metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2Dic10, Dic5.9Q16, (C5×Q8)⋊2Q8, C20⋊Q8.4C2, C20.6(C2×Q8), C2.8(D5×Q16), C4⋊C4.17D10, C5⋊1(C4.Q16), (C2×C8).14D10, C40⋊5C4.6C2, Q8⋊C4.3D5, C10.15(C2×Q16), (Q8×Dic5).4C2, C4.6(C2×Dic10), (C2×C40).14C22, (C2×Q8).100D10, Q8⋊Dic5.3C2, C10.D8.2C2, C22.188(D4×D5), C20.8Q8.3C2, C20.158(C4○D4), C4.83(D4⋊2D5), C2.14(D40⋊C2), C10.59(C8⋊C22), (C2×C20).234C23, (C2×Dic5).205D4, C10.12(C22⋊Q8), C4⋊Dic5.83C22, (Q8×C10).17C22, (C4×Dic5).24C22, C2.17(Dic5.14D4), (C2×C10).247(C2×D4), (C5×C4⋊C4).35C22, (C5×Q8⋊C4).3C2, (C2×C5⋊2C8).29C22, (C2×C4).341(C22×D5), SmallGroup(320,421)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.9Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=a5c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c-1 >
Subgroups: 342 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C4.Q16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.D8, C20.8Q8, C40⋊5C4, Q8⋊Dic5, C5×Q8⋊C4, C20⋊Q8, Q8×Dic5, Dic5.9Q16
Quotients: C1, C2, C22, D4, Q8, C23, D5, Q16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×Q16, C8⋊C22, Dic10, C22×D5, C4.Q16, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4, D40⋊C2, D5×Q16, Dic5.9Q16
►Regular action on 320 points
Generators in S320
(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)
(1 196 6 191)(2 195 7 200)(3 194 8 199)(4 193 9 198)(5 192 10 197)(11 204 16 209)(12 203 17 208)(13 202 18 207)(14 201 19 206)(15 210 20 205)(21 145 26 150)(22 144 27 149)(23 143 28 148)(24 142 29 147)(25 141 30 146)(31 190 36 185)(32 189 37 184)(33 188 38 183)(34 187 39 182)(35 186 40 181)(41 176 46 171)(42 175 47 180)(43 174 48 179)(44 173 49 178)(45 172 50 177)(51 239 56 234)(52 238 57 233)(53 237 58 232)(54 236 59 231)(55 235 60 240)(61 241 66 246)(62 250 67 245)(63 249 68 244)(64 248 69 243)(65 247 70 242)(71 230 76 225)(72 229 77 224)(73 228 78 223)(74 227 79 222)(75 226 80 221)(81 216 86 211)(82 215 87 220)(83 214 88 219)(84 213 89 218)(85 212 90 217)(91 279 96 274)(92 278 97 273)(93 277 98 272)(94 276 99 271)(95 275 100 280)(101 281 106 286)(102 290 107 285)(103 289 108 284)(104 288 109 283)(105 287 110 282)(111 270 116 265)(112 269 117 264)(113 268 118 263)(114 267 119 262)(115 266 120 261)(121 256 126 251)(122 255 127 260)(123 254 128 259)(124 253 129 258)(125 252 130 257)(131 319 136 314)(132 318 137 313)(133 317 138 312)(134 316 139 311)(135 315 140 320)(151 310 156 305)(152 309 157 304)(153 308 158 303)(154 307 159 302)(155 306 160 301)(161 293 166 298)(162 292 167 297)(163 291 168 296)(164 300 169 295)(165 299 170 294)
(1 145 18 159 39 170 43 134)(2 146 19 160 40 161 44 135)(3 147 20 151 31 162 45 136)(4 148 11 152 32 163 46 137)(5 149 12 153 33 164 47 138)(6 150 13 154 34 165 48 139)(7 141 14 155 35 166 49 140)(8 142 15 156 36 167 50 131)(9 143 16 157 37 168 41 132)(10 144 17 158 38 169 42 133)(21 207 307 182 299 174 311 196)(22 208 308 183 300 175 312 197)(23 209 309 184 291 176 313 198)(24 210 310 185 292 177 314 199)(25 201 301 186 293 178 315 200)(26 202 302 187 294 179 316 191)(27 203 303 188 295 180 317 192)(28 204 304 189 296 171 318 193)(29 205 305 190 297 172 319 194)(30 206 306 181 298 173 320 195)(51 107 90 96 76 125 62 111)(52 108 81 97 77 126 63 112)(53 109 82 98 78 127 64 113)(54 110 83 99 79 128 65 114)(55 101 84 100 80 129 66 115)(56 102 85 91 71 130 67 116)(57 103 86 92 72 121 68 117)(58 104 87 93 73 122 69 118)(59 105 88 94 74 123 70 119)(60 106 89 95 75 124 61 120)(211 273 229 251 244 269 233 284)(212 274 230 252 245 270 234 285)(213 275 221 253 246 261 235 286)(214 276 222 254 247 262 236 287)(215 277 223 255 248 263 237 288)(216 278 224 256 249 264 238 289)(217 279 225 257 250 265 239 290)(218 280 226 258 241 266 240 281)(219 271 227 259 242 267 231 282)(220 272 228 260 243 268 232 283)
(1 74 34 54)(2 75 35 55)(3 76 36 56)(4 77 37 57)(5 78 38 58)(6 79 39 59)(7 80 40 60)(8 71 31 51)(9 72 32 52)(10 73 33 53)(11 81 41 68)(12 82 42 69)(13 83 43 70)(14 84 44 61)(15 85 45 62)(16 86 46 63)(17 87 47 64)(18 88 48 65)(19 89 49 66)(20 90 50 67)(21 271 294 262)(22 272 295 263)(23 273 296 264)(24 274 297 265)(25 275 298 266)(26 276 299 267)(27 277 300 268)(28 278 291 269)(29 279 292 270)(30 280 293 261)(91 167 111 147)(92 168 112 148)(93 169 113 149)(94 170 114 150)(95 161 115 141)(96 162 116 142)(97 163 117 143)(98 164 118 144)(99 165 119 145)(100 166 120 146)(101 140 124 160)(102 131 125 151)(103 132 126 152)(104 133 127 153)(105 134 128 154)(106 135 129 155)(107 136 130 156)(108 137 121 157)(109 138 122 158)(110 139 123 159)(171 244 209 216)(172 245 210 217)(173 246 201 218)(174 247 202 219)(175 248 203 220)(176 249 204 211)(177 250 205 212)(178 241 206 213)(179 242 207 214)(180 243 208 215)(181 235 200 226)(182 236 191 227)(183 237 192 228)(184 238 193 229)(185 239 194 230)(186 240 195 221)(187 231 196 222)(188 232 197 223)(189 233 198 224)(190 234 199 225)(251 304 289 313)(252 305 290 314)(253 306 281 315)(254 307 282 316)(255 308 283 317)(256 309 284 318)(257 310 285 319)(258 301 286 320)(259 302 287 311)(260 303 288 312)
G:=sub<Sym(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), (1,196,6,191)(2,195,7,200)(3,194,8,199)(4,193,9,198)(5,192,10,197)(11,204,16,209)(12,203,17,208)(13,202,18,207)(14,201,19,206)(15,210,20,205)(21,145,26,150)(22,144,27,149)(23,143,28,148)(24,142,29,147)(25,141,30,146)(31,190,36,185)(32,189,37,184)(33,188,38,183)(34,187,39,182)(35,186,40,181)(41,176,46,171)(42,175,47,180)(43,174,48,179)(44,173,49,178)(45,172,50,177)(51,239,56,234)(52,238,57,233)(53,237,58,232)(54,236,59,231)(55,235,60,240)(61,241,66,246)(62,250,67,245)(63,249,68,244)(64,248,69,243)(65,247,70,242)(71,230,76,225)(72,229,77,224)(73,228,78,223)(74,227,79,222)(75,226,80,221)(81,216,86,211)(82,215,87,220)(83,214,88,219)(84,213,89,218)(85,212,90,217)(91,279,96,274)(92,278,97,273)(93,277,98,272)(94,276,99,271)(95,275,100,280)(101,281,106,286)(102,290,107,285)(103,289,108,284)(104,288,109,283)(105,287,110,282)(111,270,116,265)(112,269,117,264)(113,268,118,263)(114,267,119,262)(115,266,120,261)(121,256,126,251)(122,255,127,260)(123,254,128,259)(124,253,129,258)(125,252,130,257)(131,319,136,314)(132,318,137,313)(133,317,138,312)(134,316,139,311)(135,315,140,320)(151,310,156,305)(152,309,157,304)(153,308,158,303)(154,307,159,302)(155,306,160,301)(161,293,166,298)(162,292,167,297)(163,291,168,296)(164,300,169,295)(165,299,170,294), (1,145,18,159,39,170,43,134)(2,146,19,160,40,161,44,135)(3,147,20,151,31,162,45,136)(4,148,11,152,32,163,46,137)(5,149,12,153,33,164,47,138)(6,150,13,154,34,165,48,139)(7,141,14,155,35,166,49,140)(8,142,15,156,36,167,50,131)(9,143,16,157,37,168,41,132)(10,144,17,158,38,169,42,133)(21,207,307,182,299,174,311,196)(22,208,308,183,300,175,312,197)(23,209,309,184,291,176,313,198)(24,210,310,185,292,177,314,199)(25,201,301,186,293,178,315,200)(26,202,302,187,294,179,316,191)(27,203,303,188,295,180,317,192)(28,204,304,189,296,171,318,193)(29,205,305,190,297,172,319,194)(30,206,306,181,298,173,320,195)(51,107,90,96,76,125,62,111)(52,108,81,97,77,126,63,112)(53,109,82,98,78,127,64,113)(54,110,83,99,79,128,65,114)(55,101,84,100,80,129,66,115)(56,102,85,91,71,130,67,116)(57,103,86,92,72,121,68,117)(58,104,87,93,73,122,69,118)(59,105,88,94,74,123,70,119)(60,106,89,95,75,124,61,120)(211,273,229,251,244,269,233,284)(212,274,230,252,245,270,234,285)(213,275,221,253,246,261,235,286)(214,276,222,254,247,262,236,287)(215,277,223,255,248,263,237,288)(216,278,224,256,249,264,238,289)(217,279,225,257,250,265,239,290)(218,280,226,258,241,266,240,281)(219,271,227,259,242,267,231,282)(220,272,228,260,243,268,232,283), (1,74,34,54)(2,75,35,55)(3,76,36,56)(4,77,37,57)(5,78,38,58)(6,79,39,59)(7,80,40,60)(8,71,31,51)(9,72,32,52)(10,73,33,53)(11,81,41,68)(12,82,42,69)(13,83,43,70)(14,84,44,61)(15,85,45,62)(16,86,46,63)(17,87,47,64)(18,88,48,65)(19,89,49,66)(20,90,50,67)(21,271,294,262)(22,272,295,263)(23,273,296,264)(24,274,297,265)(25,275,298,266)(26,276,299,267)(27,277,300,268)(28,278,291,269)(29,279,292,270)(30,280,293,261)(91,167,111,147)(92,168,112,148)(93,169,113,149)(94,170,114,150)(95,161,115,141)(96,162,116,142)(97,163,117,143)(98,164,118,144)(99,165,119,145)(100,166,120,146)(101,140,124,160)(102,131,125,151)(103,132,126,152)(104,133,127,153)(105,134,128,154)(106,135,129,155)(107,136,130,156)(108,137,121,157)(109,138,122,158)(110,139,123,159)(171,244,209,216)(172,245,210,217)(173,246,201,218)(174,247,202,219)(175,248,203,220)(176,249,204,211)(177,250,205,212)(178,241,206,213)(179,242,207,214)(180,243,208,215)(181,235,200,226)(182,236,191,227)(183,237,192,228)(184,238,193,229)(185,239,194,230)(186,240,195,221)(187,231,196,222)(188,232,197,223)(189,233,198,224)(190,234,199,225)(251,304,289,313)(252,305,290,314)(253,306,281,315)(254,307,282,316)(255,308,283,317)(256,309,284,318)(257,310,285,319)(258,301,286,320)(259,302,287,311)(260,303,288,312)>;
G:=Group( (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), (1,196,6,191)(2,195,7,200)(3,194,8,199)(4,193,9,198)(5,192,10,197)(11,204,16,209)(12,203,17,208)(13,202,18,207)(14,201,19,206)(15,210,20,205)(21,145,26,150)(22,144,27,149)(23,143,28,148)(24,142,29,147)(25,141,30,146)(31,190,36,185)(32,189,37,184)(33,188,38,183)(34,187,39,182)(35,186,40,181)(41,176,46,171)(42,175,47,180)(43,174,48,179)(44,173,49,178)(45,172,50,177)(51,239,56,234)(52,238,57,233)(53,237,58,232)(54,236,59,231)(55,235,60,240)(61,241,66,246)(62,250,67,245)(63,249,68,244)(64,248,69,243)(65,247,70,242)(71,230,76,225)(72,229,77,224)(73,228,78,223)(74,227,79,222)(75,226,80,221)(81,216,86,211)(82,215,87,220)(83,214,88,219)(84,213,89,218)(85,212,90,217)(91,279,96,274)(92,278,97,273)(93,277,98,272)(94,276,99,271)(95,275,100,280)(101,281,106,286)(102,290,107,285)(103,289,108,284)(104,288,109,283)(105,287,110,282)(111,270,116,265)(112,269,117,264)(113,268,118,263)(114,267,119,262)(115,266,120,261)(121,256,126,251)(122,255,127,260)(123,254,128,259)(124,253,129,258)(125,252,130,257)(131,319,136,314)(132,318,137,313)(133,317,138,312)(134,316,139,311)(135,315,140,320)(151,310,156,305)(152,309,157,304)(153,308,158,303)(154,307,159,302)(155,306,160,301)(161,293,166,298)(162,292,167,297)(163,291,168,296)(164,300,169,295)(165,299,170,294), (1,145,18,159,39,170,43,134)(2,146,19,160,40,161,44,135)(3,147,20,151,31,162,45,136)(4,148,11,152,32,163,46,137)(5,149,12,153,33,164,47,138)(6,150,13,154,34,165,48,139)(7,141,14,155,35,166,49,140)(8,142,15,156,36,167,50,131)(9,143,16,157,37,168,41,132)(10,144,17,158,38,169,42,133)(21,207,307,182,299,174,311,196)(22,208,308,183,300,175,312,197)(23,209,309,184,291,176,313,198)(24,210,310,185,292,177,314,199)(25,201,301,186,293,178,315,200)(26,202,302,187,294,179,316,191)(27,203,303,188,295,180,317,192)(28,204,304,189,296,171,318,193)(29,205,305,190,297,172,319,194)(30,206,306,181,298,173,320,195)(51,107,90,96,76,125,62,111)(52,108,81,97,77,126,63,112)(53,109,82,98,78,127,64,113)(54,110,83,99,79,128,65,114)(55,101,84,100,80,129,66,115)(56,102,85,91,71,130,67,116)(57,103,86,92,72,121,68,117)(58,104,87,93,73,122,69,118)(59,105,88,94,74,123,70,119)(60,106,89,95,75,124,61,120)(211,273,229,251,244,269,233,284)(212,274,230,252,245,270,234,285)(213,275,221,253,246,261,235,286)(214,276,222,254,247,262,236,287)(215,277,223,255,248,263,237,288)(216,278,224,256,249,264,238,289)(217,279,225,257,250,265,239,290)(218,280,226,258,241,266,240,281)(219,271,227,259,242,267,231,282)(220,272,228,260,243,268,232,283), (1,74,34,54)(2,75,35,55)(3,76,36,56)(4,77,37,57)(5,78,38,58)(6,79,39,59)(7,80,40,60)(8,71,31,51)(9,72,32,52)(10,73,33,53)(11,81,41,68)(12,82,42,69)(13,83,43,70)(14,84,44,61)(15,85,45,62)(16,86,46,63)(17,87,47,64)(18,88,48,65)(19,89,49,66)(20,90,50,67)(21,271,294,262)(22,272,295,263)(23,273,296,264)(24,274,297,265)(25,275,298,266)(26,276,299,267)(27,277,300,268)(28,278,291,269)(29,279,292,270)(30,280,293,261)(91,167,111,147)(92,168,112,148)(93,169,113,149)(94,170,114,150)(95,161,115,141)(96,162,116,142)(97,163,117,143)(98,164,118,144)(99,165,119,145)(100,166,120,146)(101,140,124,160)(102,131,125,151)(103,132,126,152)(104,133,127,153)(105,134,128,154)(106,135,129,155)(107,136,130,156)(108,137,121,157)(109,138,122,158)(110,139,123,159)(171,244,209,216)(172,245,210,217)(173,246,201,218)(174,247,202,219)(175,248,203,220)(176,249,204,211)(177,250,205,212)(178,241,206,213)(179,242,207,214)(180,243,208,215)(181,235,200,226)(182,236,191,227)(183,237,192,228)(184,238,193,229)(185,239,194,230)(186,240,195,221)(187,231,196,222)(188,232,197,223)(189,233,198,224)(190,234,199,225)(251,304,289,313)(252,305,290,314)(253,306,281,315)(254,307,282,316)(255,308,283,317)(256,309,284,318)(257,310,285,319)(258,301,286,320)(259,302,287,311)(260,303,288,312) );
G=PermutationGroup([[(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)], [(1,196,6,191),(2,195,7,200),(3,194,8,199),(4,193,9,198),(5,192,10,197),(11,204,16,209),(12,203,17,208),(13,202,18,207),(14,201,19,206),(15,210,20,205),(21,145,26,150),(22,144,27,149),(23,143,28,148),(24,142,29,147),(25,141,30,146),(31,190,36,185),(32,189,37,184),(33,188,38,183),(34,187,39,182),(35,186,40,181),(41,176,46,171),(42,175,47,180),(43,174,48,179),(44,173,49,178),(45,172,50,177),(51,239,56,234),(52,238,57,233),(53,237,58,232),(54,236,59,231),(55,235,60,240),(61,241,66,246),(62,250,67,245),(63,249,68,244),(64,248,69,243),(65,247,70,242),(71,230,76,225),(72,229,77,224),(73,228,78,223),(74,227,79,222),(75,226,80,221),(81,216,86,211),(82,215,87,220),(83,214,88,219),(84,213,89,218),(85,212,90,217),(91,279,96,274),(92,278,97,273),(93,277,98,272),(94,276,99,271),(95,275,100,280),(101,281,106,286),(102,290,107,285),(103,289,108,284),(104,288,109,283),(105,287,110,282),(111,270,116,265),(112,269,117,264),(113,268,118,263),(114,267,119,262),(115,266,120,261),(121,256,126,251),(122,255,127,260),(123,254,128,259),(124,253,129,258),(125,252,130,257),(131,319,136,314),(132,318,137,313),(133,317,138,312),(134,316,139,311),(135,315,140,320),(151,310,156,305),(152,309,157,304),(153,308,158,303),(154,307,159,302),(155,306,160,301),(161,293,166,298),(162,292,167,297),(163,291,168,296),(164,300,169,295),(165,299,170,294)], [(1,145,18,159,39,170,43,134),(2,146,19,160,40,161,44,135),(3,147,20,151,31,162,45,136),(4,148,11,152,32,163,46,137),(5,149,12,153,33,164,47,138),(6,150,13,154,34,165,48,139),(7,141,14,155,35,166,49,140),(8,142,15,156,36,167,50,131),(9,143,16,157,37,168,41,132),(10,144,17,158,38,169,42,133),(21,207,307,182,299,174,311,196),(22,208,308,183,300,175,312,197),(23,209,309,184,291,176,313,198),(24,210,310,185,292,177,314,199),(25,201,301,186,293,178,315,200),(26,202,302,187,294,179,316,191),(27,203,303,188,295,180,317,192),(28,204,304,189,296,171,318,193),(29,205,305,190,297,172,319,194),(30,206,306,181,298,173,320,195),(51,107,90,96,76,125,62,111),(52,108,81,97,77,126,63,112),(53,109,82,98,78,127,64,113),(54,110,83,99,79,128,65,114),(55,101,84,100,80,129,66,115),(56,102,85,91,71,130,67,116),(57,103,86,92,72,121,68,117),(58,104,87,93,73,122,69,118),(59,105,88,94,74,123,70,119),(60,106,89,95,75,124,61,120),(211,273,229,251,244,269,233,284),(212,274,230,252,245,270,234,285),(213,275,221,253,246,261,235,286),(214,276,222,254,247,262,236,287),(215,277,223,255,248,263,237,288),(216,278,224,256,249,264,238,289),(217,279,225,257,250,265,239,290),(218,280,226,258,241,266,240,281),(219,271,227,259,242,267,231,282),(220,272,228,260,243,268,232,283)], [(1,74,34,54),(2,75,35,55),(3,76,36,56),(4,77,37,57),(5,78,38,58),(6,79,39,59),(7,80,40,60),(8,71,31,51),(9,72,32,52),(10,73,33,53),(11,81,41,68),(12,82,42,69),(13,83,43,70),(14,84,44,61),(15,85,45,62),(16,86,46,63),(17,87,47,64),(18,88,48,65),(19,89,49,66),(20,90,50,67),(21,271,294,262),(22,272,295,263),(23,273,296,264),(24,274,297,265),(25,275,298,266),(26,276,299,267),(27,277,300,268),(28,278,291,269),(29,279,292,270),(30,280,293,261),(91,167,111,147),(92,168,112,148),(93,169,113,149),(94,170,114,150),(95,161,115,141),(96,162,116,142),(97,163,117,143),(98,164,118,144),(99,165,119,145),(100,166,120,146),(101,140,124,160),(102,131,125,151),(103,132,126,152),(104,133,127,153),(105,134,128,154),(106,135,129,155),(107,136,130,156),(108,137,121,157),(109,138,122,158),(110,139,123,159),(171,244,209,216),(172,245,210,217),(173,246,201,218),(174,247,202,219),(175,248,203,220),(176,249,204,211),(177,250,205,212),(178,241,206,213),(179,242,207,214),(180,243,208,215),(181,235,200,226),(182,236,191,227),(183,237,192,228),(184,238,193,229),(185,239,194,230),(186,240,195,221),(187,231,196,222),(188,232,197,223),(189,233,198,224),(190,234,199,225),(251,304,289,313),(252,305,290,314),(253,306,281,315),(254,307,282,316),(255,308,283,317),(256,309,284,318),(257,310,285,319),(258,301,286,320),(259,302,287,311),(260,303,288,312)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | + | + | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | Q16 | C4○D4 | D10 | D10 | D10 | Dic10 | C8⋊C22 | D4⋊2D5 | D4×D5 | D40⋊C2 | D5×Q16 |
| kernel | Dic5.9Q16 | C10.D8 | C20.8Q8 | C40⋊5C4 | Q8⋊Dic5 | C5×Q8⋊C4 | C20⋊Q8 | Q8×Dic5 | C2×Dic5 | C5×Q8 | Q8⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5.9Q16 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 2 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 |
| 0 | 0 | 25 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 6 |
| 0 | 0 | 0 | 0 | 6 | 39 |
| 17 | 17 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 23 |
| 0 | 0 | 0 | 0 | 23 | 28 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 40 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,2,0,0,0,0,0,10,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,25,0,0,0,0,21,38,0,0,0,0,0,0,2,6,0,0,0,0,6,39],[17,12,0,0,0,0,17,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,13,23,0,0,0,0,23,28],[12,40,0,0,0,0,22,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;
Dic5.9Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_5._9Q_{16} % in TeX
G:=Group("Dic5.9Q16"); // GroupNames label
G:=SmallGroup(320,421);
// by ID
G=gap.SmallGroup(320,421);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,254,219,226,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=a^5*c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations