metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊3Q16, (C5×Q8).7D4, (C2×C8).36D10, C5⋊5(C4⋊2Q16), (C2×Q16).4D5, C2.17(D5×Q16), C20.184(C2×D4), (C10×Q16).8C2, C10.27(C2×Q16), (Q8×Dic5).7C2, Q8.2(C5⋊D4), (C2×Q8).118D10, C22.275(D4×D5), C20.103(C4○D4), C4.13(D4⋊2D5), (C2×C20).456C23, (C2×C40).250C22, Dic5⋊Q8.8C2, (C2×Dic5).242D4, Q8⋊Dic5.14C2, C20.8Q8.12C2, (Q8×C10).85C22, C10.119(C4⋊D4), C2.26(Q16⋊D5), C20.44D4.12C2, C10.75(C8.C22), C4⋊Dic5.179C22, (C4×Dic5).59C22, C2.28(Dic5⋊D4), (C2×Dic10).135C22, C4.46(C2×C5⋊D4), (C2×C5⋊Q16).9C2, (C2×C10).367(C2×D4), (C2×C4).544(C22×D5), (C2×C5⋊2C8).162C22, SmallGroup(320,809)
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, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 358 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C2×Q16, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C4⋊2Q16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5⋊Q16, C2×C40, C5×Q16, C2×Dic10, Q8×C10, C20.8Q8, C20.44D4, Q8⋊Dic5, C2×C5⋊Q16, Dic5⋊Q8, Q8×Dic5, C10×Q16, Dic5⋊3Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C8.C22, C5⋊D4, C22×D5, C4⋊2Q16, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×Q16, Q16⋊D5, Dic5⋊D4, 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)
(1 179 6 174)(2 178 7 173)(3 177 8 172)(4 176 9 171)(5 175 10 180)(11 170 16 165)(12 169 17 164)(13 168 18 163)(14 167 19 162)(15 166 20 161)(21 189 26 184)(22 188 27 183)(23 187 28 182)(24 186 29 181)(25 185 30 190)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 214 56 219)(52 213 57 218)(53 212 58 217)(54 211 59 216)(55 220 60 215)(61 229 66 224)(62 228 67 223)(63 227 68 222)(64 226 69 221)(65 225 70 230)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 254 96 259)(92 253 97 258)(93 252 98 257)(94 251 99 256)(95 260 100 255)(101 269 106 264)(102 268 107 263)(103 267 108 262)(104 266 109 261)(105 265 110 270)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 294 136 299)(132 293 137 298)(133 292 138 297)(134 291 139 296)(135 300 140 295)(141 309 146 304)(142 308 147 303)(143 307 148 302)(144 306 149 301)(145 305 150 310)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)
(1 156 36 148 28 165 48 136)(2 157 37 149 29 166 49 137)(3 158 38 150 30 167 50 138)(4 159 39 141 21 168 41 139)(5 160 40 142 22 169 42 140)(6 151 31 143 23 170 43 131)(7 152 32 144 24 161 44 132)(8 153 33 145 25 162 45 133)(9 154 34 146 26 163 46 134)(10 155 35 147 27 164 47 135)(11 207 299 174 319 194 302 187)(12 208 300 175 320 195 303 188)(13 209 291 176 311 196 304 189)(14 210 292 177 312 197 305 190)(15 201 293 178 313 198 306 181)(16 202 294 179 314 199 307 182)(17 203 295 180 315 200 308 183)(18 204 296 171 316 191 309 184)(19 205 297 172 317 192 310 185)(20 206 298 173 318 193 301 186)(51 111 83 91 63 123 71 103)(52 112 84 92 64 124 72 104)(53 113 85 93 65 125 73 105)(54 114 86 94 66 126 74 106)(55 115 87 95 67 127 75 107)(56 116 88 96 68 128 76 108)(57 117 89 97 69 129 77 109)(58 118 90 98 70 130 78 110)(59 119 81 99 61 121 79 101)(60 120 82 100 62 122 80 102)(211 276 244 256 224 289 231 269)(212 277 245 257 225 290 232 270)(213 278 246 258 226 281 233 261)(214 279 247 259 227 282 234 262)(215 280 248 260 228 283 235 263)(216 271 249 251 229 284 236 264)(217 272 250 252 230 285 237 265)(218 273 241 253 221 286 238 266)(219 274 242 254 222 287 239 267)(220 275 243 255 223 288 240 268)
(1 68 28 56)(2 69 29 57)(3 70 30 58)(4 61 21 59)(5 62 22 60)(6 63 23 51)(7 64 24 52)(8 65 25 53)(9 66 26 54)(10 67 27 55)(11 267 319 254)(12 268 320 255)(13 269 311 256)(14 270 312 257)(15 261 313 258)(16 262 314 259)(17 263 315 260)(18 264 316 251)(19 265 317 252)(20 266 318 253)(31 83 43 71)(32 84 44 72)(33 85 45 73)(34 86 46 74)(35 87 47 75)(36 88 48 76)(37 89 49 77)(38 90 50 78)(39 81 41 79)(40 82 42 80)(91 170 103 151)(92 161 104 152)(93 162 105 153)(94 163 106 154)(95 164 107 155)(96 165 108 156)(97 166 109 157)(98 167 110 158)(99 168 101 159)(100 169 102 160)(111 131 123 143)(112 132 124 144)(113 133 125 145)(114 134 126 146)(115 135 127 147)(116 136 128 148)(117 137 129 149)(118 138 130 150)(119 139 121 141)(120 140 122 142)(171 229 184 216)(172 230 185 217)(173 221 186 218)(174 222 187 219)(175 223 188 220)(176 224 189 211)(177 225 190 212)(178 226 181 213)(179 227 182 214)(180 228 183 215)(191 249 204 236)(192 250 205 237)(193 241 206 238)(194 242 207 239)(195 243 208 240)(196 244 209 231)(197 245 210 232)(198 246 201 233)(199 247 202 234)(200 248 203 235)(271 296 284 309)(272 297 285 310)(273 298 286 301)(274 299 287 302)(275 300 288 303)(276 291 289 304)(277 292 290 305)(278 293 281 306)(279 294 282 307)(280 295 283 308)
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), (1,179,6,174)(2,178,7,173)(3,177,8,172)(4,176,9,171)(5,175,10,180)(11,170,16,165)(12,169,17,164)(13,168,18,163)(14,167,19,162)(15,166,20,161)(21,189,26,184)(22,188,27,183)(23,187,28,182)(24,186,29,181)(25,185,30,190)(31,194,36,199)(32,193,37,198)(33,192,38,197)(34,191,39,196)(35,200,40,195)(41,209,46,204)(42,208,47,203)(43,207,48,202)(44,206,49,201)(45,205,50,210)(51,214,56,219)(52,213,57,218)(53,212,58,217)(54,211,59,216)(55,220,60,215)(61,229,66,224)(62,228,67,223)(63,227,68,222)(64,226,69,221)(65,225,70,230)(71,234,76,239)(72,233,77,238)(73,232,78,237)(74,231,79,236)(75,240,80,235)(81,249,86,244)(82,248,87,243)(83,247,88,242)(84,246,89,241)(85,245,90,250)(91,254,96,259)(92,253,97,258)(93,252,98,257)(94,251,99,256)(95,260,100,255)(101,269,106,264)(102,268,107,263)(103,267,108,262)(104,266,109,261)(105,265,110,270)(111,274,116,279)(112,273,117,278)(113,272,118,277)(114,271,119,276)(115,280,120,275)(121,289,126,284)(122,288,127,283)(123,287,128,282)(124,286,129,281)(125,285,130,290)(131,294,136,299)(132,293,137,298)(133,292,138,297)(134,291,139,296)(135,300,140,295)(141,309,146,304)(142,308,147,303)(143,307,148,302)(144,306,149,301)(145,305,150,310)(151,314,156,319)(152,313,157,318)(153,312,158,317)(154,311,159,316)(155,320,160,315), (1,156,36,148,28,165,48,136)(2,157,37,149,29,166,49,137)(3,158,38,150,30,167,50,138)(4,159,39,141,21,168,41,139)(5,160,40,142,22,169,42,140)(6,151,31,143,23,170,43,131)(7,152,32,144,24,161,44,132)(8,153,33,145,25,162,45,133)(9,154,34,146,26,163,46,134)(10,155,35,147,27,164,47,135)(11,207,299,174,319,194,302,187)(12,208,300,175,320,195,303,188)(13,209,291,176,311,196,304,189)(14,210,292,177,312,197,305,190)(15,201,293,178,313,198,306,181)(16,202,294,179,314,199,307,182)(17,203,295,180,315,200,308,183)(18,204,296,171,316,191,309,184)(19,205,297,172,317,192,310,185)(20,206,298,173,318,193,301,186)(51,111,83,91,63,123,71,103)(52,112,84,92,64,124,72,104)(53,113,85,93,65,125,73,105)(54,114,86,94,66,126,74,106)(55,115,87,95,67,127,75,107)(56,116,88,96,68,128,76,108)(57,117,89,97,69,129,77,109)(58,118,90,98,70,130,78,110)(59,119,81,99,61,121,79,101)(60,120,82,100,62,122,80,102)(211,276,244,256,224,289,231,269)(212,277,245,257,225,290,232,270)(213,278,246,258,226,281,233,261)(214,279,247,259,227,282,234,262)(215,280,248,260,228,283,235,263)(216,271,249,251,229,284,236,264)(217,272,250,252,230,285,237,265)(218,273,241,253,221,286,238,266)(219,274,242,254,222,287,239,267)(220,275,243,255,223,288,240,268), (1,68,28,56)(2,69,29,57)(3,70,30,58)(4,61,21,59)(5,62,22,60)(6,63,23,51)(7,64,24,52)(8,65,25,53)(9,66,26,54)(10,67,27,55)(11,267,319,254)(12,268,320,255)(13,269,311,256)(14,270,312,257)(15,261,313,258)(16,262,314,259)(17,263,315,260)(18,264,316,251)(19,265,317,252)(20,266,318,253)(31,83,43,71)(32,84,44,72)(33,85,45,73)(34,86,46,74)(35,87,47,75)(36,88,48,76)(37,89,49,77)(38,90,50,78)(39,81,41,79)(40,82,42,80)(91,170,103,151)(92,161,104,152)(93,162,105,153)(94,163,106,154)(95,164,107,155)(96,165,108,156)(97,166,109,157)(98,167,110,158)(99,168,101,159)(100,169,102,160)(111,131,123,143)(112,132,124,144)(113,133,125,145)(114,134,126,146)(115,135,127,147)(116,136,128,148)(117,137,129,149)(118,138,130,150)(119,139,121,141)(120,140,122,142)(171,229,184,216)(172,230,185,217)(173,221,186,218)(174,222,187,219)(175,223,188,220)(176,224,189,211)(177,225,190,212)(178,226,181,213)(179,227,182,214)(180,228,183,215)(191,249,204,236)(192,250,205,237)(193,241,206,238)(194,242,207,239)(195,243,208,240)(196,244,209,231)(197,245,210,232)(198,246,201,233)(199,247,202,234)(200,248,203,235)(271,296,284,309)(272,297,285,310)(273,298,286,301)(274,299,287,302)(275,300,288,303)(276,291,289,304)(277,292,290,305)(278,293,281,306)(279,294,282,307)(280,295,283,308)>;
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), (1,179,6,174)(2,178,7,173)(3,177,8,172)(4,176,9,171)(5,175,10,180)(11,170,16,165)(12,169,17,164)(13,168,18,163)(14,167,19,162)(15,166,20,161)(21,189,26,184)(22,188,27,183)(23,187,28,182)(24,186,29,181)(25,185,30,190)(31,194,36,199)(32,193,37,198)(33,192,38,197)(34,191,39,196)(35,200,40,195)(41,209,46,204)(42,208,47,203)(43,207,48,202)(44,206,49,201)(45,205,50,210)(51,214,56,219)(52,213,57,218)(53,212,58,217)(54,211,59,216)(55,220,60,215)(61,229,66,224)(62,228,67,223)(63,227,68,222)(64,226,69,221)(65,225,70,230)(71,234,76,239)(72,233,77,238)(73,232,78,237)(74,231,79,236)(75,240,80,235)(81,249,86,244)(82,248,87,243)(83,247,88,242)(84,246,89,241)(85,245,90,250)(91,254,96,259)(92,253,97,258)(93,252,98,257)(94,251,99,256)(95,260,100,255)(101,269,106,264)(102,268,107,263)(103,267,108,262)(104,266,109,261)(105,265,110,270)(111,274,116,279)(112,273,117,278)(113,272,118,277)(114,271,119,276)(115,280,120,275)(121,289,126,284)(122,288,127,283)(123,287,128,282)(124,286,129,281)(125,285,130,290)(131,294,136,299)(132,293,137,298)(133,292,138,297)(134,291,139,296)(135,300,140,295)(141,309,146,304)(142,308,147,303)(143,307,148,302)(144,306,149,301)(145,305,150,310)(151,314,156,319)(152,313,157,318)(153,312,158,317)(154,311,159,316)(155,320,160,315), (1,156,36,148,28,165,48,136)(2,157,37,149,29,166,49,137)(3,158,38,150,30,167,50,138)(4,159,39,141,21,168,41,139)(5,160,40,142,22,169,42,140)(6,151,31,143,23,170,43,131)(7,152,32,144,24,161,44,132)(8,153,33,145,25,162,45,133)(9,154,34,146,26,163,46,134)(10,155,35,147,27,164,47,135)(11,207,299,174,319,194,302,187)(12,208,300,175,320,195,303,188)(13,209,291,176,311,196,304,189)(14,210,292,177,312,197,305,190)(15,201,293,178,313,198,306,181)(16,202,294,179,314,199,307,182)(17,203,295,180,315,200,308,183)(18,204,296,171,316,191,309,184)(19,205,297,172,317,192,310,185)(20,206,298,173,318,193,301,186)(51,111,83,91,63,123,71,103)(52,112,84,92,64,124,72,104)(53,113,85,93,65,125,73,105)(54,114,86,94,66,126,74,106)(55,115,87,95,67,127,75,107)(56,116,88,96,68,128,76,108)(57,117,89,97,69,129,77,109)(58,118,90,98,70,130,78,110)(59,119,81,99,61,121,79,101)(60,120,82,100,62,122,80,102)(211,276,244,256,224,289,231,269)(212,277,245,257,225,290,232,270)(213,278,246,258,226,281,233,261)(214,279,247,259,227,282,234,262)(215,280,248,260,228,283,235,263)(216,271,249,251,229,284,236,264)(217,272,250,252,230,285,237,265)(218,273,241,253,221,286,238,266)(219,274,242,254,222,287,239,267)(220,275,243,255,223,288,240,268), (1,68,28,56)(2,69,29,57)(3,70,30,58)(4,61,21,59)(5,62,22,60)(6,63,23,51)(7,64,24,52)(8,65,25,53)(9,66,26,54)(10,67,27,55)(11,267,319,254)(12,268,320,255)(13,269,311,256)(14,270,312,257)(15,261,313,258)(16,262,314,259)(17,263,315,260)(18,264,316,251)(19,265,317,252)(20,266,318,253)(31,83,43,71)(32,84,44,72)(33,85,45,73)(34,86,46,74)(35,87,47,75)(36,88,48,76)(37,89,49,77)(38,90,50,78)(39,81,41,79)(40,82,42,80)(91,170,103,151)(92,161,104,152)(93,162,105,153)(94,163,106,154)(95,164,107,155)(96,165,108,156)(97,166,109,157)(98,167,110,158)(99,168,101,159)(100,169,102,160)(111,131,123,143)(112,132,124,144)(113,133,125,145)(114,134,126,146)(115,135,127,147)(116,136,128,148)(117,137,129,149)(118,138,130,150)(119,139,121,141)(120,140,122,142)(171,229,184,216)(172,230,185,217)(173,221,186,218)(174,222,187,219)(175,223,188,220)(176,224,189,211)(177,225,190,212)(178,226,181,213)(179,227,182,214)(180,228,183,215)(191,249,204,236)(192,250,205,237)(193,241,206,238)(194,242,207,239)(195,243,208,240)(196,244,209,231)(197,245,210,232)(198,246,201,233)(199,247,202,234)(200,248,203,235)(271,296,284,309)(272,297,285,310)(273,298,286,301)(274,299,287,302)(275,300,288,303)(276,291,289,304)(277,292,290,305)(278,293,281,306)(279,294,282,307)(280,295,283,308) );
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)], [(1,179,6,174),(2,178,7,173),(3,177,8,172),(4,176,9,171),(5,175,10,180),(11,170,16,165),(12,169,17,164),(13,168,18,163),(14,167,19,162),(15,166,20,161),(21,189,26,184),(22,188,27,183),(23,187,28,182),(24,186,29,181),(25,185,30,190),(31,194,36,199),(32,193,37,198),(33,192,38,197),(34,191,39,196),(35,200,40,195),(41,209,46,204),(42,208,47,203),(43,207,48,202),(44,206,49,201),(45,205,50,210),(51,214,56,219),(52,213,57,218),(53,212,58,217),(54,211,59,216),(55,220,60,215),(61,229,66,224),(62,228,67,223),(63,227,68,222),(64,226,69,221),(65,225,70,230),(71,234,76,239),(72,233,77,238),(73,232,78,237),(74,231,79,236),(75,240,80,235),(81,249,86,244),(82,248,87,243),(83,247,88,242),(84,246,89,241),(85,245,90,250),(91,254,96,259),(92,253,97,258),(93,252,98,257),(94,251,99,256),(95,260,100,255),(101,269,106,264),(102,268,107,263),(103,267,108,262),(104,266,109,261),(105,265,110,270),(111,274,116,279),(112,273,117,278),(113,272,118,277),(114,271,119,276),(115,280,120,275),(121,289,126,284),(122,288,127,283),(123,287,128,282),(124,286,129,281),(125,285,130,290),(131,294,136,299),(132,293,137,298),(133,292,138,297),(134,291,139,296),(135,300,140,295),(141,309,146,304),(142,308,147,303),(143,307,148,302),(144,306,149,301),(145,305,150,310),(151,314,156,319),(152,313,157,318),(153,312,158,317),(154,311,159,316),(155,320,160,315)], [(1,156,36,148,28,165,48,136),(2,157,37,149,29,166,49,137),(3,158,38,150,30,167,50,138),(4,159,39,141,21,168,41,139),(5,160,40,142,22,169,42,140),(6,151,31,143,23,170,43,131),(7,152,32,144,24,161,44,132),(8,153,33,145,25,162,45,133),(9,154,34,146,26,163,46,134),(10,155,35,147,27,164,47,135),(11,207,299,174,319,194,302,187),(12,208,300,175,320,195,303,188),(13,209,291,176,311,196,304,189),(14,210,292,177,312,197,305,190),(15,201,293,178,313,198,306,181),(16,202,294,179,314,199,307,182),(17,203,295,180,315,200,308,183),(18,204,296,171,316,191,309,184),(19,205,297,172,317,192,310,185),(20,206,298,173,318,193,301,186),(51,111,83,91,63,123,71,103),(52,112,84,92,64,124,72,104),(53,113,85,93,65,125,73,105),(54,114,86,94,66,126,74,106),(55,115,87,95,67,127,75,107),(56,116,88,96,68,128,76,108),(57,117,89,97,69,129,77,109),(58,118,90,98,70,130,78,110),(59,119,81,99,61,121,79,101),(60,120,82,100,62,122,80,102),(211,276,244,256,224,289,231,269),(212,277,245,257,225,290,232,270),(213,278,246,258,226,281,233,261),(214,279,247,259,227,282,234,262),(215,280,248,260,228,283,235,263),(216,271,249,251,229,284,236,264),(217,272,250,252,230,285,237,265),(218,273,241,253,221,286,238,266),(219,274,242,254,222,287,239,267),(220,275,243,255,223,288,240,268)], [(1,68,28,56),(2,69,29,57),(3,70,30,58),(4,61,21,59),(5,62,22,60),(6,63,23,51),(7,64,24,52),(8,65,25,53),(9,66,26,54),(10,67,27,55),(11,267,319,254),(12,268,320,255),(13,269,311,256),(14,270,312,257),(15,261,313,258),(16,262,314,259),(17,263,315,260),(18,264,316,251),(19,265,317,252),(20,266,318,253),(31,83,43,71),(32,84,44,72),(33,85,45,73),(34,86,46,74),(35,87,47,75),(36,88,48,76),(37,89,49,77),(38,90,50,78),(39,81,41,79),(40,82,42,80),(91,170,103,151),(92,161,104,152),(93,162,105,153),(94,163,106,154),(95,164,107,155),(96,165,108,156),(97,166,109,157),(98,167,110,158),(99,168,101,159),(100,169,102,160),(111,131,123,143),(112,132,124,144),(113,133,125,145),(114,134,126,146),(115,135,127,147),(116,136,128,148),(117,137,129,149),(118,138,130,150),(119,139,121,141),(120,140,122,142),(171,229,184,216),(172,230,185,217),(173,221,186,218),(174,222,187,219),(175,223,188,220),(176,224,189,211),(177,225,190,212),(178,226,181,213),(179,227,182,214),(180,228,183,215),(191,249,204,236),(192,250,205,237),(193,241,206,238),(194,242,207,239),(195,243,208,240),(196,244,209,231),(197,245,210,232),(198,246,201,233),(199,247,202,234),(200,248,203,235),(271,296,284,309),(272,297,285,310),(273,298,286,301),(274,299,287,302),(275,300,288,303),(276,291,289,304),(277,292,290,305),(278,293,281,306),(279,294,282,307),(280,295,283,308)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | Q16 | C4○D4 | D10 | D10 | C5⋊D4 | C8.C22 | D4⋊2D5 | D4×D5 | D5×Q16 | Q16⋊D5 |
| kernel | Dic5⋊3Q16 | C20.8Q8 | C20.44D4 | Q8⋊Dic5 | C2×C5⋊Q16 | Dic5⋊Q8 | Q8×Dic5 | C10×Q16 | C2×Dic5 | C5×Q8 | C2×Q16 | Dic5 | C20 | 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 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊3Q16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 19 | 32 |
| 17 | 6 | 0 | 0 |
| 34 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
| 33 | 1 | 0 | 0 |
| 17 | 8 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,7],[1,0,0,0,0,1,0,0,0,0,9,19,0,0,0,32],[17,34,0,0,6,0,0,0,0,0,24,40,0,0,1,17],[33,17,0,0,1,8,0,0,0,0,24,40,0,0,1,17] >;
Dic5⋊3Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_3Q_{16} % in TeX
G:=Group("Dic5:3Q16"); // GroupNames label
G:=SmallGroup(320,809);
// by ID
G=gap.SmallGroup(320,809);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,422,135,184,570,297,136,12550]);
// 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,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations