metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊4Q16, C5⋊3(C4×Q16), C5⋊Q16⋊6C4, Q8.2(C4×D5), C2.1(D5×Q16), C10.66(C4×D4), C4⋊C4.144D10, (C2×C8).205D10, Q8⋊C4.9D5, (C2×Q8).98D10, C10.12(C2×Q16), (Q8×Dic5).2C2, C22.75(D4×D5), C10.44(C4○D8), C20.46(C22×C4), (C8×Dic5).15C2, C10.D8.1C2, C20.156(C4○D4), C4.53(D4⋊2D5), (C2×C20).230C23, (C2×C40).191C22, Dic10.18(C2×C4), Dic5⋊3Q8.2C2, (C2×Dic5).271D4, C20.44D4.7C2, C4⋊Dic5.80C22, (Q8×C10).13C22, C2.3(SD16⋊3D5), C2.20(Dic5⋊4D4), (C4×Dic5).256C22, (C2×Dic10).67C22, C4.11(C2×C4×D5), C5⋊2C8.23(C2×C4), (C5×Q8).17(C2×C4), (C2×C5⋊Q16).2C2, (C2×C10).243(C2×D4), (C5×C4⋊C4).31C22, (C5×Q8⋊C4).8C2, (C2×C4).337(C22×D5), (C2×C5⋊2C8).223C22, SmallGroup(320,417)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊4Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 342 in 110 conjugacy classes, 51 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, Q16, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×C8, Q8⋊C4, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C5⋊2C8, C40, Dic10, 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⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.D8, C8×Dic5, C20.44D4, C5×Q8⋊C4, Dic5⋊3Q8, C2×C5⋊Q16, Q8×Dic5, Dic5⋊4Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, Q16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×Q16, C4○D8, C4×D5, C22×D5, C4×Q16, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, SD16⋊3D5, D5×Q16, Dic5⋊4Q16
►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)(11 157 16 152)(12 156 17 151)(13 155 18 160)(14 154 19 159)(15 153 20 158)(21 180 26 175)(22 179 27 174)(23 178 28 173)(24 177 29 172)(25 176 30 171)(31 203 36 208)(32 202 37 207)(33 201 38 206)(34 210 39 205)(35 209 40 204)(41 200 46 195)(42 199 47 194)(43 198 48 193)(44 197 49 192)(45 196 50 191)(51 223 56 228)(52 222 57 227)(53 221 58 226)(54 230 59 225)(55 229 60 224)(61 220 66 215)(62 219 67 214)(63 218 68 213)(64 217 69 212)(65 216 70 211)(71 243 76 248)(72 242 77 247)(73 241 78 246)(74 250 79 245)(75 249 80 244)(81 240 86 235)(82 239 87 234)(83 238 88 233)(84 237 89 232)(85 236 90 231)(91 263 96 268)(92 262 97 267)(93 261 98 266)(94 270 99 265)(95 269 100 264)(101 260 106 255)(102 259 107 254)(103 258 108 253)(104 257 109 252)(105 256 110 251)(111 283 116 288)(112 282 117 287)(113 281 118 286)(114 290 119 285)(115 289 120 284)(121 280 126 275)(122 279 127 274)(123 278 128 273)(124 277 129 272)(125 276 130 271)(131 303 136 308)(132 302 137 307)(133 301 138 306)(134 310 139 305)(135 309 140 304)(141 300 146 295)(142 299 147 294)(143 298 148 293)(144 297 149 292)(145 296 150 291)(161 317 166 312)(162 316 167 311)(163 315 168 320)(164 314 169 319)(165 313 170 318)
(1 16 32 298 28 318 48 307)(2 15 33 297 29 317 49 306)(3 14 34 296 30 316 50 305)(4 13 35 295 21 315 41 304)(5 12 36 294 22 314 42 303)(6 11 37 293 23 313 43 302)(7 20 38 292 24 312 44 301)(8 19 39 291 25 311 45 310)(9 18 40 300 26 320 46 309)(10 17 31 299 27 319 47 308)(51 283 87 263 67 274 71 254)(52 282 88 262 68 273 72 253)(53 281 89 261 69 272 73 252)(54 290 90 270 70 271 74 251)(55 289 81 269 61 280 75 260)(56 288 82 268 62 279 76 259)(57 287 83 267 63 278 77 258)(58 286 84 266 64 277 78 257)(59 285 85 265 65 276 79 256)(60 284 86 264 66 275 80 255)(91 219 127 248 107 228 111 239)(92 218 128 247 108 227 112 238)(93 217 129 246 109 226 113 237)(94 216 130 245 110 225 114 236)(95 215 121 244 101 224 115 235)(96 214 122 243 102 223 116 234)(97 213 123 242 103 222 117 233)(98 212 124 241 104 221 118 232)(99 211 125 250 105 230 119 231)(100 220 126 249 106 229 120 240)(131 183 151 203 147 174 164 194)(132 182 152 202 148 173 165 193)(133 181 153 201 149 172 166 192)(134 190 154 210 150 171 167 191)(135 189 155 209 141 180 168 200)(136 188 156 208 142 179 169 199)(137 187 157 207 143 178 170 198)(138 186 158 206 144 177 161 197)(139 185 159 205 145 176 162 196)(140 184 160 204 146 175 163 195)
(1 218 28 227)(2 217 29 226)(3 216 30 225)(4 215 21 224)(5 214 22 223)(6 213 23 222)(7 212 24 221)(8 211 25 230)(9 220 26 229)(10 219 27 228)(11 97 313 103)(12 96 314 102)(13 95 315 101)(14 94 316 110)(15 93 317 109)(16 92 318 108)(17 91 319 107)(18 100 320 106)(19 99 311 105)(20 98 312 104)(31 239 47 248)(32 238 48 247)(33 237 49 246)(34 236 50 245)(35 235 41 244)(36 234 42 243)(37 233 43 242)(38 232 44 241)(39 231 45 250)(40 240 46 249)(51 183 67 174)(52 182 68 173)(53 181 69 172)(54 190 70 171)(55 189 61 180)(56 188 62 179)(57 187 63 178)(58 186 64 177)(59 185 65 176)(60 184 66 175)(71 203 87 194)(72 202 88 193)(73 201 89 192)(74 210 90 191)(75 209 81 200)(76 208 82 199)(77 207 83 198)(78 206 84 197)(79 205 85 196)(80 204 86 195)(111 308 127 299)(112 307 128 298)(113 306 129 297)(114 305 130 296)(115 304 121 295)(116 303 122 294)(117 302 123 293)(118 301 124 292)(119 310 125 291)(120 309 126 300)(131 274 147 283)(132 273 148 282)(133 272 149 281)(134 271 150 290)(135 280 141 289)(136 279 142 288)(137 278 143 287)(138 277 144 286)(139 276 145 285)(140 275 146 284)(151 263 164 254)(152 262 165 253)(153 261 166 252)(154 270 167 251)(155 269 168 260)(156 268 169 259)(157 267 170 258)(158 266 161 257)(159 265 162 256)(160 264 163 255)
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)(11,157,16,152)(12,156,17,151)(13,155,18,160)(14,154,19,159)(15,153,20,158)(21,180,26,175)(22,179,27,174)(23,178,28,173)(24,177,29,172)(25,176,30,171)(31,203,36,208)(32,202,37,207)(33,201,38,206)(34,210,39,205)(35,209,40,204)(41,200,46,195)(42,199,47,194)(43,198,48,193)(44,197,49,192)(45,196,50,191)(51,223,56,228)(52,222,57,227)(53,221,58,226)(54,230,59,225)(55,229,60,224)(61,220,66,215)(62,219,67,214)(63,218,68,213)(64,217,69,212)(65,216,70,211)(71,243,76,248)(72,242,77,247)(73,241,78,246)(74,250,79,245)(75,249,80,244)(81,240,86,235)(82,239,87,234)(83,238,88,233)(84,237,89,232)(85,236,90,231)(91,263,96,268)(92,262,97,267)(93,261,98,266)(94,270,99,265)(95,269,100,264)(101,260,106,255)(102,259,107,254)(103,258,108,253)(104,257,109,252)(105,256,110,251)(111,283,116,288)(112,282,117,287)(113,281,118,286)(114,290,119,285)(115,289,120,284)(121,280,126,275)(122,279,127,274)(123,278,128,273)(124,277,129,272)(125,276,130,271)(131,303,136,308)(132,302,137,307)(133,301,138,306)(134,310,139,305)(135,309,140,304)(141,300,146,295)(142,299,147,294)(143,298,148,293)(144,297,149,292)(145,296,150,291)(161,317,166,312)(162,316,167,311)(163,315,168,320)(164,314,169,319)(165,313,170,318), (1,16,32,298,28,318,48,307)(2,15,33,297,29,317,49,306)(3,14,34,296,30,316,50,305)(4,13,35,295,21,315,41,304)(5,12,36,294,22,314,42,303)(6,11,37,293,23,313,43,302)(7,20,38,292,24,312,44,301)(8,19,39,291,25,311,45,310)(9,18,40,300,26,320,46,309)(10,17,31,299,27,319,47,308)(51,283,87,263,67,274,71,254)(52,282,88,262,68,273,72,253)(53,281,89,261,69,272,73,252)(54,290,90,270,70,271,74,251)(55,289,81,269,61,280,75,260)(56,288,82,268,62,279,76,259)(57,287,83,267,63,278,77,258)(58,286,84,266,64,277,78,257)(59,285,85,265,65,276,79,256)(60,284,86,264,66,275,80,255)(91,219,127,248,107,228,111,239)(92,218,128,247,108,227,112,238)(93,217,129,246,109,226,113,237)(94,216,130,245,110,225,114,236)(95,215,121,244,101,224,115,235)(96,214,122,243,102,223,116,234)(97,213,123,242,103,222,117,233)(98,212,124,241,104,221,118,232)(99,211,125,250,105,230,119,231)(100,220,126,249,106,229,120,240)(131,183,151,203,147,174,164,194)(132,182,152,202,148,173,165,193)(133,181,153,201,149,172,166,192)(134,190,154,210,150,171,167,191)(135,189,155,209,141,180,168,200)(136,188,156,208,142,179,169,199)(137,187,157,207,143,178,170,198)(138,186,158,206,144,177,161,197)(139,185,159,205,145,176,162,196)(140,184,160,204,146,175,163,195), (1,218,28,227)(2,217,29,226)(3,216,30,225)(4,215,21,224)(5,214,22,223)(6,213,23,222)(7,212,24,221)(8,211,25,230)(9,220,26,229)(10,219,27,228)(11,97,313,103)(12,96,314,102)(13,95,315,101)(14,94,316,110)(15,93,317,109)(16,92,318,108)(17,91,319,107)(18,100,320,106)(19,99,311,105)(20,98,312,104)(31,239,47,248)(32,238,48,247)(33,237,49,246)(34,236,50,245)(35,235,41,244)(36,234,42,243)(37,233,43,242)(38,232,44,241)(39,231,45,250)(40,240,46,249)(51,183,67,174)(52,182,68,173)(53,181,69,172)(54,190,70,171)(55,189,61,180)(56,188,62,179)(57,187,63,178)(58,186,64,177)(59,185,65,176)(60,184,66,175)(71,203,87,194)(72,202,88,193)(73,201,89,192)(74,210,90,191)(75,209,81,200)(76,208,82,199)(77,207,83,198)(78,206,84,197)(79,205,85,196)(80,204,86,195)(111,308,127,299)(112,307,128,298)(113,306,129,297)(114,305,130,296)(115,304,121,295)(116,303,122,294)(117,302,123,293)(118,301,124,292)(119,310,125,291)(120,309,126,300)(131,274,147,283)(132,273,148,282)(133,272,149,281)(134,271,150,290)(135,280,141,289)(136,279,142,288)(137,278,143,287)(138,277,144,286)(139,276,145,285)(140,275,146,284)(151,263,164,254)(152,262,165,253)(153,261,166,252)(154,270,167,251)(155,269,168,260)(156,268,169,259)(157,267,170,258)(158,266,161,257)(159,265,162,256)(160,264,163,255)>;
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)(11,157,16,152)(12,156,17,151)(13,155,18,160)(14,154,19,159)(15,153,20,158)(21,180,26,175)(22,179,27,174)(23,178,28,173)(24,177,29,172)(25,176,30,171)(31,203,36,208)(32,202,37,207)(33,201,38,206)(34,210,39,205)(35,209,40,204)(41,200,46,195)(42,199,47,194)(43,198,48,193)(44,197,49,192)(45,196,50,191)(51,223,56,228)(52,222,57,227)(53,221,58,226)(54,230,59,225)(55,229,60,224)(61,220,66,215)(62,219,67,214)(63,218,68,213)(64,217,69,212)(65,216,70,211)(71,243,76,248)(72,242,77,247)(73,241,78,246)(74,250,79,245)(75,249,80,244)(81,240,86,235)(82,239,87,234)(83,238,88,233)(84,237,89,232)(85,236,90,231)(91,263,96,268)(92,262,97,267)(93,261,98,266)(94,270,99,265)(95,269,100,264)(101,260,106,255)(102,259,107,254)(103,258,108,253)(104,257,109,252)(105,256,110,251)(111,283,116,288)(112,282,117,287)(113,281,118,286)(114,290,119,285)(115,289,120,284)(121,280,126,275)(122,279,127,274)(123,278,128,273)(124,277,129,272)(125,276,130,271)(131,303,136,308)(132,302,137,307)(133,301,138,306)(134,310,139,305)(135,309,140,304)(141,300,146,295)(142,299,147,294)(143,298,148,293)(144,297,149,292)(145,296,150,291)(161,317,166,312)(162,316,167,311)(163,315,168,320)(164,314,169,319)(165,313,170,318), (1,16,32,298,28,318,48,307)(2,15,33,297,29,317,49,306)(3,14,34,296,30,316,50,305)(4,13,35,295,21,315,41,304)(5,12,36,294,22,314,42,303)(6,11,37,293,23,313,43,302)(7,20,38,292,24,312,44,301)(8,19,39,291,25,311,45,310)(9,18,40,300,26,320,46,309)(10,17,31,299,27,319,47,308)(51,283,87,263,67,274,71,254)(52,282,88,262,68,273,72,253)(53,281,89,261,69,272,73,252)(54,290,90,270,70,271,74,251)(55,289,81,269,61,280,75,260)(56,288,82,268,62,279,76,259)(57,287,83,267,63,278,77,258)(58,286,84,266,64,277,78,257)(59,285,85,265,65,276,79,256)(60,284,86,264,66,275,80,255)(91,219,127,248,107,228,111,239)(92,218,128,247,108,227,112,238)(93,217,129,246,109,226,113,237)(94,216,130,245,110,225,114,236)(95,215,121,244,101,224,115,235)(96,214,122,243,102,223,116,234)(97,213,123,242,103,222,117,233)(98,212,124,241,104,221,118,232)(99,211,125,250,105,230,119,231)(100,220,126,249,106,229,120,240)(131,183,151,203,147,174,164,194)(132,182,152,202,148,173,165,193)(133,181,153,201,149,172,166,192)(134,190,154,210,150,171,167,191)(135,189,155,209,141,180,168,200)(136,188,156,208,142,179,169,199)(137,187,157,207,143,178,170,198)(138,186,158,206,144,177,161,197)(139,185,159,205,145,176,162,196)(140,184,160,204,146,175,163,195), (1,218,28,227)(2,217,29,226)(3,216,30,225)(4,215,21,224)(5,214,22,223)(6,213,23,222)(7,212,24,221)(8,211,25,230)(9,220,26,229)(10,219,27,228)(11,97,313,103)(12,96,314,102)(13,95,315,101)(14,94,316,110)(15,93,317,109)(16,92,318,108)(17,91,319,107)(18,100,320,106)(19,99,311,105)(20,98,312,104)(31,239,47,248)(32,238,48,247)(33,237,49,246)(34,236,50,245)(35,235,41,244)(36,234,42,243)(37,233,43,242)(38,232,44,241)(39,231,45,250)(40,240,46,249)(51,183,67,174)(52,182,68,173)(53,181,69,172)(54,190,70,171)(55,189,61,180)(56,188,62,179)(57,187,63,178)(58,186,64,177)(59,185,65,176)(60,184,66,175)(71,203,87,194)(72,202,88,193)(73,201,89,192)(74,210,90,191)(75,209,81,200)(76,208,82,199)(77,207,83,198)(78,206,84,197)(79,205,85,196)(80,204,86,195)(111,308,127,299)(112,307,128,298)(113,306,129,297)(114,305,130,296)(115,304,121,295)(116,303,122,294)(117,302,123,293)(118,301,124,292)(119,310,125,291)(120,309,126,300)(131,274,147,283)(132,273,148,282)(133,272,149,281)(134,271,150,290)(135,280,141,289)(136,279,142,288)(137,278,143,287)(138,277,144,286)(139,276,145,285)(140,275,146,284)(151,263,164,254)(152,262,165,253)(153,261,166,252)(154,270,167,251)(155,269,168,260)(156,268,169,259)(157,267,170,258)(158,266,161,257)(159,265,162,256)(160,264,163,255) );
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),(11,157,16,152),(12,156,17,151),(13,155,18,160),(14,154,19,159),(15,153,20,158),(21,180,26,175),(22,179,27,174),(23,178,28,173),(24,177,29,172),(25,176,30,171),(31,203,36,208),(32,202,37,207),(33,201,38,206),(34,210,39,205),(35,209,40,204),(41,200,46,195),(42,199,47,194),(43,198,48,193),(44,197,49,192),(45,196,50,191),(51,223,56,228),(52,222,57,227),(53,221,58,226),(54,230,59,225),(55,229,60,224),(61,220,66,215),(62,219,67,214),(63,218,68,213),(64,217,69,212),(65,216,70,211),(71,243,76,248),(72,242,77,247),(73,241,78,246),(74,250,79,245),(75,249,80,244),(81,240,86,235),(82,239,87,234),(83,238,88,233),(84,237,89,232),(85,236,90,231),(91,263,96,268),(92,262,97,267),(93,261,98,266),(94,270,99,265),(95,269,100,264),(101,260,106,255),(102,259,107,254),(103,258,108,253),(104,257,109,252),(105,256,110,251),(111,283,116,288),(112,282,117,287),(113,281,118,286),(114,290,119,285),(115,289,120,284),(121,280,126,275),(122,279,127,274),(123,278,128,273),(124,277,129,272),(125,276,130,271),(131,303,136,308),(132,302,137,307),(133,301,138,306),(134,310,139,305),(135,309,140,304),(141,300,146,295),(142,299,147,294),(143,298,148,293),(144,297,149,292),(145,296,150,291),(161,317,166,312),(162,316,167,311),(163,315,168,320),(164,314,169,319),(165,313,170,318)], [(1,16,32,298,28,318,48,307),(2,15,33,297,29,317,49,306),(3,14,34,296,30,316,50,305),(4,13,35,295,21,315,41,304),(5,12,36,294,22,314,42,303),(6,11,37,293,23,313,43,302),(7,20,38,292,24,312,44,301),(8,19,39,291,25,311,45,310),(9,18,40,300,26,320,46,309),(10,17,31,299,27,319,47,308),(51,283,87,263,67,274,71,254),(52,282,88,262,68,273,72,253),(53,281,89,261,69,272,73,252),(54,290,90,270,70,271,74,251),(55,289,81,269,61,280,75,260),(56,288,82,268,62,279,76,259),(57,287,83,267,63,278,77,258),(58,286,84,266,64,277,78,257),(59,285,85,265,65,276,79,256),(60,284,86,264,66,275,80,255),(91,219,127,248,107,228,111,239),(92,218,128,247,108,227,112,238),(93,217,129,246,109,226,113,237),(94,216,130,245,110,225,114,236),(95,215,121,244,101,224,115,235),(96,214,122,243,102,223,116,234),(97,213,123,242,103,222,117,233),(98,212,124,241,104,221,118,232),(99,211,125,250,105,230,119,231),(100,220,126,249,106,229,120,240),(131,183,151,203,147,174,164,194),(132,182,152,202,148,173,165,193),(133,181,153,201,149,172,166,192),(134,190,154,210,150,171,167,191),(135,189,155,209,141,180,168,200),(136,188,156,208,142,179,169,199),(137,187,157,207,143,178,170,198),(138,186,158,206,144,177,161,197),(139,185,159,205,145,176,162,196),(140,184,160,204,146,175,163,195)], [(1,218,28,227),(2,217,29,226),(3,216,30,225),(4,215,21,224),(5,214,22,223),(6,213,23,222),(7,212,24,221),(8,211,25,230),(9,220,26,229),(10,219,27,228),(11,97,313,103),(12,96,314,102),(13,95,315,101),(14,94,316,110),(15,93,317,109),(16,92,318,108),(17,91,319,107),(18,100,320,106),(19,99,311,105),(20,98,312,104),(31,239,47,248),(32,238,48,247),(33,237,49,246),(34,236,50,245),(35,235,41,244),(36,234,42,243),(37,233,43,242),(38,232,44,241),(39,231,45,250),(40,240,46,249),(51,183,67,174),(52,182,68,173),(53,181,69,172),(54,190,70,171),(55,189,61,180),(56,188,62,179),(57,187,63,178),(58,186,64,177),(59,185,65,176),(60,184,66,175),(71,203,87,194),(72,202,88,193),(73,201,89,192),(74,210,90,191),(75,209,81,200),(76,208,82,199),(77,207,83,198),(78,206,84,197),(79,205,85,196),(80,204,86,195),(111,308,127,299),(112,307,128,298),(113,306,129,297),(114,305,130,296),(115,304,121,295),(116,303,122,294),(117,302,123,293),(118,301,124,292),(119,310,125,291),(120,309,126,300),(131,274,147,283),(132,273,148,282),(133,272,149,281),(134,271,150,290),(135,280,141,289),(136,279,142,288),(137,278,143,287),(138,277,144,286),(139,276,145,285),(140,275,146,284),(151,263,164,254),(152,262,165,253),(153,261,166,252),(154,270,167,251),(155,269,168,260),(156,268,169,259),(157,267,170,258),(158,266,161,257),(159,265,162,256),(160,264,163,255)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 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 | 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 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 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 | C4 | D4 | D5 | Q16 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4×D5 | D4⋊2D5 | D4×D5 | SD16⋊3D5 | D5×Q16 |
| kernel | Dic5⋊4Q16 | C10.D8 | C8×Dic5 | C20.44D4 | C5×Q8⋊C4 | Dic5⋊3Q8 | C2×C5⋊Q16 | Q8×Dic5 | C5⋊Q16 | C2×Dic5 | Q8⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊4Q16 ►in GL4(𝔽41) generated by
| 35 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 35 | 23 | 0 | 0 |
| 18 | 6 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 13 | 39 | 0 | 0 |
| 2 | 28 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 12 | 17 |
| 13 | 39 | 0 | 0 |
| 2 | 28 | 0 | 0 |
| 0 | 0 | 12 | 39 |
| 0 | 0 | 11 | 29 |
G:=sub<GL(4,GF(41))| [35,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[35,18,0,0,23,6,0,0,0,0,32,0,0,0,0,32],[13,2,0,0,39,28,0,0,0,0,0,12,0,0,17,17],[13,2,0,0,39,28,0,0,0,0,12,11,0,0,39,29] >;
Dic5⋊4Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_4Q_{16} % in TeX
G:=Group("Dic5:4Q16"); // GroupNames label
G:=SmallGroup(320,417);
// by ID
G=gap.SmallGroup(320,417);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,135,268,570,297,136,12550]);
// 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=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations