metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.3Q16, Dic5.9SD16, C20⋊Q8.3C2, C2.6(D5×Q16), C4⋊C4.16D10, (C2×Q8).9D10, (C2×C8).206D10, Q8⋊C4.1D5, C10.13(C2×Q16), C2.15(D5×SD16), C5⋊2(C4.SD16), C20.13(C4○D4), C4.28(C4○D20), (C8×Dic5).16C2, C10.28(C2×SD16), C10.Q16.1C2, Q8⋊Dic5.2C2, C22.186(D4×D5), C4.54(D4⋊2D5), (C2×C20).232C23, (C2×C40).192C22, Dic5⋊Q8.4C2, (C2×Dic5).136D4, C20.44D4.8C2, C4⋊Dic5.82C22, (Q8×C10).15C22, C10.26(C4.4D4), (C4×Dic5).257C22, (C2×Dic10).68C22, C2.16(Dic5.5D4), (C2×C10).245(C2×D4), (C5×C4⋊C4).33C22, (C5×Q8⋊C4).9C2, (C2×C4).339(C22×D5), (C2×C5⋊2C8).224C22, SmallGroup(320,419)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.3Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=a5c-1 >
Subgroups: 366 in 98 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×C8, Q8⋊C4, Q8⋊C4, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4.SD16, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, Q8×C10, C10.Q16, C8×Dic5, C20.44D4, Q8⋊Dic5, C5×Q8⋊C4, C20⋊Q8, Dic5⋊Q8, Dic5.3Q16
Quotients: C1, C2, C22, D4, C23, D5, SD16, Q16, C2×D4, C4○D4, D10, C4.4D4, C2×SD16, C2×Q16, C22×D5, C4.SD16, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, D5×SD16, D5×Q16, Dic5.3Q16
►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)
(1 207 6 202)(2 206 7 201)(3 205 8 210)(4 204 9 209)(5 203 10 208)(11 143 16 148)(12 142 17 147)(13 141 18 146)(14 150 19 145)(15 149 20 144)(21 199 26 194)(22 198 27 193)(23 197 28 192)(24 196 29 191)(25 195 30 200)(31 190 36 185)(32 189 37 184)(33 188 38 183)(34 187 39 182)(35 186 40 181)(41 173 46 178)(42 172 47 177)(43 171 48 176)(44 180 49 175)(45 179 50 174)(51 250 56 245)(52 249 57 244)(53 248 58 243)(54 247 59 242)(55 246 60 241)(61 233 66 238)(62 232 67 237)(63 231 68 236)(64 240 69 235)(65 239 70 234)(71 214 76 219)(72 213 77 218)(73 212 78 217)(74 211 79 216)(75 220 80 215)(81 229 86 224)(82 228 87 223)(83 227 88 222)(84 226 89 221)(85 225 90 230)(91 290 96 285)(92 289 97 284)(93 288 98 283)(94 287 99 282)(95 286 100 281)(101 273 106 278)(102 272 107 277)(103 271 108 276)(104 280 109 275)(105 279 110 274)(111 254 116 259)(112 253 117 258)(113 252 118 257)(114 251 119 256)(115 260 120 255)(121 269 126 264)(122 268 127 263)(123 267 128 262)(124 266 129 261)(125 265 130 270)(131 314 136 319)(132 313 137 318)(133 312 138 317)(134 311 139 316)(135 320 140 315)(151 310 156 305)(152 309 157 304)(153 308 158 303)(154 307 159 302)(155 306 160 301)(161 300 166 295)(162 299 167 294)(163 298 168 293)(164 297 169 292)(165 296 170 291)
(1 165 48 134 29 159 39 143)(2 166 49 135 30 160 40 144)(3 167 50 136 21 151 31 145)(4 168 41 137 22 152 32 146)(5 169 42 138 23 153 33 147)(6 170 43 139 24 154 34 148)(7 161 44 140 25 155 35 149)(8 162 45 131 26 156 36 150)(9 163 46 132 27 157 37 141)(10 164 47 133 28 158 38 142)(11 202 291 171 316 196 307 187)(12 203 292 172 317 197 308 188)(13 204 293 173 318 198 309 189)(14 205 294 174 319 199 310 190)(15 206 295 175 320 200 301 181)(16 207 296 176 311 191 302 182)(17 208 297 177 312 192 303 183)(18 209 298 178 313 193 304 184)(19 210 299 179 314 194 305 185)(20 201 300 180 315 195 306 186)(51 116 85 91 70 125 76 110)(52 117 86 92 61 126 77 101)(53 118 87 93 62 127 78 102)(54 119 88 94 63 128 79 103)(55 120 89 95 64 129 80 104)(56 111 90 96 65 130 71 105)(57 112 81 97 66 121 72 106)(58 113 82 98 67 122 73 107)(59 114 83 99 68 123 74 108)(60 115 84 100 69 124 75 109)(211 276 242 251 227 282 236 267)(212 277 243 252 228 283 237 268)(213 278 244 253 229 284 238 269)(214 279 245 254 230 285 239 270)(215 280 246 255 221 286 240 261)(216 271 247 256 222 287 231 262)(217 272 248 257 223 288 232 263)(218 273 249 258 224 289 233 264)(219 274 250 259 225 290 234 265)(220 275 241 260 226 281 235 266)
(1 54 29 63)(2 55 30 64)(3 56 21 65)(4 57 22 66)(5 58 23 67)(6 59 24 68)(7 60 25 69)(8 51 26 70)(9 52 27 61)(10 53 28 62)(11 251 316 267)(12 252 317 268)(13 253 318 269)(14 254 319 270)(15 255 320 261)(16 256 311 262)(17 257 312 263)(18 258 313 264)(19 259 314 265)(20 260 315 266)(31 90 50 71)(32 81 41 72)(33 82 42 73)(34 83 43 74)(35 84 44 75)(36 85 45 76)(37 86 46 77)(38 87 47 78)(39 88 48 79)(40 89 49 80)(91 167 110 151)(92 168 101 152)(93 169 102 153)(94 170 103 154)(95 161 104 155)(96 162 105 156)(97 163 106 157)(98 164 107 158)(99 165 108 159)(100 166 109 160)(111 131 130 150)(112 132 121 141)(113 133 122 142)(114 134 123 143)(115 135 124 144)(116 136 125 145)(117 137 126 146)(118 138 127 147)(119 139 128 148)(120 140 129 149)(171 216 187 222)(172 217 188 223)(173 218 189 224)(174 219 190 225)(175 220 181 226)(176 211 182 227)(177 212 183 228)(178 213 184 229)(179 214 185 230)(180 215 186 221)(191 236 207 242)(192 237 208 243)(193 238 209 244)(194 239 210 245)(195 240 201 246)(196 231 202 247)(197 232 203 248)(198 233 204 249)(199 234 205 250)(200 235 206 241)(271 302 287 296)(272 303 288 297)(273 304 289 298)(274 305 290 299)(275 306 281 300)(276 307 282 291)(277 308 283 292)(278 309 284 293)(279 310 285 294)(280 301 286 295)
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), (1,207,6,202)(2,206,7,201)(3,205,8,210)(4,204,9,209)(5,203,10,208)(11,143,16,148)(12,142,17,147)(13,141,18,146)(14,150,19,145)(15,149,20,144)(21,199,26,194)(22,198,27,193)(23,197,28,192)(24,196,29,191)(25,195,30,200)(31,190,36,185)(32,189,37,184)(33,188,38,183)(34,187,39,182)(35,186,40,181)(41,173,46,178)(42,172,47,177)(43,171,48,176)(44,180,49,175)(45,179,50,174)(51,250,56,245)(52,249,57,244)(53,248,58,243)(54,247,59,242)(55,246,60,241)(61,233,66,238)(62,232,67,237)(63,231,68,236)(64,240,69,235)(65,239,70,234)(71,214,76,219)(72,213,77,218)(73,212,78,217)(74,211,79,216)(75,220,80,215)(81,229,86,224)(82,228,87,223)(83,227,88,222)(84,226,89,221)(85,225,90,230)(91,290,96,285)(92,289,97,284)(93,288,98,283)(94,287,99,282)(95,286,100,281)(101,273,106,278)(102,272,107,277)(103,271,108,276)(104,280,109,275)(105,279,110,274)(111,254,116,259)(112,253,117,258)(113,252,118,257)(114,251,119,256)(115,260,120,255)(121,269,126,264)(122,268,127,263)(123,267,128,262)(124,266,129,261)(125,265,130,270)(131,314,136,319)(132,313,137,318)(133,312,138,317)(134,311,139,316)(135,320,140,315)(151,310,156,305)(152,309,157,304)(153,308,158,303)(154,307,159,302)(155,306,160,301)(161,300,166,295)(162,299,167,294)(163,298,168,293)(164,297,169,292)(165,296,170,291), (1,165,48,134,29,159,39,143)(2,166,49,135,30,160,40,144)(3,167,50,136,21,151,31,145)(4,168,41,137,22,152,32,146)(5,169,42,138,23,153,33,147)(6,170,43,139,24,154,34,148)(7,161,44,140,25,155,35,149)(8,162,45,131,26,156,36,150)(9,163,46,132,27,157,37,141)(10,164,47,133,28,158,38,142)(11,202,291,171,316,196,307,187)(12,203,292,172,317,197,308,188)(13,204,293,173,318,198,309,189)(14,205,294,174,319,199,310,190)(15,206,295,175,320,200,301,181)(16,207,296,176,311,191,302,182)(17,208,297,177,312,192,303,183)(18,209,298,178,313,193,304,184)(19,210,299,179,314,194,305,185)(20,201,300,180,315,195,306,186)(51,116,85,91,70,125,76,110)(52,117,86,92,61,126,77,101)(53,118,87,93,62,127,78,102)(54,119,88,94,63,128,79,103)(55,120,89,95,64,129,80,104)(56,111,90,96,65,130,71,105)(57,112,81,97,66,121,72,106)(58,113,82,98,67,122,73,107)(59,114,83,99,68,123,74,108)(60,115,84,100,69,124,75,109)(211,276,242,251,227,282,236,267)(212,277,243,252,228,283,237,268)(213,278,244,253,229,284,238,269)(214,279,245,254,230,285,239,270)(215,280,246,255,221,286,240,261)(216,271,247,256,222,287,231,262)(217,272,248,257,223,288,232,263)(218,273,249,258,224,289,233,264)(219,274,250,259,225,290,234,265)(220,275,241,260,226,281,235,266), (1,54,29,63)(2,55,30,64)(3,56,21,65)(4,57,22,66)(5,58,23,67)(6,59,24,68)(7,60,25,69)(8,51,26,70)(9,52,27,61)(10,53,28,62)(11,251,316,267)(12,252,317,268)(13,253,318,269)(14,254,319,270)(15,255,320,261)(16,256,311,262)(17,257,312,263)(18,258,313,264)(19,259,314,265)(20,260,315,266)(31,90,50,71)(32,81,41,72)(33,82,42,73)(34,83,43,74)(35,84,44,75)(36,85,45,76)(37,86,46,77)(38,87,47,78)(39,88,48,79)(40,89,49,80)(91,167,110,151)(92,168,101,152)(93,169,102,153)(94,170,103,154)(95,161,104,155)(96,162,105,156)(97,163,106,157)(98,164,107,158)(99,165,108,159)(100,166,109,160)(111,131,130,150)(112,132,121,141)(113,133,122,142)(114,134,123,143)(115,135,124,144)(116,136,125,145)(117,137,126,146)(118,138,127,147)(119,139,128,148)(120,140,129,149)(171,216,187,222)(172,217,188,223)(173,218,189,224)(174,219,190,225)(175,220,181,226)(176,211,182,227)(177,212,183,228)(178,213,184,229)(179,214,185,230)(180,215,186,221)(191,236,207,242)(192,237,208,243)(193,238,209,244)(194,239,210,245)(195,240,201,246)(196,231,202,247)(197,232,203,248)(198,233,204,249)(199,234,205,250)(200,235,206,241)(271,302,287,296)(272,303,288,297)(273,304,289,298)(274,305,290,299)(275,306,281,300)(276,307,282,291)(277,308,283,292)(278,309,284,293)(279,310,285,294)(280,301,286,295)>;
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), (1,207,6,202)(2,206,7,201)(3,205,8,210)(4,204,9,209)(5,203,10,208)(11,143,16,148)(12,142,17,147)(13,141,18,146)(14,150,19,145)(15,149,20,144)(21,199,26,194)(22,198,27,193)(23,197,28,192)(24,196,29,191)(25,195,30,200)(31,190,36,185)(32,189,37,184)(33,188,38,183)(34,187,39,182)(35,186,40,181)(41,173,46,178)(42,172,47,177)(43,171,48,176)(44,180,49,175)(45,179,50,174)(51,250,56,245)(52,249,57,244)(53,248,58,243)(54,247,59,242)(55,246,60,241)(61,233,66,238)(62,232,67,237)(63,231,68,236)(64,240,69,235)(65,239,70,234)(71,214,76,219)(72,213,77,218)(73,212,78,217)(74,211,79,216)(75,220,80,215)(81,229,86,224)(82,228,87,223)(83,227,88,222)(84,226,89,221)(85,225,90,230)(91,290,96,285)(92,289,97,284)(93,288,98,283)(94,287,99,282)(95,286,100,281)(101,273,106,278)(102,272,107,277)(103,271,108,276)(104,280,109,275)(105,279,110,274)(111,254,116,259)(112,253,117,258)(113,252,118,257)(114,251,119,256)(115,260,120,255)(121,269,126,264)(122,268,127,263)(123,267,128,262)(124,266,129,261)(125,265,130,270)(131,314,136,319)(132,313,137,318)(133,312,138,317)(134,311,139,316)(135,320,140,315)(151,310,156,305)(152,309,157,304)(153,308,158,303)(154,307,159,302)(155,306,160,301)(161,300,166,295)(162,299,167,294)(163,298,168,293)(164,297,169,292)(165,296,170,291), (1,165,48,134,29,159,39,143)(2,166,49,135,30,160,40,144)(3,167,50,136,21,151,31,145)(4,168,41,137,22,152,32,146)(5,169,42,138,23,153,33,147)(6,170,43,139,24,154,34,148)(7,161,44,140,25,155,35,149)(8,162,45,131,26,156,36,150)(9,163,46,132,27,157,37,141)(10,164,47,133,28,158,38,142)(11,202,291,171,316,196,307,187)(12,203,292,172,317,197,308,188)(13,204,293,173,318,198,309,189)(14,205,294,174,319,199,310,190)(15,206,295,175,320,200,301,181)(16,207,296,176,311,191,302,182)(17,208,297,177,312,192,303,183)(18,209,298,178,313,193,304,184)(19,210,299,179,314,194,305,185)(20,201,300,180,315,195,306,186)(51,116,85,91,70,125,76,110)(52,117,86,92,61,126,77,101)(53,118,87,93,62,127,78,102)(54,119,88,94,63,128,79,103)(55,120,89,95,64,129,80,104)(56,111,90,96,65,130,71,105)(57,112,81,97,66,121,72,106)(58,113,82,98,67,122,73,107)(59,114,83,99,68,123,74,108)(60,115,84,100,69,124,75,109)(211,276,242,251,227,282,236,267)(212,277,243,252,228,283,237,268)(213,278,244,253,229,284,238,269)(214,279,245,254,230,285,239,270)(215,280,246,255,221,286,240,261)(216,271,247,256,222,287,231,262)(217,272,248,257,223,288,232,263)(218,273,249,258,224,289,233,264)(219,274,250,259,225,290,234,265)(220,275,241,260,226,281,235,266), (1,54,29,63)(2,55,30,64)(3,56,21,65)(4,57,22,66)(5,58,23,67)(6,59,24,68)(7,60,25,69)(8,51,26,70)(9,52,27,61)(10,53,28,62)(11,251,316,267)(12,252,317,268)(13,253,318,269)(14,254,319,270)(15,255,320,261)(16,256,311,262)(17,257,312,263)(18,258,313,264)(19,259,314,265)(20,260,315,266)(31,90,50,71)(32,81,41,72)(33,82,42,73)(34,83,43,74)(35,84,44,75)(36,85,45,76)(37,86,46,77)(38,87,47,78)(39,88,48,79)(40,89,49,80)(91,167,110,151)(92,168,101,152)(93,169,102,153)(94,170,103,154)(95,161,104,155)(96,162,105,156)(97,163,106,157)(98,164,107,158)(99,165,108,159)(100,166,109,160)(111,131,130,150)(112,132,121,141)(113,133,122,142)(114,134,123,143)(115,135,124,144)(116,136,125,145)(117,137,126,146)(118,138,127,147)(119,139,128,148)(120,140,129,149)(171,216,187,222)(172,217,188,223)(173,218,189,224)(174,219,190,225)(175,220,181,226)(176,211,182,227)(177,212,183,228)(178,213,184,229)(179,214,185,230)(180,215,186,221)(191,236,207,242)(192,237,208,243)(193,238,209,244)(194,239,210,245)(195,240,201,246)(196,231,202,247)(197,232,203,248)(198,233,204,249)(199,234,205,250)(200,235,206,241)(271,302,287,296)(272,303,288,297)(273,304,289,298)(274,305,290,299)(275,306,281,300)(276,307,282,291)(277,308,283,292)(278,309,284,293)(279,310,285,294)(280,301,286,295) );
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)], [(1,207,6,202),(2,206,7,201),(3,205,8,210),(4,204,9,209),(5,203,10,208),(11,143,16,148),(12,142,17,147),(13,141,18,146),(14,150,19,145),(15,149,20,144),(21,199,26,194),(22,198,27,193),(23,197,28,192),(24,196,29,191),(25,195,30,200),(31,190,36,185),(32,189,37,184),(33,188,38,183),(34,187,39,182),(35,186,40,181),(41,173,46,178),(42,172,47,177),(43,171,48,176),(44,180,49,175),(45,179,50,174),(51,250,56,245),(52,249,57,244),(53,248,58,243),(54,247,59,242),(55,246,60,241),(61,233,66,238),(62,232,67,237),(63,231,68,236),(64,240,69,235),(65,239,70,234),(71,214,76,219),(72,213,77,218),(73,212,78,217),(74,211,79,216),(75,220,80,215),(81,229,86,224),(82,228,87,223),(83,227,88,222),(84,226,89,221),(85,225,90,230),(91,290,96,285),(92,289,97,284),(93,288,98,283),(94,287,99,282),(95,286,100,281),(101,273,106,278),(102,272,107,277),(103,271,108,276),(104,280,109,275),(105,279,110,274),(111,254,116,259),(112,253,117,258),(113,252,118,257),(114,251,119,256),(115,260,120,255),(121,269,126,264),(122,268,127,263),(123,267,128,262),(124,266,129,261),(125,265,130,270),(131,314,136,319),(132,313,137,318),(133,312,138,317),(134,311,139,316),(135,320,140,315),(151,310,156,305),(152,309,157,304),(153,308,158,303),(154,307,159,302),(155,306,160,301),(161,300,166,295),(162,299,167,294),(163,298,168,293),(164,297,169,292),(165,296,170,291)], [(1,165,48,134,29,159,39,143),(2,166,49,135,30,160,40,144),(3,167,50,136,21,151,31,145),(4,168,41,137,22,152,32,146),(5,169,42,138,23,153,33,147),(6,170,43,139,24,154,34,148),(7,161,44,140,25,155,35,149),(8,162,45,131,26,156,36,150),(9,163,46,132,27,157,37,141),(10,164,47,133,28,158,38,142),(11,202,291,171,316,196,307,187),(12,203,292,172,317,197,308,188),(13,204,293,173,318,198,309,189),(14,205,294,174,319,199,310,190),(15,206,295,175,320,200,301,181),(16,207,296,176,311,191,302,182),(17,208,297,177,312,192,303,183),(18,209,298,178,313,193,304,184),(19,210,299,179,314,194,305,185),(20,201,300,180,315,195,306,186),(51,116,85,91,70,125,76,110),(52,117,86,92,61,126,77,101),(53,118,87,93,62,127,78,102),(54,119,88,94,63,128,79,103),(55,120,89,95,64,129,80,104),(56,111,90,96,65,130,71,105),(57,112,81,97,66,121,72,106),(58,113,82,98,67,122,73,107),(59,114,83,99,68,123,74,108),(60,115,84,100,69,124,75,109),(211,276,242,251,227,282,236,267),(212,277,243,252,228,283,237,268),(213,278,244,253,229,284,238,269),(214,279,245,254,230,285,239,270),(215,280,246,255,221,286,240,261),(216,271,247,256,222,287,231,262),(217,272,248,257,223,288,232,263),(218,273,249,258,224,289,233,264),(219,274,250,259,225,290,234,265),(220,275,241,260,226,281,235,266)], [(1,54,29,63),(2,55,30,64),(3,56,21,65),(4,57,22,66),(5,58,23,67),(6,59,24,68),(7,60,25,69),(8,51,26,70),(9,52,27,61),(10,53,28,62),(11,251,316,267),(12,252,317,268),(13,253,318,269),(14,254,319,270),(15,255,320,261),(16,256,311,262),(17,257,312,263),(18,258,313,264),(19,259,314,265),(20,260,315,266),(31,90,50,71),(32,81,41,72),(33,82,42,73),(34,83,43,74),(35,84,44,75),(36,85,45,76),(37,86,46,77),(38,87,47,78),(39,88,48,79),(40,89,49,80),(91,167,110,151),(92,168,101,152),(93,169,102,153),(94,170,103,154),(95,161,104,155),(96,162,105,156),(97,163,106,157),(98,164,107,158),(99,165,108,159),(100,166,109,160),(111,131,130,150),(112,132,121,141),(113,133,122,142),(114,134,123,143),(115,135,124,144),(116,136,125,145),(117,137,126,146),(118,138,127,147),(119,139,128,148),(120,140,129,149),(171,216,187,222),(172,217,188,223),(173,218,189,224),(174,219,190,225),(175,220,181,226),(176,211,182,227),(177,212,183,228),(178,213,184,229),(179,214,185,230),(180,215,186,221),(191,236,207,242),(192,237,208,243),(193,238,209,244),(194,239,210,245),(195,240,201,246),(196,231,202,247),(197,232,203,248),(198,233,204,249),(199,234,205,250),(200,235,206,241),(271,302,287,296),(272,303,288,297),(273,304,289,298),(274,305,290,299),(275,306,281,300),(276,307,282,291),(277,308,283,292),(278,309,284,293),(279,310,285,294),(280,301,286,295)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 10 | 10 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | SD16 | Q16 | C4○D4 | D10 | D10 | D10 | C4○D20 | D4⋊2D5 | D4×D5 | D5×SD16 | D5×Q16 |
| kernel | Dic5.3Q16 | C10.Q16 | C8×Dic5 | C20.44D4 | Q8⋊Dic5 | C5×Q8⋊C4 | C20⋊Q8 | Dic5⋊Q8 | C2×Dic5 | Q8⋊C4 | Dic5 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5.3Q16 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 17 | 2 | 0 | 0 | 0 | 0 |
| 19 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 11 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 25 |
| 0 | 0 | 0 | 0 | 36 | 1 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 7 | 30 |
| 35 | 5 | 0 | 0 | 0 | 0 |
| 34 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 5 | 29 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[17,19,0,0,0,0,2,24,0,0,0,0,0,0,14,27,0,0,0,0,11,27,0,0,0,0,0,0,40,36,0,0,0,0,25,1],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,6,30],[35,34,0,0,0,0,5,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,0,12,29] >;
Dic5.3Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_5._3Q_{16} % in TeX
G:=Group("Dic5.3Q16"); // GroupNames label
G:=SmallGroup(320,419);
// by ID
G=gap.SmallGroup(320,419);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,422,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations