,244),(7,113,17,103),(8,252,18,242),(9,111,19,101),(10,250,20,260),(21,294,31,284),(22,219,32,209),(23,292,33,282),(24,217,34,207),(25,290,35,300),(26,215,36,205),(27,288,37,298),(28,213,38,203),(29,286,39,296),(30,211,40,201),(41,226,51,236),(42,120,52,110),(43,224,53,234),(44,118,54,108),(45,222,55,232),(46,116,56,106),(47,240,57,230),(48,114,58,104),(49,238,59,228),(50,112,60,102),(61,307,71,317),(62,162,72,172),(63,305,73,315),(64,180,74,170),(65,303,75,313),(66,178,76,168),(67,301,77,311),(68,176,78,166),(69,319,79,309),(70,174,80,164),(81,280,91,270),(82,187,92,197),(83,278,93,268),(84,185,94,195),(85,276,95,266),(86,183,96,193),(87,274,97,264),(88,181,98,191),(89,272,99,262),(90,199,100,189),(121,173,131,163),(122,299,132,289),(123,171,133,161),(124,297,134,287),(125,169,135,179),(126,295,136,285),(127,167,137,177),(128,293,138,283),(129,165,139,175),(130,291,140,281),(141,212,151,202),(142,302,152,312),(143,210,153,220),(144,320,154,310),(145,208,155,218),(146,318,156,308),(147,206,157,216),(148,316,158,306),(149,204,159,214),(150,314,160,304),(182,241,192,251),(184,259,194,249),(186,257,196,247),(188,255,198,245),(190,253,200,243),(221,269,231,279),(223,267,233,277),(225,265,235,275),(227,263,237,273),(229,261,239,271)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | D10 | D10 | C5⋊D4 | C8⋊C22 | C8.C22 | Q8×D5 | D4⋊D10 | D4.9D10 |
| kernel | C42.68D10 | C42.D5 | C10.D8 | C20.Q8 | C20⋊2Q8 | C5×C42.C2 | C5⋊2C8 | C2×C20 | C42.C2 | C42 | C4⋊C4 | C2×C4 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.68D10 ►in GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 37 | 15 |
| 0 | 0 | 28 | 39 | 26 | 4 |
| 0 | 0 | 2 | 13 | 39 | 28 |
| 0 | 0 | 28 | 39 | 13 | 2 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 39 | 0 |
| 0 | 0 | 0 | 1 | 0 | 39 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 13 | 20 | 35 |
| 0 | 0 | 28 | 24 | 6 | 25 |
| 0 | 0 | 38 | 10 | 13 | 28 |
| 0 | 0 | 31 | 16 | 13 | 17 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 33 | 36 | 0 | 0 |
| 0 | 0 | 5 | 3 | 36 | 38 |
| 0 | 0 | 33 | 36 | 8 | 5 |
G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,2,28,2,28,0,0,13,39,13,39,0,0,37,26,39,13,0,0,15,4,28,2],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,28,28,38,31,0,0,13,24,10,16,0,0,20,6,13,13,0,0,35,25,28,17],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,5,33,5,33,0,0,3,36,3,36,0,0,0,0,36,8,0,0,0,0,38,5] >;
C42.68D10 in GAP, Magma, Sage, TeX
C_4^2._{68}D_{10} % in TeX
G:=Group("C4^2.68D10"); // GroupNames label
G:=SmallGroup(320,692);
// by ID
G=gap.SmallGroup(320,692);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,422,471,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations