metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊5Dic10, C42.123D10, C10.82- (1+4), (C5×Q8)⋊6Q8, C5⋊2(Q8⋊3Q8), C20⋊Q8.12C2, (C4×Q8).11D5, C20.45(C2×Q8), C4⋊C4.292D10, (Q8×C20).12C2, (C2×Q8).197D10, (Q8×Dic5).11C2, C4.18(C2×Dic10), C10.16(C22×Q8), (C4×C20).165C22, (C2×C20).493C23, (C2×C10).113C24, (C4×Dic10).22C2, C4.Dic10.11C2, C20.6Q8.11C2, Dic5.62(C4○D4), C4⋊Dic5.304C22, (Q8×C10).213C22, (C2×Dic5).51C23, (C4×Dic5).90C22, C2.18(C22×Dic10), C10.D4.9C22, C22.138(C23×D5), C2.11(Q8.10D10), (C2×Dic10).249C22, C2.28(D5×C4○D4), C10.143(C2×C4○D4), (C5×C4⋊C4).341C22, (C2×C4).167(C22×D5), SmallGroup(320,1241)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 550 in 200 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C4 [×6], C4 [×13], C22, C5, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], C10 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic5 [×2], Dic5 [×7], C20 [×6], C20 [×4], C2×C10, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic10 [×6], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], Q8⋊3Q8, C4×Dic5 [×6], C10.D4, C10.D4 [×9], C4⋊Dic5 [×9], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], Q8×C10, C4×Dic10 [×3], C20.6Q8 [×3], C20⋊Q8 [×3], C4.Dic10 [×3], Q8×Dic5 [×2], Q8×C20, Q8⋊5Dic10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2- (1+4), Dic10 [×4], C22×D5 [×7], Q8⋊3Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, Q8.10D10, D5×C4○D4, Q8⋊5Dic10
Generators and relations
G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >
►Regular action on 320 points
Generators in S320
(1 107 86 219)(2 108 87 220)(3 109 88 201)(4 110 89 202)(5 111 90 203)(6 112 91 204)(7 113 92 205)(8 114 93 206)(9 115 94 207)(10 116 95 208)(11 117 96 209)(12 118 97 210)(13 119 98 211)(14 120 99 212)(15 101 100 213)(16 102 81 214)(17 103 82 215)(18 104 83 216)(19 105 84 217)(20 106 85 218)(21 279 45 142)(22 280 46 143)(23 261 47 144)(24 262 48 145)(25 263 49 146)(26 264 50 147)(27 265 51 148)(28 266 52 149)(29 267 53 150)(30 268 54 151)(31 269 55 152)(32 270 56 153)(33 271 57 154)(34 272 58 155)(35 273 59 156)(36 274 60 157)(37 275 41 158)(38 276 42 159)(39 277 43 160)(40 278 44 141)(61 133 305 247)(62 134 306 248)(63 135 307 249)(64 136 308 250)(65 137 309 251)(66 138 310 252)(67 139 311 253)(68 140 312 254)(69 121 313 255)(70 122 314 256)(71 123 315 257)(72 124 316 258)(73 125 317 259)(74 126 318 260)(75 127 319 241)(76 128 320 242)(77 129 301 243)(78 130 302 244)(79 131 303 245)(80 132 304 246)(161 299 229 200)(162 300 230 181)(163 281 231 182)(164 282 232 183)(165 283 233 184)(166 284 234 185)(167 285 235 186)(168 286 236 187)(169 287 237 188)(170 288 238 189)(171 289 239 190)(172 290 240 191)(173 291 221 192)(174 292 222 193)(175 293 223 194)(176 294 224 195)(177 295 225 196)(178 296 226 197)(179 297 227 198)(180 298 228 199)
(1 245 86 131)(2 132 87 246)(3 247 88 133)(4 134 89 248)(5 249 90 135)(6 136 91 250)(7 251 92 137)(8 138 93 252)(9 253 94 139)(10 140 95 254)(11 255 96 121)(12 122 97 256)(13 257 98 123)(14 124 99 258)(15 259 100 125)(16 126 81 260)(17 241 82 127)(18 128 83 242)(19 243 84 129)(20 130 85 244)(21 284 45 185)(22 186 46 285)(23 286 47 187)(24 188 48 287)(25 288 49 189)(26 190 50 289)(27 290 51 191)(28 192 52 291)(29 292 53 193)(30 194 54 293)(31 294 55 195)(32 196 56 295)(33 296 57 197)(34 198 58 297)(35 298 59 199)(36 200 60 299)(37 300 41 181)(38 182 42 281)(39 282 43 183)(40 184 44 283)(61 109 305 201)(62 202 306 110)(63 111 307 203)(64 204 308 112)(65 113 309 205)(66 206 310 114)(67 115 311 207)(68 208 312 116)(69 117 313 209)(70 210 314 118)(71 119 315 211)(72 212 316 120)(73 101 317 213)(74 214 318 102)(75 103 319 215)(76 216 320 104)(77 105 301 217)(78 218 302 106)(79 107 303 219)(80 220 304 108)(141 165 278 233)(142 234 279 166)(143 167 280 235)(144 236 261 168)(145 169 262 237)(146 238 263 170)(147 171 264 239)(148 240 265 172)(149 173 266 221)(150 222 267 174)(151 175 268 223)(152 224 269 176)(153 177 270 225)(154 226 271 178)(155 179 272 227)(156 228 273 180)(157 161 274 229)(158 230 275 162)(159 163 276 231)(160 232 277 164)
(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)
(1 290 11 300)(2 289 12 299)(3 288 13 298)(4 287 14 297)(5 286 15 296)(6 285 16 295)(7 284 17 294)(8 283 18 293)(9 282 19 292)(10 281 20 291)(21 127 31 137)(22 126 32 136)(23 125 33 135)(24 124 34 134)(25 123 35 133)(26 122 36 132)(27 121 37 131)(28 140 38 130)(29 139 39 129)(30 138 40 128)(41 245 51 255)(42 244 52 254)(43 243 53 253)(44 242 54 252)(45 241 55 251)(46 260 56 250)(47 259 57 249)(48 258 58 248)(49 257 59 247)(50 256 60 246)(61 146 71 156)(62 145 72 155)(63 144 73 154)(64 143 74 153)(65 142 75 152)(66 141 76 151)(67 160 77 150)(68 159 78 149)(69 158 79 148)(70 157 80 147)(81 196 91 186)(82 195 92 185)(83 194 93 184)(84 193 94 183)(85 192 95 182)(86 191 96 181)(87 190 97 200)(88 189 98 199)(89 188 99 198)(90 187 100 197)(101 226 111 236)(102 225 112 235)(103 224 113 234)(104 223 114 233)(105 222 115 232)(106 221 116 231)(107 240 117 230)(108 239 118 229)(109 238 119 228)(110 237 120 227)(161 220 171 210)(162 219 172 209)(163 218 173 208)(164 217 174 207)(165 216 175 206)(166 215 176 205)(167 214 177 204)(168 213 178 203)(169 212 179 202)(170 211 180 201)(261 317 271 307)(262 316 272 306)(263 315 273 305)(264 314 274 304)(265 313 275 303)(266 312 276 302)(267 311 277 301)(268 310 278 320)(269 309 279 319)(270 308 280 318)
G:=sub<Sym(320)| (1,107,86,219)(2,108,87,220)(3,109,88,201)(4,110,89,202)(5,111,90,203)(6,112,91,204)(7,113,92,205)(8,114,93,206)(9,115,94,207)(10,116,95,208)(11,117,96,209)(12,118,97,210)(13,119,98,211)(14,120,99,212)(15,101,100,213)(16,102,81,214)(17,103,82,215)(18,104,83,216)(19,105,84,217)(20,106,85,218)(21,279,45,142)(22,280,46,143)(23,261,47,144)(24,262,48,145)(25,263,49,146)(26,264,50,147)(27,265,51,148)(28,266,52,149)(29,267,53,150)(30,268,54,151)(31,269,55,152)(32,270,56,153)(33,271,57,154)(34,272,58,155)(35,273,59,156)(36,274,60,157)(37,275,41,158)(38,276,42,159)(39,277,43,160)(40,278,44,141)(61,133,305,247)(62,134,306,248)(63,135,307,249)(64,136,308,250)(65,137,309,251)(66,138,310,252)(67,139,311,253)(68,140,312,254)(69,121,313,255)(70,122,314,256)(71,123,315,257)(72,124,316,258)(73,125,317,259)(74,126,318,260)(75,127,319,241)(76,128,320,242)(77,129,301,243)(78,130,302,244)(79,131,303,245)(80,132,304,246)(161,299,229,200)(162,300,230,181)(163,281,231,182)(164,282,232,183)(165,283,233,184)(166,284,234,185)(167,285,235,186)(168,286,236,187)(169,287,237,188)(170,288,238,189)(171,289,239,190)(172,290,240,191)(173,291,221,192)(174,292,222,193)(175,293,223,194)(176,294,224,195)(177,295,225,196)(178,296,226,197)(179,297,227,198)(180,298,228,199), (1,245,86,131)(2,132,87,246)(3,247,88,133)(4,134,89,248)(5,249,90,135)(6,136,91,250)(7,251,92,137)(8,138,93,252)(9,253,94,139)(10,140,95,254)(11,255,96,121)(12,122,97,256)(13,257,98,123)(14,124,99,258)(15,259,100,125)(16,126,81,260)(17,241,82,127)(18,128,83,242)(19,243,84,129)(20,130,85,244)(21,284,45,185)(22,186,46,285)(23,286,47,187)(24,188,48,287)(25,288,49,189)(26,190,50,289)(27,290,51,191)(28,192,52,291)(29,292,53,193)(30,194,54,293)(31,294,55,195)(32,196,56,295)(33,296,57,197)(34,198,58,297)(35,298,59,199)(36,200,60,299)(37,300,41,181)(38,182,42,281)(39,282,43,183)(40,184,44,283)(61,109,305,201)(62,202,306,110)(63,111,307,203)(64,204,308,112)(65,113,309,205)(66,206,310,114)(67,115,311,207)(68,208,312,116)(69,117,313,209)(70,210,314,118)(71,119,315,211)(72,212,316,120)(73,101,317,213)(74,214,318,102)(75,103,319,215)(76,216,320,104)(77,105,301,217)(78,218,302,106)(79,107,303,219)(80,220,304,108)(141,165,278,233)(142,234,279,166)(143,167,280,235)(144,236,261,168)(145,169,262,237)(146,238,263,170)(147,171,264,239)(148,240,265,172)(149,173,266,221)(150,222,267,174)(151,175,268,223)(152,224,269,176)(153,177,270,225)(154,226,271,178)(155,179,272,227)(156,228,273,180)(157,161,274,229)(158,230,275,162)(159,163,276,231)(160,232,277,164), (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), (1,290,11,300)(2,289,12,299)(3,288,13,298)(4,287,14,297)(5,286,15,296)(6,285,16,295)(7,284,17,294)(8,283,18,293)(9,282,19,292)(10,281,20,291)(21,127,31,137)(22,126,32,136)(23,125,33,135)(24,124,34,134)(25,123,35,133)(26,122,36,132)(27,121,37,131)(28,140,38,130)(29,139,39,129)(30,138,40,128)(41,245,51,255)(42,244,52,254)(43,243,53,253)(44,242,54,252)(45,241,55,251)(46,260,56,250)(47,259,57,249)(48,258,58,248)(49,257,59,247)(50,256,60,246)(61,146,71,156)(62,145,72,155)(63,144,73,154)(64,143,74,153)(65,142,75,152)(66,141,76,151)(67,160,77,150)(68,159,78,149)(69,158,79,148)(70,157,80,147)(81,196,91,186)(82,195,92,185)(83,194,93,184)(84,193,94,183)(85,192,95,182)(86,191,96,181)(87,190,97,200)(88,189,98,199)(89,188,99,198)(90,187,100,197)(101,226,111,236)(102,225,112,235)(103,224,113,234)(104,223,114,233)(105,222,115,232)(106,221,116,231)(107,240,117,230)(108,239,118,229)(109,238,119,228)(110,237,120,227)(161,220,171,210)(162,219,172,209)(163,218,173,208)(164,217,174,207)(165,216,175,206)(166,215,176,205)(167,214,177,204)(168,213,178,203)(169,212,179,202)(170,211,180,201)(261,317,271,307)(262,316,272,306)(263,315,273,305)(264,314,274,304)(265,313,275,303)(266,312,276,302)(267,311,277,301)(268,310,278,320)(269,309,279,319)(270,308,280,318)>;
G:=Group( (1,107,86,219)(2,108,87,220)(3,109,88,201)(4,110,89,202)(5,111,90,203)(6,112,91,204)(7,113,92,205)(8,114,93,206)(9,115,94,207)(10,116,95,208)(11,117,96,209)(12,118,97,210)(13,119,98,211)(14,120,99,212)(15,101,100,213)(16,102,81,214)(17,103,82,215)(18,104,83,216)(19,105,84,217)(20,106,85,218)(21,279,45,142)(22,280,46,143)(23,261,47,144)(24,262,48,145)(25,263,49,146)(26,264,50,147)(27,265,51,148)(28,266,52,149)(29,267,53,150)(30,268,54,151)(31,269,55,152)(32,270,56,153)(33,271,57,154)(34,272,58,155)(35,273,59,156)(36,274,60,157)(37,275,41,158)(38,276,42,159)(39,277,43,160)(40,278,44,141)(61,133,305,247)(62,134,306,248)(63,135,307,249)(64,136,308,250)(65,137,309,251)(66,138,310,252)(67,139,311,253)(68,140,312,254)(69,121,313,255)(70,122,314,256)(71,123,315,257)(72,124,316,258)(73,125,317,259)(74,126,318,260)(75,127,319,241)(76,128,320,242)(77,129,301,243)(78,130,302,244)(79,131,303,245)(80,132,304,246)(161,299,229,200)(162,300,230,181)(163,281,231,182)(164,282,232,183)(165,283,233,184)(166,284,234,185)(167,285,235,186)(168,286,236,187)(169,287,237,188)(170,288,238,189)(171,289,239,190)(172,290,240,191)(173,291,221,192)(174,292,222,193)(175,293,223,194)(176,294,224,195)(177,295,225,196)(178,296,226,197)(179,297,227,198)(180,298,228,199), (1,245,86,131)(2,132,87,246)(3,247,88,133)(4,134,89,248)(5,249,90,135)(6,136,91,250)(7,251,92,137)(8,138,93,252)(9,253,94,139)(10,140,95,254)(11,255,96,121)(12,122,97,256)(13,257,98,123)(14,124,99,258)(15,259,100,125)(16,126,81,260)(17,241,82,127)(18,128,83,242)(19,243,84,129)(20,130,85,244)(21,284,45,185)(22,186,46,285)(23,286,47,187)(24,188,48,287)(25,288,49,189)(26,190,50,289)(27,290,51,191)(28,192,52,291)(29,292,53,193)(30,194,54,293)(31,294,55,195)(32,196,56,295)(33,296,57,197)(34,198,58,297)(35,298,59,199)(36,200,60,299)(37,300,41,181)(38,182,42,281)(39,282,43,183)(40,184,44,283)(61,109,305,201)(62,202,306,110)(63,111,307,203)(64,204,308,112)(65,113,309,205)(66,206,310,114)(67,115,311,207)(68,208,312,116)(69,117,313,209)(70,210,314,118)(71,119,315,211)(72,212,316,120)(73,101,317,213)(74,214,318,102)(75,103,319,215)(76,216,320,104)(77,105,301,217)(78,218,302,106)(79,107,303,219)(80,220,304,108)(141,165,278,233)(142,234,279,166)(143,167,280,235)(144,236,261,168)(145,169,262,237)(146,238,263,170)(147,171,264,239)(148,240,265,172)(149,173,266,221)(150,222,267,174)(151,175,268,223)(152,224,269,176)(153,177,270,225)(154,226,271,178)(155,179,272,227)(156,228,273,180)(157,161,274,229)(158,230,275,162)(159,163,276,231)(160,232,277,164), (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), (1,290,11,300)(2,289,12,299)(3,288,13,298)(4,287,14,297)(5,286,15,296)(6,285,16,295)(7,284,17,294)(8,283,18,293)(9,282,19,292)(10,281,20,291)(21,127,31,137)(22,126,32,136)(23,125,33,135)(24,124,34,134)(25,123,35,133)(26,122,36,132)(27,121,37,131)(28,140,38,130)(29,139,39,129)(30,138,40,128)(41,245,51,255)(42,244,52,254)(43,243,53,253)(44,242,54,252)(45,241,55,251)(46,260,56,250)(47,259,57,249)(48,258,58,248)(49,257,59,247)(50,256,60,246)(61,146,71,156)(62,145,72,155)(63,144,73,154)(64,143,74,153)(65,142,75,152)(66,141,76,151)(67,160,77,150)(68,159,78,149)(69,158,79,148)(70,157,80,147)(81,196,91,186)(82,195,92,185)(83,194,93,184)(84,193,94,183)(85,192,95,182)(86,191,96,181)(87,190,97,200)(88,189,98,199)(89,188,99,198)(90,187,100,197)(101,226,111,236)(102,225,112,235)(103,224,113,234)(104,223,114,233)(105,222,115,232)(106,221,116,231)(107,240,117,230)(108,239,118,229)(109,238,119,228)(110,237,120,227)(161,220,171,210)(162,219,172,209)(163,218,173,208)(164,217,174,207)(165,216,175,206)(166,215,176,205)(167,214,177,204)(168,213,178,203)(169,212,179,202)(170,211,180,201)(261,317,271,307)(262,316,272,306)(263,315,273,305)(264,314,274,304)(265,313,275,303)(266,312,276,302)(267,311,277,301)(268,310,278,320)(269,309,279,319)(270,308,280,318) );
G=PermutationGroup([(1,107,86,219),(2,108,87,220),(3,109,88,201),(4,110,89,202),(5,111,90,203),(6,112,91,204),(7,113,92,205),(8,114,93,206),(9,115,94,207),(10,116,95,208),(11,117,96,209),(12,118,97,210),(13,119,98,211),(14,120,99,212),(15,101,100,213),(16,102,81,214),(17,103,82,215),(18,104,83,216),(19,105,84,217),(20,106,85,218),(21,279,45,142),(22,280,46,143),(23,261,47,144),(24,262,48,145),(25,263,49,146),(26,264,50,147),(27,265,51,148),(28,266,52,149),(29,267,53,150),(30,268,54,151),(31,269,55,152),(32,270,56,153),(33,271,57,154),(34,272,58,155),(35,273,59,156),(36,274,60,157),(37,275,41,158),(38,276,42,159),(39,277,43,160),(40,278,44,141),(61,133,305,247),(62,134,306,248),(63,135,307,249),(64,136,308,250),(65,137,309,251),(66,138,310,252),(67,139,311,253),(68,140,312,254),(69,121,313,255),(70,122,314,256),(71,123,315,257),(72,124,316,258),(73,125,317,259),(74,126,318,260),(75,127,319,241),(76,128,320,242),(77,129,301,243),(78,130,302,244),(79,131,303,245),(80,132,304,246),(161,299,229,200),(162,300,230,181),(163,281,231,182),(164,282,232,183),(165,283,233,184),(166,284,234,185),(167,285,235,186),(168,286,236,187),(169,287,237,188),(170,288,238,189),(171,289,239,190),(172,290,240,191),(173,291,221,192),(174,292,222,193),(175,293,223,194),(176,294,224,195),(177,295,225,196),(178,296,226,197),(179,297,227,198),(180,298,228,199)], [(1,245,86,131),(2,132,87,246),(3,247,88,133),(4,134,89,248),(5,249,90,135),(6,136,91,250),(7,251,92,137),(8,138,93,252),(9,253,94,139),(10,140,95,254),(11,255,96,121),(12,122,97,256),(13,257,98,123),(14,124,99,258),(15,259,100,125),(16,126,81,260),(17,241,82,127),(18,128,83,242),(19,243,84,129),(20,130,85,244),(21,284,45,185),(22,186,46,285),(23,286,47,187),(24,188,48,287),(25,288,49,189),(26,190,50,289),(27,290,51,191),(28,192,52,291),(29,292,53,193),(30,194,54,293),(31,294,55,195),(32,196,56,295),(33,296,57,197),(34,198,58,297),(35,298,59,199),(36,200,60,299),(37,300,41,181),(38,182,42,281),(39,282,43,183),(40,184,44,283),(61,109,305,201),(62,202,306,110),(63,111,307,203),(64,204,308,112),(65,113,309,205),(66,206,310,114),(67,115,311,207),(68,208,312,116),(69,117,313,209),(70,210,314,118),(71,119,315,211),(72,212,316,120),(73,101,317,213),(74,214,318,102),(75,103,319,215),(76,216,320,104),(77,105,301,217),(78,218,302,106),(79,107,303,219),(80,220,304,108),(141,165,278,233),(142,234,279,166),(143,167,280,235),(144,236,261,168),(145,169,262,237),(146,238,263,170),(147,171,264,239),(148,240,265,172),(149,173,266,221),(150,222,267,174),(151,175,268,223),(152,224,269,176),(153,177,270,225),(154,226,271,178),(155,179,272,227),(156,228,273,180),(157,161,274,229),(158,230,275,162),(159,163,276,231),(160,232,277,164)], [(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)], [(1,290,11,300),(2,289,12,299),(3,288,13,298),(4,287,14,297),(5,286,15,296),(6,285,16,295),(7,284,17,294),(8,283,18,293),(9,282,19,292),(10,281,20,291),(21,127,31,137),(22,126,32,136),(23,125,33,135),(24,124,34,134),(25,123,35,133),(26,122,36,132),(27,121,37,131),(28,140,38,130),(29,139,39,129),(30,138,40,128),(41,245,51,255),(42,244,52,254),(43,243,53,253),(44,242,54,252),(45,241,55,251),(46,260,56,250),(47,259,57,249),(48,258,58,248),(49,257,59,247),(50,256,60,246),(61,146,71,156),(62,145,72,155),(63,144,73,154),(64,143,74,153),(65,142,75,152),(66,141,76,151),(67,160,77,150),(68,159,78,149),(69,158,79,148),(70,157,80,147),(81,196,91,186),(82,195,92,185),(83,194,93,184),(84,193,94,183),(85,192,95,182),(86,191,96,181),(87,190,97,200),(88,189,98,199),(89,188,99,198),(90,187,100,197),(101,226,111,236),(102,225,112,235),(103,224,113,234),(104,223,114,233),(105,222,115,232),(106,221,116,231),(107,240,117,230),(108,239,118,229),(109,238,119,228),(110,237,120,227),(161,220,171,210),(162,219,172,209),(163,218,173,208),(164,217,174,207),(165,216,175,206),(166,215,176,205),(167,214,177,204),(168,213,178,203),(169,212,179,202),(170,211,180,201),(261,317,271,307),(262,316,272,306),(263,315,273,305),(264,314,274,304),(265,313,275,303),(266,312,276,302),(267,311,277,301),(268,310,278,320),(269,309,279,319),(270,308,280,318)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 35 |
| 0 | 0 | 0 | 0 | 20 | 23 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 9 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 14 | 27 | 0 | 0 | 0 | 0 |
| 11 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 24 | 0 | 0 |
| 0 | 0 | 23 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,20,0,0,0,0,35,23],[0,1,0,0,0,0,40,7,0,0,0,0,0,0,40,18,0,0,0,0,9,1,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[14,11,0,0,0,0,27,27,0,0,0,0,0,0,12,23,0,0,0,0,24,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | Dic10 | 2- (1+4) | Q8.10D10 | D5×C4○D4 |
| kernel | Q8⋊5Dic10 | C4×Dic10 | C20.6Q8 | C20⋊Q8 | C4.Dic10 | Q8×Dic5 | Q8×C20 | C5×Q8 | C4×Q8 | Dic5 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_5Dic_{10} % in TeX
G:=Group("Q8:5Dic10"); // GroupNames label
G:=SmallGroup(320,1241);
// by ID
G=gap.SmallGroup(320,1241);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=c^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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