Copied to
clipboard

## G = C42.279D10order 320 = 26·5

### 2nd central extension by C42 of D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.279D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C4×C5⋊2C8 — C42.279D10
 Lower central C5 — C10 — C42.279D10
 Upper central C1 — C42 — C4×C8

Generators and relations for C42.279D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a-1b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 134 in 66 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C4, C22, C5, C8, C2×C4, C10, C42, C2×C8, C2×C8, C20, C2×C10, C4×C8, C4×C8, C52C8, C52C8, C40, C2×C20, C8⋊C8, C2×C52C8, C4×C20, C2×C40, C4×C52C8, C4×C40, C42.279D10
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C42, C2×C8, M4(2), Dic5, D10, C4×C8, C8⋊C4, C4×D5, C2×Dic5, C8⋊C8, C8×D5, C8⋊D5, C4.Dic5, C4×Dic5, C42.D5, C8×Dic5, C408C4, C42.279D10

Smallest permutation representation of C42.279D10
Regular action on 320 points
Generators in S320
```(1 140 178 316)(2 141 179 317)(3 142 180 318)(4 143 181 319)(5 144 182 320)(6 145 183 281)(7 146 184 282)(8 147 185 283)(9 148 186 284)(10 149 187 285)(11 150 188 286)(12 151 189 287)(13 152 190 288)(14 153 191 289)(15 154 192 290)(16 155 193 291)(17 156 194 292)(18 157 195 293)(19 158 196 294)(20 159 197 295)(21 160 198 296)(22 121 199 297)(23 122 200 298)(24 123 161 299)(25 124 162 300)(26 125 163 301)(27 126 164 302)(28 127 165 303)(29 128 166 304)(30 129 167 305)(31 130 168 306)(32 131 169 307)(33 132 170 308)(34 133 171 309)(35 134 172 310)(36 135 173 311)(37 136 174 312)(38 137 175 313)(39 138 176 314)(40 139 177 315)(41 238 82 267)(42 239 83 268)(43 240 84 269)(44 201 85 270)(45 202 86 271)(46 203 87 272)(47 204 88 273)(48 205 89 274)(49 206 90 275)(50 207 91 276)(51 208 92 277)(52 209 93 278)(53 210 94 279)(54 211 95 280)(55 212 96 241)(56 213 97 242)(57 214 98 243)(58 215 99 244)(59 216 100 245)(60 217 101 246)(61 218 102 247)(62 219 103 248)(63 220 104 249)(64 221 105 250)(65 222 106 251)(66 223 107 252)(67 224 108 253)(68 225 109 254)(69 226 110 255)(70 227 111 256)(71 228 112 257)(72 229 113 258)(73 230 114 259)(74 231 115 260)(75 232 116 261)(76 233 117 262)(77 234 118 263)(78 235 119 264)(79 236 120 265)(80 237 81 266)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)(161 171 181 191)(162 172 182 192)(163 173 183 193)(164 174 184 194)(165 175 185 195)(166 176 186 196)(167 177 187 197)(168 178 188 198)(169 179 189 199)(170 180 190 200)(201 211 221 231)(202 212 222 232)(203 213 223 233)(204 214 224 234)(205 215 225 235)(206 216 226 236)(207 217 227 237)(208 218 228 238)(209 219 229 239)(210 220 230 240)(241 251 261 271)(242 252 262 272)(243 253 263 273)(244 254 264 274)(245 255 265 275)(246 256 266 276)(247 257 267 277)(248 258 268 278)(249 259 269 279)(250 260 270 280)(281 291 301 311)(282 292 302 312)(283 293 303 313)(284 294 304 314)(285 295 305 315)(286 296 306 316)(287 297 307 317)(288 298 308 318)(289 299 309 319)(290 300 310 320)
(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)
(1 210 286 63 198 259 130 84)(2 248 287 113 199 239 131 52)(3 228 288 41 200 277 132 102)(4 266 289 91 161 217 133 70)(5 206 290 59 162 255 134 120)(6 244 291 109 163 235 135 48)(7 224 292 77 164 273 136 98)(8 262 293 87 165 213 137 66)(9 202 294 55 166 251 138 116)(10 280 295 105 167 231 139 44)(11 220 296 73 168 269 140 94)(12 258 297 83 169 209 141 62)(13 238 298 51 170 247 142 112)(14 276 299 101 171 227 143 80)(15 216 300 69 172 265 144 90)(16 254 301 119 173 205 145 58)(17 234 302 47 174 243 146 108)(18 272 303 97 175 223 147 76)(19 212 304 65 176 261 148 86)(20 250 305 115 177 201 149 54)(21 230 306 43 178 279 150 104)(22 268 307 93 179 219 151 72)(23 208 308 61 180 257 152 82)(24 246 309 111 181 237 153 50)(25 226 310 79 182 275 154 100)(26 264 311 89 183 215 155 68)(27 204 312 57 184 253 156 118)(28 242 313 107 185 233 157 46)(29 222 314 75 186 271 158 96)(30 260 315 85 187 211 159 64)(31 240 316 53 188 249 160 114)(32 278 317 103 189 229 121 42)(33 218 318 71 190 267 122 92)(34 256 319 81 191 207 123 60)(35 236 320 49 192 245 124 110)(36 274 281 99 193 225 125 78)(37 214 282 67 194 263 126 88)(38 252 283 117 195 203 127 56)(39 232 284 45 196 241 128 106)(40 270 285 95 197 221 129 74)```

`G:=sub<Sym(320)| (1,140,178,316)(2,141,179,317)(3,142,180,318)(4,143,181,319)(5,144,182,320)(6,145,183,281)(7,146,184,282)(8,147,185,283)(9,148,186,284)(10,149,187,285)(11,150,188,286)(12,151,189,287)(13,152,190,288)(14,153,191,289)(15,154,192,290)(16,155,193,291)(17,156,194,292)(18,157,195,293)(19,158,196,294)(20,159,197,295)(21,160,198,296)(22,121,199,297)(23,122,200,298)(24,123,161,299)(25,124,162,300)(26,125,163,301)(27,126,164,302)(28,127,165,303)(29,128,166,304)(30,129,167,305)(31,130,168,306)(32,131,169,307)(33,132,170,308)(34,133,171,309)(35,134,172,310)(36,135,173,311)(37,136,174,312)(38,137,175,313)(39,138,176,314)(40,139,177,315)(41,238,82,267)(42,239,83,268)(43,240,84,269)(44,201,85,270)(45,202,86,271)(46,203,87,272)(47,204,88,273)(48,205,89,274)(49,206,90,275)(50,207,91,276)(51,208,92,277)(52,209,93,278)(53,210,94,279)(54,211,95,280)(55,212,96,241)(56,213,97,242)(57,214,98,243)(58,215,99,244)(59,216,100,245)(60,217,101,246)(61,218,102,247)(62,219,103,248)(63,220,104,249)(64,221,105,250)(65,222,106,251)(66,223,107,252)(67,224,108,253)(68,225,109,254)(69,226,110,255)(70,227,111,256)(71,228,112,257)(72,229,113,258)(73,230,114,259)(74,231,115,260)(75,232,116,261)(76,233,117,262)(77,234,118,263)(78,235,119,264)(79,236,120,265)(80,237,81,266), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160)(161,171,181,191)(162,172,182,192)(163,173,183,193)(164,174,184,194)(165,175,185,195)(166,176,186,196)(167,177,187,197)(168,178,188,198)(169,179,189,199)(170,180,190,200)(201,211,221,231)(202,212,222,232)(203,213,223,233)(204,214,224,234)(205,215,225,235)(206,216,226,236)(207,217,227,237)(208,218,228,238)(209,219,229,239)(210,220,230,240)(241,251,261,271)(242,252,262,272)(243,253,263,273)(244,254,264,274)(245,255,265,275)(246,256,266,276)(247,257,267,277)(248,258,268,278)(249,259,269,279)(250,260,270,280)(281,291,301,311)(282,292,302,312)(283,293,303,313)(284,294,304,314)(285,295,305,315)(286,296,306,316)(287,297,307,317)(288,298,308,318)(289,299,309,319)(290,300,310,320), (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), (1,210,286,63,198,259,130,84)(2,248,287,113,199,239,131,52)(3,228,288,41,200,277,132,102)(4,266,289,91,161,217,133,70)(5,206,290,59,162,255,134,120)(6,244,291,109,163,235,135,48)(7,224,292,77,164,273,136,98)(8,262,293,87,165,213,137,66)(9,202,294,55,166,251,138,116)(10,280,295,105,167,231,139,44)(11,220,296,73,168,269,140,94)(12,258,297,83,169,209,141,62)(13,238,298,51,170,247,142,112)(14,276,299,101,171,227,143,80)(15,216,300,69,172,265,144,90)(16,254,301,119,173,205,145,58)(17,234,302,47,174,243,146,108)(18,272,303,97,175,223,147,76)(19,212,304,65,176,261,148,86)(20,250,305,115,177,201,149,54)(21,230,306,43,178,279,150,104)(22,268,307,93,179,219,151,72)(23,208,308,61,180,257,152,82)(24,246,309,111,181,237,153,50)(25,226,310,79,182,275,154,100)(26,264,311,89,183,215,155,68)(27,204,312,57,184,253,156,118)(28,242,313,107,185,233,157,46)(29,222,314,75,186,271,158,96)(30,260,315,85,187,211,159,64)(31,240,316,53,188,249,160,114)(32,278,317,103,189,229,121,42)(33,218,318,71,190,267,122,92)(34,256,319,81,191,207,123,60)(35,236,320,49,192,245,124,110)(36,274,281,99,193,225,125,78)(37,214,282,67,194,263,126,88)(38,252,283,117,195,203,127,56)(39,232,284,45,196,241,128,106)(40,270,285,95,197,221,129,74)>;`

`G:=Group( (1,140,178,316)(2,141,179,317)(3,142,180,318)(4,143,181,319)(5,144,182,320)(6,145,183,281)(7,146,184,282)(8,147,185,283)(9,148,186,284)(10,149,187,285)(11,150,188,286)(12,151,189,287)(13,152,190,288)(14,153,191,289)(15,154,192,290)(16,155,193,291)(17,156,194,292)(18,157,195,293)(19,158,196,294)(20,159,197,295)(21,160,198,296)(22,121,199,297)(23,122,200,298)(24,123,161,299)(25,124,162,300)(26,125,163,301)(27,126,164,302)(28,127,165,303)(29,128,166,304)(30,129,167,305)(31,130,168,306)(32,131,169,307)(33,132,170,308)(34,133,171,309)(35,134,172,310)(36,135,173,311)(37,136,174,312)(38,137,175,313)(39,138,176,314)(40,139,177,315)(41,238,82,267)(42,239,83,268)(43,240,84,269)(44,201,85,270)(45,202,86,271)(46,203,87,272)(47,204,88,273)(48,205,89,274)(49,206,90,275)(50,207,91,276)(51,208,92,277)(52,209,93,278)(53,210,94,279)(54,211,95,280)(55,212,96,241)(56,213,97,242)(57,214,98,243)(58,215,99,244)(59,216,100,245)(60,217,101,246)(61,218,102,247)(62,219,103,248)(63,220,104,249)(64,221,105,250)(65,222,106,251)(66,223,107,252)(67,224,108,253)(68,225,109,254)(69,226,110,255)(70,227,111,256)(71,228,112,257)(72,229,113,258)(73,230,114,259)(74,231,115,260)(75,232,116,261)(76,233,117,262)(77,234,118,263)(78,235,119,264)(79,236,120,265)(80,237,81,266), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160)(161,171,181,191)(162,172,182,192)(163,173,183,193)(164,174,184,194)(165,175,185,195)(166,176,186,196)(167,177,187,197)(168,178,188,198)(169,179,189,199)(170,180,190,200)(201,211,221,231)(202,212,222,232)(203,213,223,233)(204,214,224,234)(205,215,225,235)(206,216,226,236)(207,217,227,237)(208,218,228,238)(209,219,229,239)(210,220,230,240)(241,251,261,271)(242,252,262,272)(243,253,263,273)(244,254,264,274)(245,255,265,275)(246,256,266,276)(247,257,267,277)(248,258,268,278)(249,259,269,279)(250,260,270,280)(281,291,301,311)(282,292,302,312)(283,293,303,313)(284,294,304,314)(285,295,305,315)(286,296,306,316)(287,297,307,317)(288,298,308,318)(289,299,309,319)(290,300,310,320), (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), (1,210,286,63,198,259,130,84)(2,248,287,113,199,239,131,52)(3,228,288,41,200,277,132,102)(4,266,289,91,161,217,133,70)(5,206,290,59,162,255,134,120)(6,244,291,109,163,235,135,48)(7,224,292,77,164,273,136,98)(8,262,293,87,165,213,137,66)(9,202,294,55,166,251,138,116)(10,280,295,105,167,231,139,44)(11,220,296,73,168,269,140,94)(12,258,297,83,169,209,141,62)(13,238,298,51,170,247,142,112)(14,276,299,101,171,227,143,80)(15,216,300,69,172,265,144,90)(16,254,301,119,173,205,145,58)(17,234,302,47,174,243,146,108)(18,272,303,97,175,223,147,76)(19,212,304,65,176,261,148,86)(20,250,305,115,177,201,149,54)(21,230,306,43,178,279,150,104)(22,268,307,93,179,219,151,72)(23,208,308,61,180,257,152,82)(24,246,309,111,181,237,153,50)(25,226,310,79,182,275,154,100)(26,264,311,89,183,215,155,68)(27,204,312,57,184,253,156,118)(28,242,313,107,185,233,157,46)(29,222,314,75,186,271,158,96)(30,260,315,85,187,211,159,64)(31,240,316,53,188,249,160,114)(32,278,317,103,189,229,121,42)(33,218,318,71,190,267,122,92)(34,256,319,81,191,207,123,60)(35,236,320,49,192,245,124,110)(36,274,281,99,193,225,125,78)(37,214,282,67,194,263,126,88)(38,252,283,117,195,203,127,56)(39,232,284,45,196,241,128,106)(40,270,285,95,197,221,129,74) );`

`G=PermutationGroup([[(1,140,178,316),(2,141,179,317),(3,142,180,318),(4,143,181,319),(5,144,182,320),(6,145,183,281),(7,146,184,282),(8,147,185,283),(9,148,186,284),(10,149,187,285),(11,150,188,286),(12,151,189,287),(13,152,190,288),(14,153,191,289),(15,154,192,290),(16,155,193,291),(17,156,194,292),(18,157,195,293),(19,158,196,294),(20,159,197,295),(21,160,198,296),(22,121,199,297),(23,122,200,298),(24,123,161,299),(25,124,162,300),(26,125,163,301),(27,126,164,302),(28,127,165,303),(29,128,166,304),(30,129,167,305),(31,130,168,306),(32,131,169,307),(33,132,170,308),(34,133,171,309),(35,134,172,310),(36,135,173,311),(37,136,174,312),(38,137,175,313),(39,138,176,314),(40,139,177,315),(41,238,82,267),(42,239,83,268),(43,240,84,269),(44,201,85,270),(45,202,86,271),(46,203,87,272),(47,204,88,273),(48,205,89,274),(49,206,90,275),(50,207,91,276),(51,208,92,277),(52,209,93,278),(53,210,94,279),(54,211,95,280),(55,212,96,241),(56,213,97,242),(57,214,98,243),(58,215,99,244),(59,216,100,245),(60,217,101,246),(61,218,102,247),(62,219,103,248),(63,220,104,249),(64,221,105,250),(65,222,106,251),(66,223,107,252),(67,224,108,253),(68,225,109,254),(69,226,110,255),(70,227,111,256),(71,228,112,257),(72,229,113,258),(73,230,114,259),(74,231,115,260),(75,232,116,261),(76,233,117,262),(77,234,118,263),(78,235,119,264),(79,236,120,265),(80,237,81,266)], [(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160),(161,171,181,191),(162,172,182,192),(163,173,183,193),(164,174,184,194),(165,175,185,195),(166,176,186,196),(167,177,187,197),(168,178,188,198),(169,179,189,199),(170,180,190,200),(201,211,221,231),(202,212,222,232),(203,213,223,233),(204,214,224,234),(205,215,225,235),(206,216,226,236),(207,217,227,237),(208,218,228,238),(209,219,229,239),(210,220,230,240),(241,251,261,271),(242,252,262,272),(243,253,263,273),(244,254,264,274),(245,255,265,275),(246,256,266,276),(247,257,267,277),(248,258,268,278),(249,259,269,279),(250,260,270,280),(281,291,301,311),(282,292,302,312),(283,293,303,313),(284,294,304,314),(285,295,305,315),(286,296,306,316),(287,297,307,317),(288,298,308,318),(289,299,309,319),(290,300,310,320)], [(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)], [(1,210,286,63,198,259,130,84),(2,248,287,113,199,239,131,52),(3,228,288,41,200,277,132,102),(4,266,289,91,161,217,133,70),(5,206,290,59,162,255,134,120),(6,244,291,109,163,235,135,48),(7,224,292,77,164,273,136,98),(8,262,293,87,165,213,137,66),(9,202,294,55,166,251,138,116),(10,280,295,105,167,231,139,44),(11,220,296,73,168,269,140,94),(12,258,297,83,169,209,141,62),(13,238,298,51,170,247,142,112),(14,276,299,101,171,227,143,80),(15,216,300,69,172,265,144,90),(16,254,301,119,173,205,145,58),(17,234,302,47,174,243,146,108),(18,272,303,97,175,223,147,76),(19,212,304,65,176,261,148,86),(20,250,305,115,177,201,149,54),(21,230,306,43,178,279,150,104),(22,268,307,93,179,219,151,72),(23,208,308,61,180,257,152,82),(24,246,309,111,181,237,153,50),(25,226,310,79,182,275,154,100),(26,264,311,89,183,215,155,68),(27,204,312,57,184,253,156,118),(28,242,313,107,185,233,157,46),(29,222,314,75,186,271,158,96),(30,260,315,85,187,211,159,64),(31,240,316,53,188,249,160,114),(32,278,317,103,189,229,121,42),(33,218,318,71,190,267,122,92),(34,256,319,81,191,207,123,60),(35,236,320,49,192,245,124,110),(36,274,281,99,193,225,125,78),(37,214,282,67,194,263,126,88),(38,252,283,117,195,203,127,56),(39,232,284,45,196,241,128,106),(40,270,285,95,197,221,129,74)]])`

104 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 5A 5B 8A ··· 8H 8I ··· 8X 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 1 ··· 1 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C4 C4 C8 D5 M4(2) D10 Dic5 C4×D5 C8×D5 C8⋊D5 C4.Dic5 kernel C42.279D10 C4×C5⋊2C8 C4×C40 C2×C5⋊2C8 C2×C40 C5⋊2C8 C4×C8 C20 C42 C2×C8 C2×C4 C4 C4 C4 # reps 1 2 1 8 4 16 2 8 2 4 8 16 16 16

Matrix representation of C42.279D10 in GL3(𝔽41) generated by

 1 0 0 0 32 0 0 0 32
,
 9 0 0 0 9 0 0 0 9
,
 3 0 0 0 23 9 0 4 19
,
 3 0 0 0 28 22 0 37 13
`G:=sub<GL(3,GF(41))| [1,0,0,0,32,0,0,0,32],[9,0,0,0,9,0,0,0,9],[3,0,0,0,23,4,0,9,19],[3,0,0,0,28,37,0,22,13] >;`

C42.279D10 in GAP, Magma, Sage, TeX

`C_4^2._{279}D_{10}`
`% in TeX`

`G:=Group("C4^2.279D10");`
`// GroupNames label`

`G:=SmallGroup(320,12);`
`// by ID`

`G=gap.SmallGroup(320,12);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,184,80,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=a^-1*b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;`
`// generators/relations`

׿
×
𝔽