metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.215D10, C4.32(Q8×D5), C5⋊2C8.5Q8, C4⋊C4.73D10, C20.32(C2×Q8), C5⋊4(C8.5Q8), (C2×C20).274D4, C10.28(C4⋊Q8), C42.C2.3D5, C20.6Q8.7C2, C10.108(C4○D8), (C2×C20).382C23, (C4×C20).112C22, C10.D8.13C2, C20.Q8.14C2, C2.8(Dic5⋊Q8), C4⋊Dic5.152C22, C2.27(D4.8D10), (C4×C5⋊2C8).10C2, (C2×C10).513(C2×D4), (C5×C42.C2).2C2, (C2×C4).111(C5⋊D4), (C5×C4⋊C4).120C22, (C2×C4).480(C22×D5), C22.186(C2×C5⋊D4), (C2×C5⋊2C8).262C22, SmallGroup(320,691)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.215D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc9 >
Subgroups: 254 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C4×C8, C4.Q8, C2.D8, C42.C2, C42.C2, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C8.5Q8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C4×C5⋊2C8, C10.D8, C20.Q8, C20.6Q8, C5×C42.C2, C42.215D10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C4○D8, C5⋊D4, C22×D5, C8.5Q8, Q8×D5, C2×C5⋊D4, Dic5⋊Q8, D4.8D10, C42.215D10
►Regular action on 320 points
Generators in S320
(1 253 11 243)(2 27 12 37)(3 255 13 245)(4 29 14 39)(5 257 15 247)(6 31 16 21)(7 259 17 249)(8 33 18 23)(9 241 19 251)(10 35 20 25)(22 309 32 319)(24 311 34 301)(26 313 36 303)(28 315 38 305)(30 317 40 307)(41 275 51 265)(42 121 52 131)(43 277 53 267)(44 123 54 133)(45 279 55 269)(46 125 56 135)(47 261 57 271)(48 127 58 137)(49 263 59 273)(50 129 60 139)(61 140 71 130)(62 276 72 266)(63 122 73 132)(64 278 74 268)(65 124 75 134)(66 280 76 270)(67 126 77 136)(68 262 78 272)(69 128 79 138)(70 264 80 274)(81 159 91 149)(82 221 92 231)(83 141 93 151)(84 223 94 233)(85 143 95 153)(86 225 96 235)(87 145 97 155)(88 227 98 237)(89 147 99 157)(90 229 100 239)(101 186 111 196)(102 171 112 161)(103 188 113 198)(104 173 114 163)(105 190 115 200)(106 175 116 165)(107 192 117 182)(108 177 118 167)(109 194 119 184)(110 179 120 169)(142 218 152 208)(144 220 154 210)(146 202 156 212)(148 204 158 214)(150 206 160 216)(162 289 172 299)(164 291 174 281)(166 293 176 283)(168 295 178 285)(170 297 180 287)(181 292 191 282)(183 294 193 284)(185 296 195 286)(187 298 197 288)(189 300 199 290)(201 236 211 226)(203 238 213 228)(205 240 215 230)(207 222 217 232)(209 224 219 234)(242 302 252 312)(244 304 254 314)(246 306 256 316)(248 308 258 318)(250 310 260 320)
(1 159 313 230)(2 231 314 160)(3 141 315 232)(4 233 316 142)(5 143 317 234)(6 235 318 144)(7 145 319 236)(8 237 320 146)(9 147 301 238)(10 239 302 148)(11 149 303 240)(12 221 304 150)(13 151 305 222)(14 223 306 152)(15 153 307 224)(16 225 308 154)(17 155 309 226)(18 227 310 156)(19 157 311 228)(20 229 312 158)(21 96 258 210)(22 211 259 97)(23 98 260 212)(24 213 241 99)(25 100 242 214)(26 215 243 81)(27 82 244 216)(28 217 245 83)(29 84 246 218)(30 219 247 85)(31 86 248 220)(32 201 249 87)(33 88 250 202)(34 203 251 89)(35 90 252 204)(36 205 253 91)(37 92 254 206)(38 207 255 93)(39 94 256 208)(40 209 257 95)(41 109 71 285)(42 286 72 110)(43 111 73 287)(44 288 74 112)(45 113 75 289)(46 290 76 114)(47 115 77 291)(48 292 78 116)(49 117 79 293)(50 294 80 118)(51 119 61 295)(52 296 62 120)(53 101 63 297)(54 298 64 102)(55 103 65 299)(56 300 66 104)(57 105 67 281)(58 282 68 106)(59 107 69 283)(60 284 70 108)(121 185 266 179)(122 180 267 186)(123 187 268 161)(124 162 269 188)(125 189 270 163)(126 164 271 190)(127 191 272 165)(128 166 273 192)(129 193 274 167)(130 168 275 194)(131 195 276 169)(132 170 277 196)(133 197 278 171)(134 172 279 198)(135 199 280 173)(136 174 261 200)(137 181 262 175)(138 176 263 182)(139 183 264 177)(140 178 265 184)
(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)
(1 183 11 193)(2 273 12 263)(3 181 13 191)(4 271 14 261)(5 199 15 189)(6 269 16 279)(7 197 17 187)(8 267 18 277)(9 195 19 185)(10 265 20 275)(21 75 31 65)(22 298 32 288)(23 73 33 63)(24 296 34 286)(25 71 35 61)(26 294 36 284)(27 69 37 79)(28 292 38 282)(29 67 39 77)(30 290 40 300)(41 252 51 242)(42 213 52 203)(43 250 53 260)(44 211 54 201)(45 248 55 258)(46 209 56 219)(47 246 57 256)(48 207 58 217)(49 244 59 254)(50 205 60 215)(62 89 72 99)(64 87 74 97)(66 85 76 95)(68 83 78 93)(70 81 80 91)(82 107 92 117)(84 105 94 115)(86 103 96 113)(88 101 98 111)(90 119 100 109)(102 249 112 259)(104 247 114 257)(106 245 116 255)(108 243 118 253)(110 241 120 251)(121 147 131 157)(122 310 132 320)(123 145 133 155)(124 308 134 318)(125 143 135 153)(126 306 136 316)(127 141 137 151)(128 304 138 314)(129 159 139 149)(130 302 140 312)(142 190 152 200)(144 188 154 198)(146 186 156 196)(148 184 158 194)(150 182 160 192)(161 319 171 309)(162 225 172 235)(163 317 173 307)(164 223 174 233)(165 315 175 305)(166 221 176 231)(167 313 177 303)(168 239 178 229)(169 311 179 301)(170 237 180 227)(202 297 212 287)(204 295 214 285)(206 293 216 283)(208 291 218 281)(210 289 220 299)(222 272 232 262)(224 270 234 280)(226 268 236 278)(228 266 238 276)(230 264 240 274)
G:=sub<Sym(320)| (1,253,11,243)(2,27,12,37)(3,255,13,245)(4,29,14,39)(5,257,15,247)(6,31,16,21)(7,259,17,249)(8,33,18,23)(9,241,19,251)(10,35,20,25)(22,309,32,319)(24,311,34,301)(26,313,36,303)(28,315,38,305)(30,317,40,307)(41,275,51,265)(42,121,52,131)(43,277,53,267)(44,123,54,133)(45,279,55,269)(46,125,56,135)(47,261,57,271)(48,127,58,137)(49,263,59,273)(50,129,60,139)(61,140,71,130)(62,276,72,266)(63,122,73,132)(64,278,74,268)(65,124,75,134)(66,280,76,270)(67,126,77,136)(68,262,78,272)(69,128,79,138)(70,264,80,274)(81,159,91,149)(82,221,92,231)(83,141,93,151)(84,223,94,233)(85,143,95,153)(86,225,96,235)(87,145,97,155)(88,227,98,237)(89,147,99,157)(90,229,100,239)(101,186,111,196)(102,171,112,161)(103,188,113,198)(104,173,114,163)(105,190,115,200)(106,175,116,165)(107,192,117,182)(108,177,118,167)(109,194,119,184)(110,179,120,169)(142,218,152,208)(144,220,154,210)(146,202,156,212)(148,204,158,214)(150,206,160,216)(162,289,172,299)(164,291,174,281)(166,293,176,283)(168,295,178,285)(170,297,180,287)(181,292,191,282)(183,294,193,284)(185,296,195,286)(187,298,197,288)(189,300,199,290)(201,236,211,226)(203,238,213,228)(205,240,215,230)(207,222,217,232)(209,224,219,234)(242,302,252,312)(244,304,254,314)(246,306,256,316)(248,308,258,318)(250,310,260,320), (1,159,313,230)(2,231,314,160)(3,141,315,232)(4,233,316,142)(5,143,317,234)(6,235,318,144)(7,145,319,236)(8,237,320,146)(9,147,301,238)(10,239,302,148)(11,149,303,240)(12,221,304,150)(13,151,305,222)(14,223,306,152)(15,153,307,224)(16,225,308,154)(17,155,309,226)(18,227,310,156)(19,157,311,228)(20,229,312,158)(21,96,258,210)(22,211,259,97)(23,98,260,212)(24,213,241,99)(25,100,242,214)(26,215,243,81)(27,82,244,216)(28,217,245,83)(29,84,246,218)(30,219,247,85)(31,86,248,220)(32,201,249,87)(33,88,250,202)(34,203,251,89)(35,90,252,204)(36,205,253,91)(37,92,254,206)(38,207,255,93)(39,94,256,208)(40,209,257,95)(41,109,71,285)(42,286,72,110)(43,111,73,287)(44,288,74,112)(45,113,75,289)(46,290,76,114)(47,115,77,291)(48,292,78,116)(49,117,79,293)(50,294,80,118)(51,119,61,295)(52,296,62,120)(53,101,63,297)(54,298,64,102)(55,103,65,299)(56,300,66,104)(57,105,67,281)(58,282,68,106)(59,107,69,283)(60,284,70,108)(121,185,266,179)(122,180,267,186)(123,187,268,161)(124,162,269,188)(125,189,270,163)(126,164,271,190)(127,191,272,165)(128,166,273,192)(129,193,274,167)(130,168,275,194)(131,195,276,169)(132,170,277,196)(133,197,278,171)(134,172,279,198)(135,199,280,173)(136,174,261,200)(137,181,262,175)(138,176,263,182)(139,183,264,177)(140,178,265,184), (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), (1,183,11,193)(2,273,12,263)(3,181,13,191)(4,271,14,261)(5,199,15,189)(6,269,16,279)(7,197,17,187)(8,267,18,277)(9,195,19,185)(10,265,20,275)(21,75,31,65)(22,298,32,288)(23,73,33,63)(24,296,34,286)(25,71,35,61)(26,294,36,284)(27,69,37,79)(28,292,38,282)(29,67,39,77)(30,290,40,300)(41,252,51,242)(42,213,52,203)(43,250,53,260)(44,211,54,201)(45,248,55,258)(46,209,56,219)(47,246,57,256)(48,207,58,217)(49,244,59,254)(50,205,60,215)(62,89,72,99)(64,87,74,97)(66,85,76,95)(68,83,78,93)(70,81,80,91)(82,107,92,117)(84,105,94,115)(86,103,96,113)(88,101,98,111)(90,119,100,109)(102,249,112,259)(104,247,114,257)(106,245,116,255)(108,243,118,253)(110,241,120,251)(121,147,131,157)(122,310,132,320)(123,145,133,155)(124,308,134,318)(125,143,135,153)(126,306,136,316)(127,141,137,151)(128,304,138,314)(129,159,139,149)(130,302,140,312)(142,190,152,200)(144,188,154,198)(146,186,156,196)(148,184,158,194)(150,182,160,192)(161,319,171,309)(162,225,172,235)(163,317,173,307)(164,223,174,233)(165,315,175,305)(166,221,176,231)(167,313,177,303)(168,239,178,229)(169,311,179,301)(170,237,180,227)(202,297,212,287)(204,295,214,285)(206,293,216,283)(208,291,218,281)(210,289,220,299)(222,272,232,262)(224,270,234,280)(226,268,236,278)(228,266,238,276)(230,264,240,274)>;
G:=Group( (1,253,11,243)(2,27,12,37)(3,255,13,245)(4,29,14,39)(5,257,15,247)(6,31,16,21)(7,259,17,249)(8,33,18,23)(9,241,19,251)(10,35,20,25)(22,309,32,319)(24,311,34,301)(26,313,36,303)(28,315,38,305)(30,317,40,307)(41,275,51,265)(42,121,52,131)(43,277,53,267)(44,123,54,133)(45,279,55,269)(46,125,56,135)(47,261,57,271)(48,127,58,137)(49,263,59,273)(50,129,60,139)(61,140,71,130)(62,276,72,266)(63,122,73,132)(64,278,74,268)(65,124,75,134)(66,280,76,270)(67,126,77,136)(68,262,78,272)(69,128,79,138)(70,264,80,274)(81,159,91,149)(82,221,92,231)(83,141,93,151)(84,223,94,233)(85,143,95,153)(86,225,96,235)(87,145,97,155)(88,227,98,237)(89,147,99,157)(90,229,100,239)(101,186,111,196)(102,171,112,161)(103,188,113,198)(104,173,114,163)(105,190,115,200)(106,175,116,165)(107,192,117,182)(108,177,118,167)(109,194,119,184)(110,179,120,169)(142,218,152,208)(144,220,154,210)(146,202,156,212)(148,204,158,214)(150,206,160,216)(162,289,172,299)(164,291,174,281)(166,293,176,283)(168,295,178,285)(170,297,180,287)(181,292,191,282)(183,294,193,284)(185,296,195,286)(187,298,197,288)(189,300,199,290)(201,236,211,226)(203,238,213,228)(205,240,215,230)(207,222,217,232)(209,224,219,234)(242,302,252,312)(244,304,254,314)(246,306,256,316)(248,308,258,318)(250,310,260,320), (1,159,313,230)(2,231,314,160)(3,141,315,232)(4,233,316,142)(5,143,317,234)(6,235,318,144)(7,145,319,236)(8,237,320,146)(9,147,301,238)(10,239,302,148)(11,149,303,240)(12,221,304,150)(13,151,305,222)(14,223,306,152)(15,153,307,224)(16,225,308,154)(17,155,309,226)(18,227,310,156)(19,157,311,228)(20,229,312,158)(21,96,258,210)(22,211,259,97)(23,98,260,212)(24,213,241,99)(25,100,242,214)(26,215,243,81)(27,82,244,216)(28,217,245,83)(29,84,246,218)(30,219,247,85)(31,86,248,220)(32,201,249,87)(33,88,250,202)(34,203,251,89)(35,90,252,204)(36,205,253,91)(37,92,254,206)(38,207,255,93)(39,94,256,208)(40,209,257,95)(41,109,71,285)(42,286,72,110)(43,111,73,287)(44,288,74,112)(45,113,75,289)(46,290,76,114)(47,115,77,291)(48,292,78,116)(49,117,79,293)(50,294,80,118)(51,119,61,295)(52,296,62,120)(53,101,63,297)(54,298,64,102)(55,103,65,299)(56,300,66,104)(57,105,67,281)(58,282,68,106)(59,107,69,283)(60,284,70,108)(121,185,266,179)(122,180,267,186)(123,187,268,161)(124,162,269,188)(125,189,270,163)(126,164,271,190)(127,191,272,165)(128,166,273,192)(129,193,274,167)(130,168,275,194)(131,195,276,169)(132,170,277,196)(133,197,278,171)(134,172,279,198)(135,199,280,173)(136,174,261,200)(137,181,262,175)(138,176,263,182)(139,183,264,177)(140,178,265,184), (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), (1,183,11,193)(2,273,12,263)(3,181,13,191)(4,271,14,261)(5,199,15,189)(6,269,16,279)(7,197,17,187)(8,267,18,277)(9,195,19,185)(10,265,20,275)(21,75,31,65)(22,298,32,288)(23,73,33,63)(24,296,34,286)(25,71,35,61)(26,294,36,284)(27,69,37,79)(28,292,38,282)(29,67,39,77)(30,290,40,300)(41,252,51,242)(42,213,52,203)(43,250,53,260)(44,211,54,201)(45,248,55,258)(46,209,56,219)(47,246,57,256)(48,207,58,217)(49,244,59,254)(50,205,60,215)(62,89,72,99)(64,87,74,97)(66,85,76,95)(68,83,78,93)(70,81,80,91)(82,107,92,117)(84,105,94,115)(86,103,96,113)(88,101,98,111)(90,119,100,109)(102,249,112,259)(104,247,114,257)(106,245,116,255)(108,243,118,253)(110,241,120,251)(121,147,131,157)(122,310,132,320)(123,145,133,155)(124,308,134,318)(125,143,135,153)(126,306,136,316)(127,141,137,151)(128,304,138,314)(129,159,139,149)(130,302,140,312)(142,190,152,200)(144,188,154,198)(146,186,156,196)(148,184,158,194)(150,182,160,192)(161,319,171,309)(162,225,172,235)(163,317,173,307)(164,223,174,233)(165,315,175,305)(166,221,176,231)(167,313,177,303)(168,239,178,229)(169,311,179,301)(170,237,180,227)(202,297,212,287)(204,295,214,285)(206,293,216,283)(208,291,218,281)(210,289,220,299)(222,272,232,262)(224,270,234,280)(226,268,236,278)(228,266,238,276)(230,264,240,274) );
G=PermutationGroup([[(1,253,11,243),(2,27,12,37),(3,255,13,245),(4,29,14,39),(5,257,15,247),(6,31,16,21),(7,259,17,249),(8,33,18,23),(9,241,19,251),(10,35,20,25),(22,309,32,319),(24,311,34,301),(26,313,36,303),(28,315,38,305),(30,317,40,307),(41,275,51,265),(42,121,52,131),(43,277,53,267),(44,123,54,133),(45,279,55,269),(46,125,56,135),(47,261,57,271),(48,127,58,137),(49,263,59,273),(50,129,60,139),(61,140,71,130),(62,276,72,266),(63,122,73,132),(64,278,74,268),(65,124,75,134),(66,280,76,270),(67,126,77,136),(68,262,78,272),(69,128,79,138),(70,264,80,274),(81,159,91,149),(82,221,92,231),(83,141,93,151),(84,223,94,233),(85,143,95,153),(86,225,96,235),(87,145,97,155),(88,227,98,237),(89,147,99,157),(90,229,100,239),(101,186,111,196),(102,171,112,161),(103,188,113,198),(104,173,114,163),(105,190,115,200),(106,175,116,165),(107,192,117,182),(108,177,118,167),(109,194,119,184),(110,179,120,169),(142,218,152,208),(144,220,154,210),(146,202,156,212),(148,204,158,214),(150,206,160,216),(162,289,172,299),(164,291,174,281),(166,293,176,283),(168,295,178,285),(170,297,180,287),(181,292,191,282),(183,294,193,284),(185,296,195,286),(187,298,197,288),(189,300,199,290),(201,236,211,226),(203,238,213,228),(205,240,215,230),(207,222,217,232),(209,224,219,234),(242,302,252,312),(244,304,254,314),(246,306,256,316),(248,308,258,318),(250,310,260,320)], [(1,159,313,230),(2,231,314,160),(3,141,315,232),(4,233,316,142),(5,143,317,234),(6,235,318,144),(7,145,319,236),(8,237,320,146),(9,147,301,238),(10,239,302,148),(11,149,303,240),(12,221,304,150),(13,151,305,222),(14,223,306,152),(15,153,307,224),(16,225,308,154),(17,155,309,226),(18,227,310,156),(19,157,311,228),(20,229,312,158),(21,96,258,210),(22,211,259,97),(23,98,260,212),(24,213,241,99),(25,100,242,214),(26,215,243,81),(27,82,244,216),(28,217,245,83),(29,84,246,218),(30,219,247,85),(31,86,248,220),(32,201,249,87),(33,88,250,202),(34,203,251,89),(35,90,252,204),(36,205,253,91),(37,92,254,206),(38,207,255,93),(39,94,256,208),(40,209,257,95),(41,109,71,285),(42,286,72,110),(43,111,73,287),(44,288,74,112),(45,113,75,289),(46,290,76,114),(47,115,77,291),(48,292,78,116),(49,117,79,293),(50,294,80,118),(51,119,61,295),(52,296,62,120),(53,101,63,297),(54,298,64,102),(55,103,65,299),(56,300,66,104),(57,105,67,281),(58,282,68,106),(59,107,69,283),(60,284,70,108),(121,185,266,179),(122,180,267,186),(123,187,268,161),(124,162,269,188),(125,189,270,163),(126,164,271,190),(127,191,272,165),(128,166,273,192),(129,193,274,167),(130,168,275,194),(131,195,276,169),(132,170,277,196),(133,197,278,171),(134,172,279,198),(135,199,280,173),(136,174,261,200),(137,181,262,175),(138,176,263,182),(139,183,264,177),(140,178,265,184)], [(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)], [(1,183,11,193),(2,273,12,263),(3,181,13,191),(4,271,14,261),(5,199,15,189),(6,269,16,279),(7,197,17,187),(8,267,18,277),(9,195,19,185),(10,265,20,275),(21,75,31,65),(22,298,32,288),(23,73,33,63),(24,296,34,286),(25,71,35,61),(26,294,36,284),(27,69,37,79),(28,292,38,282),(29,67,39,77),(30,290,40,300),(41,252,51,242),(42,213,52,203),(43,250,53,260),(44,211,54,201),(45,248,55,258),(46,209,56,219),(47,246,57,256),(48,207,58,217),(49,244,59,254),(50,205,60,215),(62,89,72,99),(64,87,74,97),(66,85,76,95),(68,83,78,93),(70,81,80,91),(82,107,92,117),(84,105,94,115),(86,103,96,113),(88,101,98,111),(90,119,100,109),(102,249,112,259),(104,247,114,257),(106,245,116,255),(108,243,118,253),(110,241,120,251),(121,147,131,157),(122,310,132,320),(123,145,133,155),(124,308,134,318),(125,143,135,153),(126,306,136,316),(127,141,137,151),(128,304,138,314),(129,159,139,149),(130,302,140,312),(142,190,152,200),(144,188,154,198),(146,186,156,196),(148,184,158,194),(150,182,160,192),(161,319,171,309),(162,225,172,235),(163,317,173,307),(164,223,174,233),(165,315,175,305),(166,221,176,231),(167,313,177,303),(168,239,178,229),(169,311,179,301),(170,237,180,227),(202,297,212,287),(204,295,214,285),(206,293,216,283),(208,291,218,281),(210,289,220,299),(222,272,232,262),(224,270,234,280),(226,268,236,278),(228,266,238,276),(230,264,240,274)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | D10 | D10 | C4○D8 | C5⋊D4 | Q8×D5 | D4.8D10 |
| kernel | C42.215D10 | C4×C5⋊2C8 | C10.D8 | C20.Q8 | C20.6Q8 | C5×C42.C2 | C5⋊2C8 | C2×C20 | C42.C2 | C42 | C4⋊C4 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C42.215D10 ►in GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 4 |
| 0 | 0 | 0 | 0 | 35 | 33 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 17 | 0 | 0 | 0 | 0 |
| 17 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 38 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 37 |
| 0 | 0 | 0 | 0 | 29 | 19 |
| 25 | 27 | 0 | 0 | 0 | 0 |
| 27 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 21 | 0 | 0 |
| 0 | 0 | 3 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 38 |
| 0 | 0 | 0 | 0 | 36 | 5 |
G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,35,0,0,0,0,4,33],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,17,0,0,0,0,17,9,0,0,0,0,0,0,24,3,0,0,0,0,38,3,0,0,0,0,0,0,22,29,0,0,0,0,37,19],[25,27,0,0,0,0,27,16,0,0,0,0,0,0,15,3,0,0,0,0,21,26,0,0,0,0,0,0,36,36,0,0,0,0,38,5] >;
C42.215D10 in GAP, Magma, Sage, TeX
C_4^2._{215}D_{10} % in TeX
G:=Group("C4^2.215D10"); // GroupNames label
G:=SmallGroup(320,691);
// by ID
G=gap.SmallGroup(320,691);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,422,471,58,438,102,12550]);
// 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=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations