metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊2Q16, Dic10.10D4, C4.90(D4×D5), C2.7(D5×Q16), (C2×C8).13D10, C5⋊1(C4⋊2Q16), C4⋊C4.145D10, C4.5(C4○D20), C20.114(C2×D4), (C2×Q8).10D10, Q8⋊C4.2D5, C10.14(C2×Q16), C20.14(C4○D4), (C2×C40).13C22, (C2×Dic20).3C2, C10.Q16.2C2, C22.187(D4×D5), C20.8Q8.2C2, C10.21(C4⋊D4), (C2×C20).233C23, Dic5⋊Q8.5C2, (C2×Dic5).204D4, Dic5⋊3Q8.3C2, (Q8×C10).16C22, C2.24(D10⋊D4), C2.14(SD16⋊D5), C10.32(C8.C22), (C4×Dic5).23C22, (C2×Dic10).69C22, (C2×C5⋊Q16).3C2, (C2×C10).246(C2×D4), (C5×C4⋊C4).34C22, (C5×Q8⋊C4).2C2, (C2×C5⋊2C8).28C22, (C2×C4).340(C22×D5), SmallGroup(320,420)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=cac-1=a-1, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 390 in 108 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, Q16, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C5⋊2C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4⋊2Q16, Dic20, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.Q16, C20.8Q8, C5×Q8⋊C4, Dic5⋊3Q8, C2×Dic20, C2×C5⋊Q16, Dic5⋊Q8, Dic5⋊Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C8.C22, C22×D5, C4⋊2Q16, C4○D20, D4×D5, D10⋊D4, SD16⋊D5, D5×Q16, Dic5⋊Q16
►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)
(1 182 6 187)(2 181 7 186)(3 190 8 185)(4 189 9 184)(5 188 10 183)(11 155 16 160)(12 154 17 159)(13 153 18 158)(14 152 19 157)(15 151 20 156)(21 172 26 177)(22 171 27 176)(23 180 28 175)(24 179 29 174)(25 178 30 173)(31 201 36 206)(32 210 37 205)(33 209 38 204)(34 208 39 203)(35 207 40 202)(41 192 46 197)(42 191 47 196)(43 200 48 195)(44 199 49 194)(45 198 50 193)(51 221 56 226)(52 230 57 225)(53 229 58 224)(54 228 59 223)(55 227 60 222)(61 212 66 217)(62 211 67 216)(63 220 68 215)(64 219 69 214)(65 218 70 213)(71 241 76 246)(72 250 77 245)(73 249 78 244)(74 248 79 243)(75 247 80 242)(81 232 86 237)(82 231 87 236)(83 240 88 235)(84 239 89 234)(85 238 90 233)(91 261 96 266)(92 270 97 265)(93 269 98 264)(94 268 99 263)(95 267 100 262)(101 252 106 257)(102 251 107 256)(103 260 108 255)(104 259 109 254)(105 258 110 253)(111 281 116 286)(112 290 117 285)(113 289 118 284)(114 288 119 283)(115 287 120 282)(121 272 126 277)(122 271 127 276)(123 280 128 275)(124 279 129 274)(125 278 130 273)(131 301 136 306)(132 310 137 305)(133 309 138 304)(134 308 139 303)(135 307 140 302)(141 292 146 297)(142 291 147 296)(143 300 148 295)(144 299 149 294)(145 298 150 293)(161 319 166 314)(162 318 167 313)(163 317 168 312)(164 316 169 311)(165 315 170 320)
(1 11 35 300 28 315 43 307)(2 20 36 299 29 314 44 306)(3 19 37 298 30 313 45 305)(4 18 38 297 21 312 46 304)(5 17 39 296 22 311 47 303)(6 16 40 295 23 320 48 302)(7 15 31 294 24 319 49 301)(8 14 32 293 25 318 50 310)(9 13 33 292 26 317 41 309)(10 12 34 291 27 316 42 308)(51 281 84 266 69 274 76 259)(52 290 85 265 70 273 77 258)(53 289 86 264 61 272 78 257)(54 288 87 263 62 271 79 256)(55 287 88 262 63 280 80 255)(56 286 89 261 64 279 71 254)(57 285 90 270 65 278 72 253)(58 284 81 269 66 277 73 252)(59 283 82 268 67 276 74 251)(60 282 83 267 68 275 75 260)(91 219 124 241 109 226 116 234)(92 218 125 250 110 225 117 233)(93 217 126 249 101 224 118 232)(94 216 127 248 102 223 119 231)(95 215 128 247 103 222 120 240)(96 214 129 246 104 221 111 239)(97 213 130 245 105 230 112 238)(98 212 121 244 106 229 113 237)(99 211 122 243 107 228 114 236)(100 220 123 242 108 227 115 235)(131 186 156 201 149 179 161 194)(132 185 157 210 150 178 162 193)(133 184 158 209 141 177 163 192)(134 183 159 208 142 176 164 191)(135 182 160 207 143 175 165 200)(136 181 151 206 144 174 166 199)(137 190 152 205 145 173 167 198)(138 189 153 204 146 172 168 197)(139 188 154 203 147 171 169 196)(140 187 155 202 148 180 170 195)
(1 68 28 60)(2 69 29 51)(3 70 30 52)(4 61 21 53)(5 62 22 54)(6 63 23 55)(7 64 24 56)(8 65 25 57)(9 66 26 58)(10 67 27 59)(11 267 315 260)(12 268 316 251)(13 269 317 252)(14 270 318 253)(15 261 319 254)(16 262 320 255)(17 263 311 256)(18 264 312 257)(19 265 313 258)(20 266 314 259)(31 89 49 71)(32 90 50 72)(33 81 41 73)(34 82 42 74)(35 83 43 75)(36 84 44 76)(37 85 45 77)(38 86 46 78)(39 87 47 79)(40 88 48 80)(91 166 109 151)(92 167 110 152)(93 168 101 153)(94 169 102 154)(95 170 103 155)(96 161 104 156)(97 162 105 157)(98 163 106 158)(99 164 107 159)(100 165 108 160)(111 131 129 149)(112 132 130 150)(113 133 121 141)(114 134 122 142)(115 135 123 143)(116 136 124 144)(117 137 125 145)(118 138 126 146)(119 139 127 147)(120 140 128 148)(171 223 188 216)(172 224 189 217)(173 225 190 218)(174 226 181 219)(175 227 182 220)(176 228 183 211)(177 229 184 212)(178 230 185 213)(179 221 186 214)(180 222 187 215)(191 243 208 236)(192 244 209 237)(193 245 210 238)(194 246 201 239)(195 247 202 240)(196 248 203 231)(197 249 204 232)(198 250 205 233)(199 241 206 234)(200 242 207 235)(271 296 288 303)(272 297 289 304)(273 298 290 305)(274 299 281 306)(275 300 282 307)(276 291 283 308)(277 292 284 309)(278 293 285 310)(279 294 286 301)(280 295 287 302)
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), (1,182,6,187)(2,181,7,186)(3,190,8,185)(4,189,9,184)(5,188,10,183)(11,155,16,160)(12,154,17,159)(13,153,18,158)(14,152,19,157)(15,151,20,156)(21,172,26,177)(22,171,27,176)(23,180,28,175)(24,179,29,174)(25,178,30,173)(31,201,36,206)(32,210,37,205)(33,209,38,204)(34,208,39,203)(35,207,40,202)(41,192,46,197)(42,191,47,196)(43,200,48,195)(44,199,49,194)(45,198,50,193)(51,221,56,226)(52,230,57,225)(53,229,58,224)(54,228,59,223)(55,227,60,222)(61,212,66,217)(62,211,67,216)(63,220,68,215)(64,219,69,214)(65,218,70,213)(71,241,76,246)(72,250,77,245)(73,249,78,244)(74,248,79,243)(75,247,80,242)(81,232,86,237)(82,231,87,236)(83,240,88,235)(84,239,89,234)(85,238,90,233)(91,261,96,266)(92,270,97,265)(93,269,98,264)(94,268,99,263)(95,267,100,262)(101,252,106,257)(102,251,107,256)(103,260,108,255)(104,259,109,254)(105,258,110,253)(111,281,116,286)(112,290,117,285)(113,289,118,284)(114,288,119,283)(115,287,120,282)(121,272,126,277)(122,271,127,276)(123,280,128,275)(124,279,129,274)(125,278,130,273)(131,301,136,306)(132,310,137,305)(133,309,138,304)(134,308,139,303)(135,307,140,302)(141,292,146,297)(142,291,147,296)(143,300,148,295)(144,299,149,294)(145,298,150,293)(161,319,166,314)(162,318,167,313)(163,317,168,312)(164,316,169,311)(165,315,170,320), (1,11,35,300,28,315,43,307)(2,20,36,299,29,314,44,306)(3,19,37,298,30,313,45,305)(4,18,38,297,21,312,46,304)(5,17,39,296,22,311,47,303)(6,16,40,295,23,320,48,302)(7,15,31,294,24,319,49,301)(8,14,32,293,25,318,50,310)(9,13,33,292,26,317,41,309)(10,12,34,291,27,316,42,308)(51,281,84,266,69,274,76,259)(52,290,85,265,70,273,77,258)(53,289,86,264,61,272,78,257)(54,288,87,263,62,271,79,256)(55,287,88,262,63,280,80,255)(56,286,89,261,64,279,71,254)(57,285,90,270,65,278,72,253)(58,284,81,269,66,277,73,252)(59,283,82,268,67,276,74,251)(60,282,83,267,68,275,75,260)(91,219,124,241,109,226,116,234)(92,218,125,250,110,225,117,233)(93,217,126,249,101,224,118,232)(94,216,127,248,102,223,119,231)(95,215,128,247,103,222,120,240)(96,214,129,246,104,221,111,239)(97,213,130,245,105,230,112,238)(98,212,121,244,106,229,113,237)(99,211,122,243,107,228,114,236)(100,220,123,242,108,227,115,235)(131,186,156,201,149,179,161,194)(132,185,157,210,150,178,162,193)(133,184,158,209,141,177,163,192)(134,183,159,208,142,176,164,191)(135,182,160,207,143,175,165,200)(136,181,151,206,144,174,166,199)(137,190,152,205,145,173,167,198)(138,189,153,204,146,172,168,197)(139,188,154,203,147,171,169,196)(140,187,155,202,148,180,170,195), (1,68,28,60)(2,69,29,51)(3,70,30,52)(4,61,21,53)(5,62,22,54)(6,63,23,55)(7,64,24,56)(8,65,25,57)(9,66,26,58)(10,67,27,59)(11,267,315,260)(12,268,316,251)(13,269,317,252)(14,270,318,253)(15,261,319,254)(16,262,320,255)(17,263,311,256)(18,264,312,257)(19,265,313,258)(20,266,314,259)(31,89,49,71)(32,90,50,72)(33,81,41,73)(34,82,42,74)(35,83,43,75)(36,84,44,76)(37,85,45,77)(38,86,46,78)(39,87,47,79)(40,88,48,80)(91,166,109,151)(92,167,110,152)(93,168,101,153)(94,169,102,154)(95,170,103,155)(96,161,104,156)(97,162,105,157)(98,163,106,158)(99,164,107,159)(100,165,108,160)(111,131,129,149)(112,132,130,150)(113,133,121,141)(114,134,122,142)(115,135,123,143)(116,136,124,144)(117,137,125,145)(118,138,126,146)(119,139,127,147)(120,140,128,148)(171,223,188,216)(172,224,189,217)(173,225,190,218)(174,226,181,219)(175,227,182,220)(176,228,183,211)(177,229,184,212)(178,230,185,213)(179,221,186,214)(180,222,187,215)(191,243,208,236)(192,244,209,237)(193,245,210,238)(194,246,201,239)(195,247,202,240)(196,248,203,231)(197,249,204,232)(198,250,205,233)(199,241,206,234)(200,242,207,235)(271,296,288,303)(272,297,289,304)(273,298,290,305)(274,299,281,306)(275,300,282,307)(276,291,283,308)(277,292,284,309)(278,293,285,310)(279,294,286,301)(280,295,287,302)>;
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), (1,182,6,187)(2,181,7,186)(3,190,8,185)(4,189,9,184)(5,188,10,183)(11,155,16,160)(12,154,17,159)(13,153,18,158)(14,152,19,157)(15,151,20,156)(21,172,26,177)(22,171,27,176)(23,180,28,175)(24,179,29,174)(25,178,30,173)(31,201,36,206)(32,210,37,205)(33,209,38,204)(34,208,39,203)(35,207,40,202)(41,192,46,197)(42,191,47,196)(43,200,48,195)(44,199,49,194)(45,198,50,193)(51,221,56,226)(52,230,57,225)(53,229,58,224)(54,228,59,223)(55,227,60,222)(61,212,66,217)(62,211,67,216)(63,220,68,215)(64,219,69,214)(65,218,70,213)(71,241,76,246)(72,250,77,245)(73,249,78,244)(74,248,79,243)(75,247,80,242)(81,232,86,237)(82,231,87,236)(83,240,88,235)(84,239,89,234)(85,238,90,233)(91,261,96,266)(92,270,97,265)(93,269,98,264)(94,268,99,263)(95,267,100,262)(101,252,106,257)(102,251,107,256)(103,260,108,255)(104,259,109,254)(105,258,110,253)(111,281,116,286)(112,290,117,285)(113,289,118,284)(114,288,119,283)(115,287,120,282)(121,272,126,277)(122,271,127,276)(123,280,128,275)(124,279,129,274)(125,278,130,273)(131,301,136,306)(132,310,137,305)(133,309,138,304)(134,308,139,303)(135,307,140,302)(141,292,146,297)(142,291,147,296)(143,300,148,295)(144,299,149,294)(145,298,150,293)(161,319,166,314)(162,318,167,313)(163,317,168,312)(164,316,169,311)(165,315,170,320), (1,11,35,300,28,315,43,307)(2,20,36,299,29,314,44,306)(3,19,37,298,30,313,45,305)(4,18,38,297,21,312,46,304)(5,17,39,296,22,311,47,303)(6,16,40,295,23,320,48,302)(7,15,31,294,24,319,49,301)(8,14,32,293,25,318,50,310)(9,13,33,292,26,317,41,309)(10,12,34,291,27,316,42,308)(51,281,84,266,69,274,76,259)(52,290,85,265,70,273,77,258)(53,289,86,264,61,272,78,257)(54,288,87,263,62,271,79,256)(55,287,88,262,63,280,80,255)(56,286,89,261,64,279,71,254)(57,285,90,270,65,278,72,253)(58,284,81,269,66,277,73,252)(59,283,82,268,67,276,74,251)(60,282,83,267,68,275,75,260)(91,219,124,241,109,226,116,234)(92,218,125,250,110,225,117,233)(93,217,126,249,101,224,118,232)(94,216,127,248,102,223,119,231)(95,215,128,247,103,222,120,240)(96,214,129,246,104,221,111,239)(97,213,130,245,105,230,112,238)(98,212,121,244,106,229,113,237)(99,211,122,243,107,228,114,236)(100,220,123,242,108,227,115,235)(131,186,156,201,149,179,161,194)(132,185,157,210,150,178,162,193)(133,184,158,209,141,177,163,192)(134,183,159,208,142,176,164,191)(135,182,160,207,143,175,165,200)(136,181,151,206,144,174,166,199)(137,190,152,205,145,173,167,198)(138,189,153,204,146,172,168,197)(139,188,154,203,147,171,169,196)(140,187,155,202,148,180,170,195), (1,68,28,60)(2,69,29,51)(3,70,30,52)(4,61,21,53)(5,62,22,54)(6,63,23,55)(7,64,24,56)(8,65,25,57)(9,66,26,58)(10,67,27,59)(11,267,315,260)(12,268,316,251)(13,269,317,252)(14,270,318,253)(15,261,319,254)(16,262,320,255)(17,263,311,256)(18,264,312,257)(19,265,313,258)(20,266,314,259)(31,89,49,71)(32,90,50,72)(33,81,41,73)(34,82,42,74)(35,83,43,75)(36,84,44,76)(37,85,45,77)(38,86,46,78)(39,87,47,79)(40,88,48,80)(91,166,109,151)(92,167,110,152)(93,168,101,153)(94,169,102,154)(95,170,103,155)(96,161,104,156)(97,162,105,157)(98,163,106,158)(99,164,107,159)(100,165,108,160)(111,131,129,149)(112,132,130,150)(113,133,121,141)(114,134,122,142)(115,135,123,143)(116,136,124,144)(117,137,125,145)(118,138,126,146)(119,139,127,147)(120,140,128,148)(171,223,188,216)(172,224,189,217)(173,225,190,218)(174,226,181,219)(175,227,182,220)(176,228,183,211)(177,229,184,212)(178,230,185,213)(179,221,186,214)(180,222,187,215)(191,243,208,236)(192,244,209,237)(193,245,210,238)(194,246,201,239)(195,247,202,240)(196,248,203,231)(197,249,204,232)(198,250,205,233)(199,241,206,234)(200,242,207,235)(271,296,288,303)(272,297,289,304)(273,298,290,305)(274,299,281,306)(275,300,282,307)(276,291,283,308)(277,292,284,309)(278,293,285,310)(279,294,286,301)(280,295,287,302) );
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)], [(1,182,6,187),(2,181,7,186),(3,190,8,185),(4,189,9,184),(5,188,10,183),(11,155,16,160),(12,154,17,159),(13,153,18,158),(14,152,19,157),(15,151,20,156),(21,172,26,177),(22,171,27,176),(23,180,28,175),(24,179,29,174),(25,178,30,173),(31,201,36,206),(32,210,37,205),(33,209,38,204),(34,208,39,203),(35,207,40,202),(41,192,46,197),(42,191,47,196),(43,200,48,195),(44,199,49,194),(45,198,50,193),(51,221,56,226),(52,230,57,225),(53,229,58,224),(54,228,59,223),(55,227,60,222),(61,212,66,217),(62,211,67,216),(63,220,68,215),(64,219,69,214),(65,218,70,213),(71,241,76,246),(72,250,77,245),(73,249,78,244),(74,248,79,243),(75,247,80,242),(81,232,86,237),(82,231,87,236),(83,240,88,235),(84,239,89,234),(85,238,90,233),(91,261,96,266),(92,270,97,265),(93,269,98,264),(94,268,99,263),(95,267,100,262),(101,252,106,257),(102,251,107,256),(103,260,108,255),(104,259,109,254),(105,258,110,253),(111,281,116,286),(112,290,117,285),(113,289,118,284),(114,288,119,283),(115,287,120,282),(121,272,126,277),(122,271,127,276),(123,280,128,275),(124,279,129,274),(125,278,130,273),(131,301,136,306),(132,310,137,305),(133,309,138,304),(134,308,139,303),(135,307,140,302),(141,292,146,297),(142,291,147,296),(143,300,148,295),(144,299,149,294),(145,298,150,293),(161,319,166,314),(162,318,167,313),(163,317,168,312),(164,316,169,311),(165,315,170,320)], [(1,11,35,300,28,315,43,307),(2,20,36,299,29,314,44,306),(3,19,37,298,30,313,45,305),(4,18,38,297,21,312,46,304),(5,17,39,296,22,311,47,303),(6,16,40,295,23,320,48,302),(7,15,31,294,24,319,49,301),(8,14,32,293,25,318,50,310),(9,13,33,292,26,317,41,309),(10,12,34,291,27,316,42,308),(51,281,84,266,69,274,76,259),(52,290,85,265,70,273,77,258),(53,289,86,264,61,272,78,257),(54,288,87,263,62,271,79,256),(55,287,88,262,63,280,80,255),(56,286,89,261,64,279,71,254),(57,285,90,270,65,278,72,253),(58,284,81,269,66,277,73,252),(59,283,82,268,67,276,74,251),(60,282,83,267,68,275,75,260),(91,219,124,241,109,226,116,234),(92,218,125,250,110,225,117,233),(93,217,126,249,101,224,118,232),(94,216,127,248,102,223,119,231),(95,215,128,247,103,222,120,240),(96,214,129,246,104,221,111,239),(97,213,130,245,105,230,112,238),(98,212,121,244,106,229,113,237),(99,211,122,243,107,228,114,236),(100,220,123,242,108,227,115,235),(131,186,156,201,149,179,161,194),(132,185,157,210,150,178,162,193),(133,184,158,209,141,177,163,192),(134,183,159,208,142,176,164,191),(135,182,160,207,143,175,165,200),(136,181,151,206,144,174,166,199),(137,190,152,205,145,173,167,198),(138,189,153,204,146,172,168,197),(139,188,154,203,147,171,169,196),(140,187,155,202,148,180,170,195)], [(1,68,28,60),(2,69,29,51),(3,70,30,52),(4,61,21,53),(5,62,22,54),(6,63,23,55),(7,64,24,56),(8,65,25,57),(9,66,26,58),(10,67,27,59),(11,267,315,260),(12,268,316,251),(13,269,317,252),(14,270,318,253),(15,261,319,254),(16,262,320,255),(17,263,311,256),(18,264,312,257),(19,265,313,258),(20,266,314,259),(31,89,49,71),(32,90,50,72),(33,81,41,73),(34,82,42,74),(35,83,43,75),(36,84,44,76),(37,85,45,77),(38,86,46,78),(39,87,47,79),(40,88,48,80),(91,166,109,151),(92,167,110,152),(93,168,101,153),(94,169,102,154),(95,170,103,155),(96,161,104,156),(97,162,105,157),(98,163,106,158),(99,164,107,159),(100,165,108,160),(111,131,129,149),(112,132,130,150),(113,133,121,141),(114,134,122,142),(115,135,123,143),(116,136,124,144),(117,137,125,145),(118,138,126,146),(119,139,127,147),(120,140,128,148),(171,223,188,216),(172,224,189,217),(173,225,190,218),(174,226,181,219),(175,227,182,220),(176,228,183,211),(177,229,184,212),(178,230,185,213),(179,221,186,214),(180,222,187,215),(191,243,208,236),(192,244,209,237),(193,245,210,238),(194,246,201,239),(195,247,202,240),(196,248,203,231),(197,249,204,232),(198,250,205,233),(199,241,206,234),(200,242,207,235),(271,296,288,303),(272,297,289,304),(273,298,290,305),(274,299,281,306),(275,300,282,307),(276,291,283,308),(277,292,284,309),(278,293,285,310),(279,294,286,301),(280,295,287,302)]])
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 | D4 | D5 | Q16 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8.C22 | D4×D5 | D4×D5 | SD16⋊D5 | D5×Q16 |
| kernel | Dic5⋊Q16 | C10.Q16 | C20.8Q8 | C5×Q8⋊C4 | Dic5⋊3Q8 | C2×Dic20 | C2×C5⋊Q16 | Dic5⋊Q8 | Dic10 | C2×Dic5 | Q8⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | 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⋊Q16 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 16 | 0 | 0 |
| 13 | 23 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 28 | 14 | 0 | 0 |
| 23 | 13 | 0 | 0 |
| 0 | 0 | 0 | 26 |
| 0 | 0 | 11 | 17 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 23 | 16 |
| 0 | 0 | 13 | 18 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[18,13,0,0,16,23,0,0,0,0,40,0,0,0,0,40],[28,23,0,0,14,13,0,0,0,0,0,11,0,0,26,17],[17,1,0,0,40,24,0,0,0,0,23,13,0,0,16,18] >;
Dic5⋊Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes Q_{16} % in TeX
G:=Group("Dic5:Q16"); // GroupNames label
G:=SmallGroup(320,420);
// by ID
G=gap.SmallGroup(320,420);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,422,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=c^4,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations