direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4.6Q16, C20.28Q16, C20.42SD16, C4⋊C8.3C10, C4⋊Q8.2C10, C4.6(C5×Q16), (C2×Q8).2C20, C4.7(C5×SD16), (C2×C20).505D4, C42.5(C2×C10), (Q8×C10).15C4, (C4×C20).245C22, C10.24(Q8⋊C4), C10.21(C4.D4), (C5×C4⋊C8).9C2, (C5×C4⋊Q8).17C2, (C2×C4).13(C2×C20), (C2×C4).111(C5×D4), C2.5(C5×Q8⋊C4), C2.5(C5×C4.D4), (C2×C20).353(C2×C4), C22.41(C5×C22⋊C4), (C2×C10).192(C22⋊C4), SmallGroup(320,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.6Q16
G = < a,b,c,d | a5=b4=c8=1, d2=b2c4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c-1 >
Subgroups: 114 in 64 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C4⋊C8, C4⋊Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C4.6Q16, C4×C20, C5×C4⋊C4, C2×C40, Q8×C10, C5×C4⋊C8, C5×C4⋊Q8, C5×C4.6Q16
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, SD16, Q16, C20, C2×C10, C4.D4, Q8⋊C4, C2×C20, C5×D4, C4.6Q16, C5×C22⋊C4, C5×SD16, C5×Q16, C5×C4.D4, C5×Q8⋊C4, C5×C4.6Q16
►Regular action on 320 points
Generators in S320
(1 279 271 263 255)(2 280 272 264 256)(3 273 265 257 249)(4 274 266 258 250)(5 275 267 259 251)(6 276 268 260 252)(7 277 269 261 253)(8 278 270 262 254)(9 41 33 25 17)(10 42 34 26 18)(11 43 35 27 19)(12 44 36 28 20)(13 45 37 29 21)(14 46 38 30 22)(15 47 39 31 23)(16 48 40 32 24)(49 81 73 65 57)(50 82 74 66 58)(51 83 75 67 59)(52 84 76 68 60)(53 85 77 69 61)(54 86 78 70 62)(55 87 79 71 63)(56 88 80 72 64)(89 121 113 105 97)(90 122 114 106 98)(91 123 115 107 99)(92 124 116 108 100)(93 125 117 109 101)(94 126 118 110 102)(95 127 119 111 103)(96 128 120 112 104)(129 161 153 145 137)(130 162 154 146 138)(131 163 155 147 139)(132 164 156 148 140)(133 165 157 149 141)(134 166 158 150 142)(135 167 159 151 143)(136 168 160 152 144)(169 201 193 185 177)(170 202 194 186 178)(171 203 195 187 179)(172 204 196 188 180)(173 205 197 189 181)(174 206 198 190 182)(175 207 199 191 183)(176 208 200 192 184)(209 241 233 225 217)(210 242 234 226 218)(211 243 235 227 219)(212 244 236 228 220)(213 245 237 229 221)(214 246 238 230 222)(215 247 239 231 223)(216 248 240 232 224)(281 313 305 297 289)(282 314 306 298 290)(283 315 307 299 291)(284 316 308 300 292)(285 317 309 301 293)(286 318 310 302 294)(287 319 311 303 295)(288 320 312 304 296)
(1 89 11 173)(2 174 12 90)(3 91 13 175)(4 176 14 92)(5 93 15 169)(6 170 16 94)(7 95 9 171)(8 172 10 96)(17 179 253 103)(18 104 254 180)(19 181 255 97)(20 98 256 182)(21 183 249 99)(22 100 250 184)(23 177 251 101)(24 102 252 178)(25 187 261 111)(26 112 262 188)(27 189 263 105)(28 106 264 190)(29 191 257 107)(30 108 258 192)(31 185 259 109)(32 110 260 186)(33 195 269 119)(34 120 270 196)(35 197 271 113)(36 114 272 198)(37 199 265 115)(38 116 266 200)(39 193 267 117)(40 118 268 194)(41 203 277 127)(42 128 278 204)(43 205 279 121)(44 122 280 206)(45 207 273 123)(46 124 274 208)(47 201 275 125)(48 126 276 202)(49 131 285 215)(50 216 286 132)(51 133 287 209)(52 210 288 134)(53 135 281 211)(54 212 282 136)(55 129 283 213)(56 214 284 130)(57 139 293 223)(58 224 294 140)(59 141 295 217)(60 218 296 142)(61 143 289 219)(62 220 290 144)(63 137 291 221)(64 222 292 138)(65 147 301 231)(66 232 302 148)(67 149 303 225)(68 226 304 150)(69 151 297 227)(70 228 298 152)(71 145 299 229)(72 230 300 146)(73 155 309 239)(74 240 310 156)(75 157 311 233)(76 234 312 158)(77 159 305 235)(78 236 306 160)(79 153 307 237)(80 238 308 154)(81 163 317 247)(82 248 318 164)(83 165 319 241)(84 242 320 166)(85 167 313 243)(86 244 314 168)(87 161 315 245)(88 246 316 162)
(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 51 15 283)(2 216 16 136)(3 49 9 281)(4 214 10 134)(5 55 11 287)(6 212 12 132)(7 53 13 285)(8 210 14 130)(17 289 249 57)(18 142 250 222)(19 295 251 63)(20 140 252 220)(21 293 253 61)(22 138 254 218)(23 291 255 59)(24 144 256 224)(25 297 257 65)(26 150 258 230)(27 303 259 71)(28 148 260 228)(29 301 261 69)(30 146 262 226)(31 299 263 67)(32 152 264 232)(33 305 265 73)(34 158 266 238)(35 311 267 79)(36 156 268 236)(37 309 269 77)(38 154 270 234)(39 307 271 75)(40 160 272 240)(41 313 273 81)(42 166 274 246)(43 319 275 87)(44 164 276 244)(45 317 277 85)(46 162 278 242)(47 315 279 83)(48 168 280 248)(50 94 282 174)(52 92 284 172)(54 90 286 170)(56 96 288 176)(58 102 290 182)(60 100 292 180)(62 98 294 178)(64 104 296 184)(66 110 298 190)(68 108 300 188)(70 106 302 186)(72 112 304 192)(74 118 306 198)(76 116 308 196)(78 114 310 194)(80 120 312 200)(82 126 314 206)(84 124 316 204)(86 122 318 202)(88 128 320 208)(89 209 169 129)(91 215 171 135)(93 213 173 133)(95 211 175 131)(97 217 177 137)(99 223 179 143)(101 221 181 141)(103 219 183 139)(105 225 185 145)(107 231 187 151)(109 229 189 149)(111 227 191 147)(113 233 193 153)(115 239 195 159)(117 237 197 157)(119 235 199 155)(121 241 201 161)(123 247 203 167)(125 245 205 165)(127 243 207 163)
G:=sub<Sym(320)| (1,279,271,263,255)(2,280,272,264,256)(3,273,265,257,249)(4,274,266,258,250)(5,275,267,259,251)(6,276,268,260,252)(7,277,269,261,253)(8,278,270,262,254)(9,41,33,25,17)(10,42,34,26,18)(11,43,35,27,19)(12,44,36,28,20)(13,45,37,29,21)(14,46,38,30,22)(15,47,39,31,23)(16,48,40,32,24)(49,81,73,65,57)(50,82,74,66,58)(51,83,75,67,59)(52,84,76,68,60)(53,85,77,69,61)(54,86,78,70,62)(55,87,79,71,63)(56,88,80,72,64)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,161,153,145,137)(130,162,154,146,138)(131,163,155,147,139)(132,164,156,148,140)(133,165,157,149,141)(134,166,158,150,142)(135,167,159,151,143)(136,168,160,152,144)(169,201,193,185,177)(170,202,194,186,178)(171,203,195,187,179)(172,204,196,188,180)(173,205,197,189,181)(174,206,198,190,182)(175,207,199,191,183)(176,208,200,192,184)(209,241,233,225,217)(210,242,234,226,218)(211,243,235,227,219)(212,244,236,228,220)(213,245,237,229,221)(214,246,238,230,222)(215,247,239,231,223)(216,248,240,232,224)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,89,11,173)(2,174,12,90)(3,91,13,175)(4,176,14,92)(5,93,15,169)(6,170,16,94)(7,95,9,171)(8,172,10,96)(17,179,253,103)(18,104,254,180)(19,181,255,97)(20,98,256,182)(21,183,249,99)(22,100,250,184)(23,177,251,101)(24,102,252,178)(25,187,261,111)(26,112,262,188)(27,189,263,105)(28,106,264,190)(29,191,257,107)(30,108,258,192)(31,185,259,109)(32,110,260,186)(33,195,269,119)(34,120,270,196)(35,197,271,113)(36,114,272,198)(37,199,265,115)(38,116,266,200)(39,193,267,117)(40,118,268,194)(41,203,277,127)(42,128,278,204)(43,205,279,121)(44,122,280,206)(45,207,273,123)(46,124,274,208)(47,201,275,125)(48,126,276,202)(49,131,285,215)(50,216,286,132)(51,133,287,209)(52,210,288,134)(53,135,281,211)(54,212,282,136)(55,129,283,213)(56,214,284,130)(57,139,293,223)(58,224,294,140)(59,141,295,217)(60,218,296,142)(61,143,289,219)(62,220,290,144)(63,137,291,221)(64,222,292,138)(65,147,301,231)(66,232,302,148)(67,149,303,225)(68,226,304,150)(69,151,297,227)(70,228,298,152)(71,145,299,229)(72,230,300,146)(73,155,309,239)(74,240,310,156)(75,157,311,233)(76,234,312,158)(77,159,305,235)(78,236,306,160)(79,153,307,237)(80,238,308,154)(81,163,317,247)(82,248,318,164)(83,165,319,241)(84,242,320,166)(85,167,313,243)(86,244,314,168)(87,161,315,245)(88,246,316,162), (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), (1,51,15,283)(2,216,16,136)(3,49,9,281)(4,214,10,134)(5,55,11,287)(6,212,12,132)(7,53,13,285)(8,210,14,130)(17,289,249,57)(18,142,250,222)(19,295,251,63)(20,140,252,220)(21,293,253,61)(22,138,254,218)(23,291,255,59)(24,144,256,224)(25,297,257,65)(26,150,258,230)(27,303,259,71)(28,148,260,228)(29,301,261,69)(30,146,262,226)(31,299,263,67)(32,152,264,232)(33,305,265,73)(34,158,266,238)(35,311,267,79)(36,156,268,236)(37,309,269,77)(38,154,270,234)(39,307,271,75)(40,160,272,240)(41,313,273,81)(42,166,274,246)(43,319,275,87)(44,164,276,244)(45,317,277,85)(46,162,278,242)(47,315,279,83)(48,168,280,248)(50,94,282,174)(52,92,284,172)(54,90,286,170)(56,96,288,176)(58,102,290,182)(60,100,292,180)(62,98,294,178)(64,104,296,184)(66,110,298,190)(68,108,300,188)(70,106,302,186)(72,112,304,192)(74,118,306,198)(76,116,308,196)(78,114,310,194)(80,120,312,200)(82,126,314,206)(84,124,316,204)(86,122,318,202)(88,128,320,208)(89,209,169,129)(91,215,171,135)(93,213,173,133)(95,211,175,131)(97,217,177,137)(99,223,179,143)(101,221,181,141)(103,219,183,139)(105,225,185,145)(107,231,187,151)(109,229,189,149)(111,227,191,147)(113,233,193,153)(115,239,195,159)(117,237,197,157)(119,235,199,155)(121,241,201,161)(123,247,203,167)(125,245,205,165)(127,243,207,163)>;
G:=Group( (1,279,271,263,255)(2,280,272,264,256)(3,273,265,257,249)(4,274,266,258,250)(5,275,267,259,251)(6,276,268,260,252)(7,277,269,261,253)(8,278,270,262,254)(9,41,33,25,17)(10,42,34,26,18)(11,43,35,27,19)(12,44,36,28,20)(13,45,37,29,21)(14,46,38,30,22)(15,47,39,31,23)(16,48,40,32,24)(49,81,73,65,57)(50,82,74,66,58)(51,83,75,67,59)(52,84,76,68,60)(53,85,77,69,61)(54,86,78,70,62)(55,87,79,71,63)(56,88,80,72,64)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,161,153,145,137)(130,162,154,146,138)(131,163,155,147,139)(132,164,156,148,140)(133,165,157,149,141)(134,166,158,150,142)(135,167,159,151,143)(136,168,160,152,144)(169,201,193,185,177)(170,202,194,186,178)(171,203,195,187,179)(172,204,196,188,180)(173,205,197,189,181)(174,206,198,190,182)(175,207,199,191,183)(176,208,200,192,184)(209,241,233,225,217)(210,242,234,226,218)(211,243,235,227,219)(212,244,236,228,220)(213,245,237,229,221)(214,246,238,230,222)(215,247,239,231,223)(216,248,240,232,224)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,89,11,173)(2,174,12,90)(3,91,13,175)(4,176,14,92)(5,93,15,169)(6,170,16,94)(7,95,9,171)(8,172,10,96)(17,179,253,103)(18,104,254,180)(19,181,255,97)(20,98,256,182)(21,183,249,99)(22,100,250,184)(23,177,251,101)(24,102,252,178)(25,187,261,111)(26,112,262,188)(27,189,263,105)(28,106,264,190)(29,191,257,107)(30,108,258,192)(31,185,259,109)(32,110,260,186)(33,195,269,119)(34,120,270,196)(35,197,271,113)(36,114,272,198)(37,199,265,115)(38,116,266,200)(39,193,267,117)(40,118,268,194)(41,203,277,127)(42,128,278,204)(43,205,279,121)(44,122,280,206)(45,207,273,123)(46,124,274,208)(47,201,275,125)(48,126,276,202)(49,131,285,215)(50,216,286,132)(51,133,287,209)(52,210,288,134)(53,135,281,211)(54,212,282,136)(55,129,283,213)(56,214,284,130)(57,139,293,223)(58,224,294,140)(59,141,295,217)(60,218,296,142)(61,143,289,219)(62,220,290,144)(63,137,291,221)(64,222,292,138)(65,147,301,231)(66,232,302,148)(67,149,303,225)(68,226,304,150)(69,151,297,227)(70,228,298,152)(71,145,299,229)(72,230,300,146)(73,155,309,239)(74,240,310,156)(75,157,311,233)(76,234,312,158)(77,159,305,235)(78,236,306,160)(79,153,307,237)(80,238,308,154)(81,163,317,247)(82,248,318,164)(83,165,319,241)(84,242,320,166)(85,167,313,243)(86,244,314,168)(87,161,315,245)(88,246,316,162), (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), (1,51,15,283)(2,216,16,136)(3,49,9,281)(4,214,10,134)(5,55,11,287)(6,212,12,132)(7,53,13,285)(8,210,14,130)(17,289,249,57)(18,142,250,222)(19,295,251,63)(20,140,252,220)(21,293,253,61)(22,138,254,218)(23,291,255,59)(24,144,256,224)(25,297,257,65)(26,150,258,230)(27,303,259,71)(28,148,260,228)(29,301,261,69)(30,146,262,226)(31,299,263,67)(32,152,264,232)(33,305,265,73)(34,158,266,238)(35,311,267,79)(36,156,268,236)(37,309,269,77)(38,154,270,234)(39,307,271,75)(40,160,272,240)(41,313,273,81)(42,166,274,246)(43,319,275,87)(44,164,276,244)(45,317,277,85)(46,162,278,242)(47,315,279,83)(48,168,280,248)(50,94,282,174)(52,92,284,172)(54,90,286,170)(56,96,288,176)(58,102,290,182)(60,100,292,180)(62,98,294,178)(64,104,296,184)(66,110,298,190)(68,108,300,188)(70,106,302,186)(72,112,304,192)(74,118,306,198)(76,116,308,196)(78,114,310,194)(80,120,312,200)(82,126,314,206)(84,124,316,204)(86,122,318,202)(88,128,320,208)(89,209,169,129)(91,215,171,135)(93,213,173,133)(95,211,175,131)(97,217,177,137)(99,223,179,143)(101,221,181,141)(103,219,183,139)(105,225,185,145)(107,231,187,151)(109,229,189,149)(111,227,191,147)(113,233,193,153)(115,239,195,159)(117,237,197,157)(119,235,199,155)(121,241,201,161)(123,247,203,167)(125,245,205,165)(127,243,207,163) );
G=PermutationGroup([[(1,279,271,263,255),(2,280,272,264,256),(3,273,265,257,249),(4,274,266,258,250),(5,275,267,259,251),(6,276,268,260,252),(7,277,269,261,253),(8,278,270,262,254),(9,41,33,25,17),(10,42,34,26,18),(11,43,35,27,19),(12,44,36,28,20),(13,45,37,29,21),(14,46,38,30,22),(15,47,39,31,23),(16,48,40,32,24),(49,81,73,65,57),(50,82,74,66,58),(51,83,75,67,59),(52,84,76,68,60),(53,85,77,69,61),(54,86,78,70,62),(55,87,79,71,63),(56,88,80,72,64),(89,121,113,105,97),(90,122,114,106,98),(91,123,115,107,99),(92,124,116,108,100),(93,125,117,109,101),(94,126,118,110,102),(95,127,119,111,103),(96,128,120,112,104),(129,161,153,145,137),(130,162,154,146,138),(131,163,155,147,139),(132,164,156,148,140),(133,165,157,149,141),(134,166,158,150,142),(135,167,159,151,143),(136,168,160,152,144),(169,201,193,185,177),(170,202,194,186,178),(171,203,195,187,179),(172,204,196,188,180),(173,205,197,189,181),(174,206,198,190,182),(175,207,199,191,183),(176,208,200,192,184),(209,241,233,225,217),(210,242,234,226,218),(211,243,235,227,219),(212,244,236,228,220),(213,245,237,229,221),(214,246,238,230,222),(215,247,239,231,223),(216,248,240,232,224),(281,313,305,297,289),(282,314,306,298,290),(283,315,307,299,291),(284,316,308,300,292),(285,317,309,301,293),(286,318,310,302,294),(287,319,311,303,295),(288,320,312,304,296)], [(1,89,11,173),(2,174,12,90),(3,91,13,175),(4,176,14,92),(5,93,15,169),(6,170,16,94),(7,95,9,171),(8,172,10,96),(17,179,253,103),(18,104,254,180),(19,181,255,97),(20,98,256,182),(21,183,249,99),(22,100,250,184),(23,177,251,101),(24,102,252,178),(25,187,261,111),(26,112,262,188),(27,189,263,105),(28,106,264,190),(29,191,257,107),(30,108,258,192),(31,185,259,109),(32,110,260,186),(33,195,269,119),(34,120,270,196),(35,197,271,113),(36,114,272,198),(37,199,265,115),(38,116,266,200),(39,193,267,117),(40,118,268,194),(41,203,277,127),(42,128,278,204),(43,205,279,121),(44,122,280,206),(45,207,273,123),(46,124,274,208),(47,201,275,125),(48,126,276,202),(49,131,285,215),(50,216,286,132),(51,133,287,209),(52,210,288,134),(53,135,281,211),(54,212,282,136),(55,129,283,213),(56,214,284,130),(57,139,293,223),(58,224,294,140),(59,141,295,217),(60,218,296,142),(61,143,289,219),(62,220,290,144),(63,137,291,221),(64,222,292,138),(65,147,301,231),(66,232,302,148),(67,149,303,225),(68,226,304,150),(69,151,297,227),(70,228,298,152),(71,145,299,229),(72,230,300,146),(73,155,309,239),(74,240,310,156),(75,157,311,233),(76,234,312,158),(77,159,305,235),(78,236,306,160),(79,153,307,237),(80,238,308,154),(81,163,317,247),(82,248,318,164),(83,165,319,241),(84,242,320,166),(85,167,313,243),(86,244,314,168),(87,161,315,245),(88,246,316,162)], [(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)], [(1,51,15,283),(2,216,16,136),(3,49,9,281),(4,214,10,134),(5,55,11,287),(6,212,12,132),(7,53,13,285),(8,210,14,130),(17,289,249,57),(18,142,250,222),(19,295,251,63),(20,140,252,220),(21,293,253,61),(22,138,254,218),(23,291,255,59),(24,144,256,224),(25,297,257,65),(26,150,258,230),(27,303,259,71),(28,148,260,228),(29,301,261,69),(30,146,262,226),(31,299,263,67),(32,152,264,232),(33,305,265,73),(34,158,266,238),(35,311,267,79),(36,156,268,236),(37,309,269,77),(38,154,270,234),(39,307,271,75),(40,160,272,240),(41,313,273,81),(42,166,274,246),(43,319,275,87),(44,164,276,244),(45,317,277,85),(46,162,278,242),(47,315,279,83),(48,168,280,248),(50,94,282,174),(52,92,284,172),(54,90,286,170),(56,96,288,176),(58,102,290,182),(60,100,292,180),(62,98,294,178),(64,104,296,184),(66,110,298,190),(68,108,300,188),(70,106,302,186),(72,112,304,192),(74,118,306,198),(76,116,308,196),(78,114,310,194),(80,120,312,200),(82,126,314,206),(84,124,316,204),(86,122,318,202),(88,128,320,208),(89,209,169,129),(91,215,171,135),(93,213,173,133),(95,211,175,131),(97,217,177,137),(99,223,179,143),(101,221,181,141),(103,219,183,139),(105,225,185,145),(107,231,187,151),(109,229,189,149),(111,227,191,147),(113,233,193,153),(115,239,195,159),(117,237,197,157),(119,235,199,155),(121,241,201,161),(123,247,203,167),(125,245,205,165),(127,243,207,163)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | 20R | 20S | 20T | 20U | ··· | 20AB | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | SD16 | Q16 | C5×D4 | C5×SD16 | C5×Q16 | C4.D4 | C5×C4.D4 |
| kernel | C5×C4.6Q16 | C5×C4⋊C8 | C5×C4⋊Q8 | Q8×C10 | C4.6Q16 | C4⋊C8 | C4⋊Q8 | C2×Q8 | C2×C20 | C20 | C20 | C2×C4 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 4 | 4 | 8 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C4.6Q16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 32 |
| 3 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 31 | 3 |
| 0 | 0 | 8 | 10 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 11 | 12 |
| 0 | 0 | 24 | 30 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,2,32],[3,0,0,0,0,27,0,0,0,0,31,8,0,0,3,10],[0,40,0,0,1,0,0,0,0,0,11,24,0,0,12,30] >;
C5×C4.6Q16 in GAP, Magma, Sage, TeX
C_5\times C_4._6Q_{16} % in TeX
G:=Group("C5xC4.6Q16"); // GroupNames label
G:=SmallGroup(320,138);
// by ID
G=gap.SmallGroup(320,138);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,2530,248,4911,242]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=1,d^2=b^2*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^-1>;
// generators/relations