direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4.SD16, C20.20Q16, C20.33SD16, (C4×C8).4C10, (C4×C40).7C2, C4⋊Q8.7C10, C4.3(C5×Q16), C2.9(C10×Q16), C4.4(C5×SD16), (C2×C20).420D4, C10.56(C2×Q16), C42.77(C2×C10), Q8⋊C4.1C10, C2.15(C10×SD16), C10.95(C2×SD16), C20.269(C4○D4), (C2×C20).943C23, (C2×C40).367C22, (C4×C20).361C22, C22.108(D4×C10), C10.72(C4.4D4), (Q8×C10).172C22, C4.14(C5×C4○D4), (C2×C4).76(C5×D4), (C5×C4⋊Q8).22C2, C4⋊C4.18(C2×C10), (C2×C8).69(C2×C10), (C2×C10).664(C2×D4), (C5×Q8⋊C4).1C2, (C2×Q8).16(C2×C10), C2.10(C5×C4.4D4), (C5×C4⋊C4).238C22, (C2×C4).118(C22×C10), SmallGroup(320,988)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.SD16
G = < a,b,c,d | a5=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c3 >
Subgroups: 162 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C20, C20, C2×C10, C4×C8, Q8⋊C4, C4⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C4.SD16, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, Q8×C10, C4×C40, C5×Q8⋊C4, C5×C4⋊Q8, C5×C4.SD16
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, Q16, C2×D4, C4○D4, C2×C10, C4.4D4, C2×SD16, C2×Q16, C5×D4, C22×C10, C4.SD16, C5×SD16, C5×Q16, D4×C10, C5×C4○D4, C5×C4.4D4, C10×SD16, C10×Q16, C5×C4.SD16
►Regular action on 320 points
Generators in S320
(1 139 63 131 55)(2 140 64 132 56)(3 141 57 133 49)(4 142 58 134 50)(5 143 59 135 51)(6 144 60 136 52)(7 137 61 129 53)(8 138 62 130 54)(9 109 101 96 17)(10 110 102 89 18)(11 111 103 90 19)(12 112 104 91 20)(13 105 97 92 21)(14 106 98 93 22)(15 107 99 94 23)(16 108 100 95 24)(25 126 117 46 38)(26 127 118 47 39)(27 128 119 48 40)(28 121 120 41 33)(29 122 113 42 34)(30 123 114 43 35)(31 124 115 44 36)(32 125 116 45 37)(65 165 157 73 149)(66 166 158 74 150)(67 167 159 75 151)(68 168 160 76 152)(69 161 153 77 145)(70 162 154 78 146)(71 163 155 79 147)(72 164 156 80 148)(81 314 234 306 226)(82 315 235 307 227)(83 316 236 308 228)(84 317 237 309 229)(85 318 238 310 230)(86 319 239 311 231)(87 320 240 312 232)(88 313 233 305 225)(169 246 202 274 194)(170 247 203 275 195)(171 248 204 276 196)(172 241 205 277 197)(173 242 206 278 198)(174 243 207 279 199)(175 244 208 280 200)(176 245 201 273 193)(177 249 265 185 257)(178 250 266 186 258)(179 251 267 187 259)(180 252 268 188 260)(181 253 269 189 261)(182 254 270 190 262)(183 255 271 191 263)(184 256 272 192 264)(209 281 297 217 289)(210 282 298 218 290)(211 283 299 219 291)(212 284 300 220 292)(213 285 301 221 293)(214 286 302 222 294)(215 287 303 223 295)(216 288 304 224 296)
(1 66 9 31)(2 67 10 32)(3 68 11 25)(4 69 12 26)(5 70 13 27)(6 71 14 28)(7 72 15 29)(8 65 16 30)(17 36 55 150)(18 37 56 151)(19 38 49 152)(20 39 50 145)(21 40 51 146)(22 33 52 147)(23 34 53 148)(24 35 54 149)(41 136 79 93)(42 129 80 94)(43 130 73 95)(44 131 74 96)(45 132 75 89)(46 133 76 90)(47 134 77 91)(48 135 78 92)(57 160 103 117)(58 153 104 118)(59 154 97 119)(60 155 98 120)(61 156 99 113)(62 157 100 114)(63 158 101 115)(64 159 102 116)(81 265 169 297)(82 266 170 298)(83 267 171 299)(84 268 172 300)(85 269 173 301)(86 270 174 302)(87 271 175 303)(88 272 176 304)(105 128 143 162)(106 121 144 163)(107 122 137 164)(108 123 138 165)(109 124 139 166)(110 125 140 167)(111 126 141 168)(112 127 142 161)(177 274 209 306)(178 275 210 307)(179 276 211 308)(180 277 212 309)(181 278 213 310)(182 279 214 311)(183 280 215 312)(184 273 216 305)(185 246 217 314)(186 247 218 315)(187 248 219 316)(188 241 220 317)(189 242 221 318)(190 243 222 319)(191 244 223 320)(192 245 224 313)(193 288 225 256)(194 281 226 249)(195 282 227 250)(196 283 228 251)(197 284 229 252)(198 285 230 253)(199 286 231 254)(200 287 232 255)(201 296 233 264)(202 289 234 257)(203 290 235 258)(204 291 236 259)(205 292 237 260)(206 293 238 261)(207 294 239 262)(208 295 240 263)
(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)
(1 250 9 282)(2 285 10 253)(3 256 11 288)(4 283 12 251)(5 254 13 286)(6 281 14 249)(7 252 15 284)(8 287 16 255)(17 210 55 178)(18 181 56 213)(19 216 49 184)(20 179 50 211)(21 214 51 182)(22 177 52 209)(23 212 53 180)(24 183 54 215)(25 193 68 225)(26 228 69 196)(27 199 70 231)(28 226 71 194)(29 197 72 229)(30 232 65 200)(31 195 66 227)(32 230 67 198)(33 306 147 274)(34 277 148 309)(35 312 149 280)(36 275 150 307)(37 310 151 278)(38 273 152 305)(39 308 145 276)(40 279 146 311)(41 234 79 202)(42 205 80 237)(43 240 73 208)(44 203 74 235)(45 238 75 206)(46 201 76 233)(47 236 77 204)(48 207 78 239)(57 192 103 224)(58 219 104 187)(59 190 97 222)(60 217 98 185)(61 188 99 220)(62 223 100 191)(63 186 101 218)(64 221 102 189)(81 163 169 121)(82 124 170 166)(83 161 171 127)(84 122 172 164)(85 167 173 125)(86 128 174 162)(87 165 175 123)(88 126 176 168)(89 261 132 293)(90 296 133 264)(91 259 134 291)(92 294 135 262)(93 257 136 289)(94 292 129 260)(95 263 130 295)(96 290 131 258)(105 302 143 270)(106 265 144 297)(107 300 137 268)(108 271 138 303)(109 298 139 266)(110 269 140 301)(111 304 141 272)(112 267 142 299)(113 241 156 317)(114 320 157 244)(115 247 158 315)(116 318 159 242)(117 245 160 313)(118 316 153 248)(119 243 154 319)(120 314 155 246)
G:=sub<Sym(320)| (1,139,63,131,55)(2,140,64,132,56)(3,141,57,133,49)(4,142,58,134,50)(5,143,59,135,51)(6,144,60,136,52)(7,137,61,129,53)(8,138,62,130,54)(9,109,101,96,17)(10,110,102,89,18)(11,111,103,90,19)(12,112,104,91,20)(13,105,97,92,21)(14,106,98,93,22)(15,107,99,94,23)(16,108,100,95,24)(25,126,117,46,38)(26,127,118,47,39)(27,128,119,48,40)(28,121,120,41,33)(29,122,113,42,34)(30,123,114,43,35)(31,124,115,44,36)(32,125,116,45,37)(65,165,157,73,149)(66,166,158,74,150)(67,167,159,75,151)(68,168,160,76,152)(69,161,153,77,145)(70,162,154,78,146)(71,163,155,79,147)(72,164,156,80,148)(81,314,234,306,226)(82,315,235,307,227)(83,316,236,308,228)(84,317,237,309,229)(85,318,238,310,230)(86,319,239,311,231)(87,320,240,312,232)(88,313,233,305,225)(169,246,202,274,194)(170,247,203,275,195)(171,248,204,276,196)(172,241,205,277,197)(173,242,206,278,198)(174,243,207,279,199)(175,244,208,280,200)(176,245,201,273,193)(177,249,265,185,257)(178,250,266,186,258)(179,251,267,187,259)(180,252,268,188,260)(181,253,269,189,261)(182,254,270,190,262)(183,255,271,191,263)(184,256,272,192,264)(209,281,297,217,289)(210,282,298,218,290)(211,283,299,219,291)(212,284,300,220,292)(213,285,301,221,293)(214,286,302,222,294)(215,287,303,223,295)(216,288,304,224,296), (1,66,9,31)(2,67,10,32)(3,68,11,25)(4,69,12,26)(5,70,13,27)(6,71,14,28)(7,72,15,29)(8,65,16,30)(17,36,55,150)(18,37,56,151)(19,38,49,152)(20,39,50,145)(21,40,51,146)(22,33,52,147)(23,34,53,148)(24,35,54,149)(41,136,79,93)(42,129,80,94)(43,130,73,95)(44,131,74,96)(45,132,75,89)(46,133,76,90)(47,134,77,91)(48,135,78,92)(57,160,103,117)(58,153,104,118)(59,154,97,119)(60,155,98,120)(61,156,99,113)(62,157,100,114)(63,158,101,115)(64,159,102,116)(81,265,169,297)(82,266,170,298)(83,267,171,299)(84,268,172,300)(85,269,173,301)(86,270,174,302)(87,271,175,303)(88,272,176,304)(105,128,143,162)(106,121,144,163)(107,122,137,164)(108,123,138,165)(109,124,139,166)(110,125,140,167)(111,126,141,168)(112,127,142,161)(177,274,209,306)(178,275,210,307)(179,276,211,308)(180,277,212,309)(181,278,213,310)(182,279,214,311)(183,280,215,312)(184,273,216,305)(185,246,217,314)(186,247,218,315)(187,248,219,316)(188,241,220,317)(189,242,221,318)(190,243,222,319)(191,244,223,320)(192,245,224,313)(193,288,225,256)(194,281,226,249)(195,282,227,250)(196,283,228,251)(197,284,229,252)(198,285,230,253)(199,286,231,254)(200,287,232,255)(201,296,233,264)(202,289,234,257)(203,290,235,258)(204,291,236,259)(205,292,237,260)(206,293,238,261)(207,294,239,262)(208,295,240,263), (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), (1,250,9,282)(2,285,10,253)(3,256,11,288)(4,283,12,251)(5,254,13,286)(6,281,14,249)(7,252,15,284)(8,287,16,255)(17,210,55,178)(18,181,56,213)(19,216,49,184)(20,179,50,211)(21,214,51,182)(22,177,52,209)(23,212,53,180)(24,183,54,215)(25,193,68,225)(26,228,69,196)(27,199,70,231)(28,226,71,194)(29,197,72,229)(30,232,65,200)(31,195,66,227)(32,230,67,198)(33,306,147,274)(34,277,148,309)(35,312,149,280)(36,275,150,307)(37,310,151,278)(38,273,152,305)(39,308,145,276)(40,279,146,311)(41,234,79,202)(42,205,80,237)(43,240,73,208)(44,203,74,235)(45,238,75,206)(46,201,76,233)(47,236,77,204)(48,207,78,239)(57,192,103,224)(58,219,104,187)(59,190,97,222)(60,217,98,185)(61,188,99,220)(62,223,100,191)(63,186,101,218)(64,221,102,189)(81,163,169,121)(82,124,170,166)(83,161,171,127)(84,122,172,164)(85,167,173,125)(86,128,174,162)(87,165,175,123)(88,126,176,168)(89,261,132,293)(90,296,133,264)(91,259,134,291)(92,294,135,262)(93,257,136,289)(94,292,129,260)(95,263,130,295)(96,290,131,258)(105,302,143,270)(106,265,144,297)(107,300,137,268)(108,271,138,303)(109,298,139,266)(110,269,140,301)(111,304,141,272)(112,267,142,299)(113,241,156,317)(114,320,157,244)(115,247,158,315)(116,318,159,242)(117,245,160,313)(118,316,153,248)(119,243,154,319)(120,314,155,246)>;
G:=Group( (1,139,63,131,55)(2,140,64,132,56)(3,141,57,133,49)(4,142,58,134,50)(5,143,59,135,51)(6,144,60,136,52)(7,137,61,129,53)(8,138,62,130,54)(9,109,101,96,17)(10,110,102,89,18)(11,111,103,90,19)(12,112,104,91,20)(13,105,97,92,21)(14,106,98,93,22)(15,107,99,94,23)(16,108,100,95,24)(25,126,117,46,38)(26,127,118,47,39)(27,128,119,48,40)(28,121,120,41,33)(29,122,113,42,34)(30,123,114,43,35)(31,124,115,44,36)(32,125,116,45,37)(65,165,157,73,149)(66,166,158,74,150)(67,167,159,75,151)(68,168,160,76,152)(69,161,153,77,145)(70,162,154,78,146)(71,163,155,79,147)(72,164,156,80,148)(81,314,234,306,226)(82,315,235,307,227)(83,316,236,308,228)(84,317,237,309,229)(85,318,238,310,230)(86,319,239,311,231)(87,320,240,312,232)(88,313,233,305,225)(169,246,202,274,194)(170,247,203,275,195)(171,248,204,276,196)(172,241,205,277,197)(173,242,206,278,198)(174,243,207,279,199)(175,244,208,280,200)(176,245,201,273,193)(177,249,265,185,257)(178,250,266,186,258)(179,251,267,187,259)(180,252,268,188,260)(181,253,269,189,261)(182,254,270,190,262)(183,255,271,191,263)(184,256,272,192,264)(209,281,297,217,289)(210,282,298,218,290)(211,283,299,219,291)(212,284,300,220,292)(213,285,301,221,293)(214,286,302,222,294)(215,287,303,223,295)(216,288,304,224,296), (1,66,9,31)(2,67,10,32)(3,68,11,25)(4,69,12,26)(5,70,13,27)(6,71,14,28)(7,72,15,29)(8,65,16,30)(17,36,55,150)(18,37,56,151)(19,38,49,152)(20,39,50,145)(21,40,51,146)(22,33,52,147)(23,34,53,148)(24,35,54,149)(41,136,79,93)(42,129,80,94)(43,130,73,95)(44,131,74,96)(45,132,75,89)(46,133,76,90)(47,134,77,91)(48,135,78,92)(57,160,103,117)(58,153,104,118)(59,154,97,119)(60,155,98,120)(61,156,99,113)(62,157,100,114)(63,158,101,115)(64,159,102,116)(81,265,169,297)(82,266,170,298)(83,267,171,299)(84,268,172,300)(85,269,173,301)(86,270,174,302)(87,271,175,303)(88,272,176,304)(105,128,143,162)(106,121,144,163)(107,122,137,164)(108,123,138,165)(109,124,139,166)(110,125,140,167)(111,126,141,168)(112,127,142,161)(177,274,209,306)(178,275,210,307)(179,276,211,308)(180,277,212,309)(181,278,213,310)(182,279,214,311)(183,280,215,312)(184,273,216,305)(185,246,217,314)(186,247,218,315)(187,248,219,316)(188,241,220,317)(189,242,221,318)(190,243,222,319)(191,244,223,320)(192,245,224,313)(193,288,225,256)(194,281,226,249)(195,282,227,250)(196,283,228,251)(197,284,229,252)(198,285,230,253)(199,286,231,254)(200,287,232,255)(201,296,233,264)(202,289,234,257)(203,290,235,258)(204,291,236,259)(205,292,237,260)(206,293,238,261)(207,294,239,262)(208,295,240,263), (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), (1,250,9,282)(2,285,10,253)(3,256,11,288)(4,283,12,251)(5,254,13,286)(6,281,14,249)(7,252,15,284)(8,287,16,255)(17,210,55,178)(18,181,56,213)(19,216,49,184)(20,179,50,211)(21,214,51,182)(22,177,52,209)(23,212,53,180)(24,183,54,215)(25,193,68,225)(26,228,69,196)(27,199,70,231)(28,226,71,194)(29,197,72,229)(30,232,65,200)(31,195,66,227)(32,230,67,198)(33,306,147,274)(34,277,148,309)(35,312,149,280)(36,275,150,307)(37,310,151,278)(38,273,152,305)(39,308,145,276)(40,279,146,311)(41,234,79,202)(42,205,80,237)(43,240,73,208)(44,203,74,235)(45,238,75,206)(46,201,76,233)(47,236,77,204)(48,207,78,239)(57,192,103,224)(58,219,104,187)(59,190,97,222)(60,217,98,185)(61,188,99,220)(62,223,100,191)(63,186,101,218)(64,221,102,189)(81,163,169,121)(82,124,170,166)(83,161,171,127)(84,122,172,164)(85,167,173,125)(86,128,174,162)(87,165,175,123)(88,126,176,168)(89,261,132,293)(90,296,133,264)(91,259,134,291)(92,294,135,262)(93,257,136,289)(94,292,129,260)(95,263,130,295)(96,290,131,258)(105,302,143,270)(106,265,144,297)(107,300,137,268)(108,271,138,303)(109,298,139,266)(110,269,140,301)(111,304,141,272)(112,267,142,299)(113,241,156,317)(114,320,157,244)(115,247,158,315)(116,318,159,242)(117,245,160,313)(118,316,153,248)(119,243,154,319)(120,314,155,246) );
G=PermutationGroup([[(1,139,63,131,55),(2,140,64,132,56),(3,141,57,133,49),(4,142,58,134,50),(5,143,59,135,51),(6,144,60,136,52),(7,137,61,129,53),(8,138,62,130,54),(9,109,101,96,17),(10,110,102,89,18),(11,111,103,90,19),(12,112,104,91,20),(13,105,97,92,21),(14,106,98,93,22),(15,107,99,94,23),(16,108,100,95,24),(25,126,117,46,38),(26,127,118,47,39),(27,128,119,48,40),(28,121,120,41,33),(29,122,113,42,34),(30,123,114,43,35),(31,124,115,44,36),(32,125,116,45,37),(65,165,157,73,149),(66,166,158,74,150),(67,167,159,75,151),(68,168,160,76,152),(69,161,153,77,145),(70,162,154,78,146),(71,163,155,79,147),(72,164,156,80,148),(81,314,234,306,226),(82,315,235,307,227),(83,316,236,308,228),(84,317,237,309,229),(85,318,238,310,230),(86,319,239,311,231),(87,320,240,312,232),(88,313,233,305,225),(169,246,202,274,194),(170,247,203,275,195),(171,248,204,276,196),(172,241,205,277,197),(173,242,206,278,198),(174,243,207,279,199),(175,244,208,280,200),(176,245,201,273,193),(177,249,265,185,257),(178,250,266,186,258),(179,251,267,187,259),(180,252,268,188,260),(181,253,269,189,261),(182,254,270,190,262),(183,255,271,191,263),(184,256,272,192,264),(209,281,297,217,289),(210,282,298,218,290),(211,283,299,219,291),(212,284,300,220,292),(213,285,301,221,293),(214,286,302,222,294),(215,287,303,223,295),(216,288,304,224,296)], [(1,66,9,31),(2,67,10,32),(3,68,11,25),(4,69,12,26),(5,70,13,27),(6,71,14,28),(7,72,15,29),(8,65,16,30),(17,36,55,150),(18,37,56,151),(19,38,49,152),(20,39,50,145),(21,40,51,146),(22,33,52,147),(23,34,53,148),(24,35,54,149),(41,136,79,93),(42,129,80,94),(43,130,73,95),(44,131,74,96),(45,132,75,89),(46,133,76,90),(47,134,77,91),(48,135,78,92),(57,160,103,117),(58,153,104,118),(59,154,97,119),(60,155,98,120),(61,156,99,113),(62,157,100,114),(63,158,101,115),(64,159,102,116),(81,265,169,297),(82,266,170,298),(83,267,171,299),(84,268,172,300),(85,269,173,301),(86,270,174,302),(87,271,175,303),(88,272,176,304),(105,128,143,162),(106,121,144,163),(107,122,137,164),(108,123,138,165),(109,124,139,166),(110,125,140,167),(111,126,141,168),(112,127,142,161),(177,274,209,306),(178,275,210,307),(179,276,211,308),(180,277,212,309),(181,278,213,310),(182,279,214,311),(183,280,215,312),(184,273,216,305),(185,246,217,314),(186,247,218,315),(187,248,219,316),(188,241,220,317),(189,242,221,318),(190,243,222,319),(191,244,223,320),(192,245,224,313),(193,288,225,256),(194,281,226,249),(195,282,227,250),(196,283,228,251),(197,284,229,252),(198,285,230,253),(199,286,231,254),(200,287,232,255),(201,296,233,264),(202,289,234,257),(203,290,235,258),(204,291,236,259),(205,292,237,260),(206,293,238,261),(207,294,239,262),(208,295,240,263)], [(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)], [(1,250,9,282),(2,285,10,253),(3,256,11,288),(4,283,12,251),(5,254,13,286),(6,281,14,249),(7,252,15,284),(8,287,16,255),(17,210,55,178),(18,181,56,213),(19,216,49,184),(20,179,50,211),(21,214,51,182),(22,177,52,209),(23,212,53,180),(24,183,54,215),(25,193,68,225),(26,228,69,196),(27,199,70,231),(28,226,71,194),(29,197,72,229),(30,232,65,200),(31,195,66,227),(32,230,67,198),(33,306,147,274),(34,277,148,309),(35,312,149,280),(36,275,150,307),(37,310,151,278),(38,273,152,305),(39,308,145,276),(40,279,146,311),(41,234,79,202),(42,205,80,237),(43,240,73,208),(44,203,74,235),(45,238,75,206),(46,201,76,233),(47,236,77,204),(48,207,78,239),(57,192,103,224),(58,219,104,187),(59,190,97,222),(60,217,98,185),(61,188,99,220),(62,223,100,191),(63,186,101,218),(64,221,102,189),(81,163,169,121),(82,124,170,166),(83,161,171,127),(84,122,172,164),(85,167,173,125),(86,128,174,162),(87,165,175,123),(88,126,176,168),(89,261,132,293),(90,296,133,264),(91,259,134,291),(92,294,135,262),(93,257,136,289),(94,292,129,260),(95,263,130,295),(96,290,131,258),(105,302,143,270),(106,265,144,297),(107,300,137,268),(108,271,138,303),(109,298,139,266),(110,269,140,301),(111,304,141,272),(112,267,142,299),(113,241,156,317),(114,320,157,244),(115,247,158,315),(116,318,159,242),(117,245,160,313),(118,316,153,248),(119,243,154,319),(120,314,155,246)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | SD16 | Q16 | C4○D4 | C5×D4 | C5×SD16 | C5×Q16 | C5×C4○D4 |
| kernel | C5×C4.SD16 | C4×C40 | C5×Q8⋊C4 | C5×C4⋊Q8 | C4.SD16 | C4×C8 | Q8⋊C4 | C4⋊Q8 | C2×C20 | C20 | C20 | C20 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 2 | 4 | 4 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C5×C4.SD16 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 15 | 15 | 0 | 0 |
| 26 | 15 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 23 | 28 | 0 | 0 |
| 28 | 18 | 0 | 0 |
| 0 | 0 | 34 | 14 |
| 0 | 0 | 14 | 7 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[15,26,0,0,15,15,0,0,0,0,12,12,0,0,29,12],[23,28,0,0,28,18,0,0,0,0,34,14,0,0,14,7] >;
C5×C4.SD16 in GAP, Magma, Sage, TeX
C_5\times C_4.{\rm SD}_{16} % in TeX
G:=Group("C5xC4.SD16"); // GroupNames label
G:=SmallGroup(320,988);
// by ID
G=gap.SmallGroup(320,988);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,568,1766,226,7004,172,10085,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^3>;
// generators/relations