metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊5Q16, Dic20⋊12C4, C5⋊4(C4×Q16), C8.15(C4×D5), C2.3(D5×Q16), C40.57(C2×C4), C10.83(C4×D4), C2.D8.9D5, C4⋊C4.167D10, (C2×C8).226D10, C10.20(C2×Q16), (C8×Dic5).1C2, C22.88(D4×D5), C20.36(C4○D4), C10.26(C4○D8), C2.3(D8⋊3D5), (C2×C40).78C22, C4.8(Q8⋊2D5), (C2×Dic20).9C2, C10.Q16.7C2, (C2×C20).289C23, C20.106(C22×C4), (C2×Dic5).276D4, Dic5⋊3Q8.7C2, Dic10.24(C2×C4), C2.13(D20⋊8C4), (C2×Dic10).90C22, (C4×Dic5).263C22, C4.44(C2×C4×D5), (C5×C2.D8).4C2, (C2×C10).294(C2×D4), (C5×C4⋊C4).82C22, (C2×C4).392(C22×D5), (C2×C5⋊2C8).238C22, SmallGroup(320,500)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊5Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 358 in 110 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C5⋊2C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Q16, Dic20, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C5×C4⋊C4, C2×C40, C2×Dic10, C10.Q16, C8×Dic5, C5×C2.D8, Dic5⋊3Q8, C2×Dic20, Dic5⋊5Q16
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, Q8⋊2D5, D20⋊8C4, D8⋊3D5, D5×Q16, Dic5⋊5Q16
►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)(11 153 16 158)(12 152 17 157)(13 151 18 156)(14 160 19 155)(15 159 20 154)(21 174 26 179)(22 173 27 178)(23 172 28 177)(24 171 29 176)(25 180 30 175)(31 209 36 204)(32 208 37 203)(33 207 38 202)(34 206 39 201)(35 205 40 210)(41 194 46 199)(42 193 47 198)(43 192 48 197)(44 191 49 196)(45 200 50 195)(51 229 56 224)(52 228 57 223)(53 227 58 222)(54 226 59 221)(55 225 60 230)(61 214 66 219)(62 213 67 218)(63 212 68 217)(64 211 69 216)(65 220 70 215)(71 249 76 244)(72 248 77 243)(73 247 78 242)(74 246 79 241)(75 245 80 250)(81 234 86 239)(82 233 87 238)(83 232 88 237)(84 231 89 236)(85 240 90 235)(91 269 96 264)(92 268 97 263)(93 267 98 262)(94 266 99 261)(95 265 100 270)(101 254 106 259)(102 253 107 258)(103 252 108 257)(104 251 109 256)(105 260 110 255)(111 289 116 284)(112 288 117 283)(113 287 118 282)(114 286 119 281)(115 285 120 290)(121 274 126 279)(122 273 127 278)(123 272 128 277)(124 271 129 276)(125 280 130 275)(131 309 136 304)(132 308 137 303)(133 307 138 302)(134 306 139 301)(135 305 140 310)(141 294 146 299)(142 293 147 298)(143 292 148 297)(144 291 149 296)(145 300 150 295)(161 311 166 316)(162 320 167 315)(163 319 168 314)(164 318 169 313)(165 317 170 312)
(1 158 38 148 28 165 48 138)(2 159 39 149 29 166 49 139)(3 160 40 150 30 167 50 140)(4 151 31 141 21 168 41 131)(5 152 32 142 22 169 42 132)(6 153 33 143 23 170 43 133)(7 154 34 144 24 161 44 134)(8 155 35 145 25 162 45 135)(9 156 36 146 26 163 46 136)(10 157 37 147 27 164 47 137)(11 202 297 177 317 197 302 182)(12 203 298 178 318 198 303 183)(13 204 299 179 319 199 304 184)(14 205 300 180 320 200 305 185)(15 206 291 171 311 191 306 186)(16 207 292 172 312 192 307 187)(17 208 293 173 313 193 308 188)(18 209 294 174 314 194 309 189)(19 210 295 175 315 195 310 190)(20 201 296 176 316 196 301 181)(51 116 81 96 61 126 71 106)(52 117 82 97 62 127 72 107)(53 118 83 98 63 128 73 108)(54 119 84 99 64 129 74 109)(55 120 85 100 65 130 75 110)(56 111 86 91 66 121 76 101)(57 112 87 92 67 122 77 102)(58 113 88 93 68 123 78 103)(59 114 89 94 69 124 79 104)(60 115 90 95 70 125 80 105)(211 276 246 256 226 281 231 261)(212 277 247 257 227 282 232 262)(213 278 248 258 228 283 233 263)(214 279 249 259 229 284 234 264)(215 280 250 260 230 285 235 265)(216 271 241 251 221 286 236 266)(217 272 242 252 222 287 237 267)(218 273 243 253 223 288 238 268)(219 274 244 254 224 289 239 269)(220 275 245 255 225 290 240 270)
(1 217 28 222)(2 216 29 221)(3 215 30 230)(4 214 21 229)(5 213 22 228)(6 212 23 227)(7 211 24 226)(8 220 25 225)(9 219 26 224)(10 218 27 223)(11 98 317 108)(12 97 318 107)(13 96 319 106)(14 95 320 105)(15 94 311 104)(16 93 312 103)(17 92 313 102)(18 91 314 101)(19 100 315 110)(20 99 316 109)(31 234 41 249)(32 233 42 248)(33 232 43 247)(34 231 44 246)(35 240 45 245)(36 239 46 244)(37 238 47 243)(38 237 48 242)(39 236 49 241)(40 235 50 250)(51 184 61 179)(52 183 62 178)(53 182 63 177)(54 181 64 176)(55 190 65 175)(56 189 66 174)(57 188 67 173)(58 187 68 172)(59 186 69 171)(60 185 70 180)(71 204 81 199)(72 203 82 198)(73 202 83 197)(74 201 84 196)(75 210 85 195)(76 209 86 194)(77 208 87 193)(78 207 88 192)(79 206 89 191)(80 205 90 200)(111 309 121 294)(112 308 122 293)(113 307 123 292)(114 306 124 291)(115 305 125 300)(116 304 126 299)(117 303 127 298)(118 302 128 297)(119 301 129 296)(120 310 130 295)(131 279 141 284)(132 278 142 283)(133 277 143 282)(134 276 144 281)(135 275 145 290)(136 274 146 289)(137 273 147 288)(138 272 148 287)(139 271 149 286)(140 280 150 285)(151 264 168 259)(152 263 169 258)(153 262 170 257)(154 261 161 256)(155 270 162 255)(156 269 163 254)(157 268 164 253)(158 267 165 252)(159 266 166 251)(160 265 167 260)
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)(11,153,16,158)(12,152,17,157)(13,151,18,156)(14,160,19,155)(15,159,20,154)(21,174,26,179)(22,173,27,178)(23,172,28,177)(24,171,29,176)(25,180,30,175)(31,209,36,204)(32,208,37,203)(33,207,38,202)(34,206,39,201)(35,205,40,210)(41,194,46,199)(42,193,47,198)(43,192,48,197)(44,191,49,196)(45,200,50,195)(51,229,56,224)(52,228,57,223)(53,227,58,222)(54,226,59,221)(55,225,60,230)(61,214,66,219)(62,213,67,218)(63,212,68,217)(64,211,69,216)(65,220,70,215)(71,249,76,244)(72,248,77,243)(73,247,78,242)(74,246,79,241)(75,245,80,250)(81,234,86,239)(82,233,87,238)(83,232,88,237)(84,231,89,236)(85,240,90,235)(91,269,96,264)(92,268,97,263)(93,267,98,262)(94,266,99,261)(95,265,100,270)(101,254,106,259)(102,253,107,258)(103,252,108,257)(104,251,109,256)(105,260,110,255)(111,289,116,284)(112,288,117,283)(113,287,118,282)(114,286,119,281)(115,285,120,290)(121,274,126,279)(122,273,127,278)(123,272,128,277)(124,271,129,276)(125,280,130,275)(131,309,136,304)(132,308,137,303)(133,307,138,302)(134,306,139,301)(135,305,140,310)(141,294,146,299)(142,293,147,298)(143,292,148,297)(144,291,149,296)(145,300,150,295)(161,311,166,316)(162,320,167,315)(163,319,168,314)(164,318,169,313)(165,317,170,312), (1,158,38,148,28,165,48,138)(2,159,39,149,29,166,49,139)(3,160,40,150,30,167,50,140)(4,151,31,141,21,168,41,131)(5,152,32,142,22,169,42,132)(6,153,33,143,23,170,43,133)(7,154,34,144,24,161,44,134)(8,155,35,145,25,162,45,135)(9,156,36,146,26,163,46,136)(10,157,37,147,27,164,47,137)(11,202,297,177,317,197,302,182)(12,203,298,178,318,198,303,183)(13,204,299,179,319,199,304,184)(14,205,300,180,320,200,305,185)(15,206,291,171,311,191,306,186)(16,207,292,172,312,192,307,187)(17,208,293,173,313,193,308,188)(18,209,294,174,314,194,309,189)(19,210,295,175,315,195,310,190)(20,201,296,176,316,196,301,181)(51,116,81,96,61,126,71,106)(52,117,82,97,62,127,72,107)(53,118,83,98,63,128,73,108)(54,119,84,99,64,129,74,109)(55,120,85,100,65,130,75,110)(56,111,86,91,66,121,76,101)(57,112,87,92,67,122,77,102)(58,113,88,93,68,123,78,103)(59,114,89,94,69,124,79,104)(60,115,90,95,70,125,80,105)(211,276,246,256,226,281,231,261)(212,277,247,257,227,282,232,262)(213,278,248,258,228,283,233,263)(214,279,249,259,229,284,234,264)(215,280,250,260,230,285,235,265)(216,271,241,251,221,286,236,266)(217,272,242,252,222,287,237,267)(218,273,243,253,223,288,238,268)(219,274,244,254,224,289,239,269)(220,275,245,255,225,290,240,270), (1,217,28,222)(2,216,29,221)(3,215,30,230)(4,214,21,229)(5,213,22,228)(6,212,23,227)(7,211,24,226)(8,220,25,225)(9,219,26,224)(10,218,27,223)(11,98,317,108)(12,97,318,107)(13,96,319,106)(14,95,320,105)(15,94,311,104)(16,93,312,103)(17,92,313,102)(18,91,314,101)(19,100,315,110)(20,99,316,109)(31,234,41,249)(32,233,42,248)(33,232,43,247)(34,231,44,246)(35,240,45,245)(36,239,46,244)(37,238,47,243)(38,237,48,242)(39,236,49,241)(40,235,50,250)(51,184,61,179)(52,183,62,178)(53,182,63,177)(54,181,64,176)(55,190,65,175)(56,189,66,174)(57,188,67,173)(58,187,68,172)(59,186,69,171)(60,185,70,180)(71,204,81,199)(72,203,82,198)(73,202,83,197)(74,201,84,196)(75,210,85,195)(76,209,86,194)(77,208,87,193)(78,207,88,192)(79,206,89,191)(80,205,90,200)(111,309,121,294)(112,308,122,293)(113,307,123,292)(114,306,124,291)(115,305,125,300)(116,304,126,299)(117,303,127,298)(118,302,128,297)(119,301,129,296)(120,310,130,295)(131,279,141,284)(132,278,142,283)(133,277,143,282)(134,276,144,281)(135,275,145,290)(136,274,146,289)(137,273,147,288)(138,272,148,287)(139,271,149,286)(140,280,150,285)(151,264,168,259)(152,263,169,258)(153,262,170,257)(154,261,161,256)(155,270,162,255)(156,269,163,254)(157,268,164,253)(158,267,165,252)(159,266,166,251)(160,265,167,260)>;
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)(11,153,16,158)(12,152,17,157)(13,151,18,156)(14,160,19,155)(15,159,20,154)(21,174,26,179)(22,173,27,178)(23,172,28,177)(24,171,29,176)(25,180,30,175)(31,209,36,204)(32,208,37,203)(33,207,38,202)(34,206,39,201)(35,205,40,210)(41,194,46,199)(42,193,47,198)(43,192,48,197)(44,191,49,196)(45,200,50,195)(51,229,56,224)(52,228,57,223)(53,227,58,222)(54,226,59,221)(55,225,60,230)(61,214,66,219)(62,213,67,218)(63,212,68,217)(64,211,69,216)(65,220,70,215)(71,249,76,244)(72,248,77,243)(73,247,78,242)(74,246,79,241)(75,245,80,250)(81,234,86,239)(82,233,87,238)(83,232,88,237)(84,231,89,236)(85,240,90,235)(91,269,96,264)(92,268,97,263)(93,267,98,262)(94,266,99,261)(95,265,100,270)(101,254,106,259)(102,253,107,258)(103,252,108,257)(104,251,109,256)(105,260,110,255)(111,289,116,284)(112,288,117,283)(113,287,118,282)(114,286,119,281)(115,285,120,290)(121,274,126,279)(122,273,127,278)(123,272,128,277)(124,271,129,276)(125,280,130,275)(131,309,136,304)(132,308,137,303)(133,307,138,302)(134,306,139,301)(135,305,140,310)(141,294,146,299)(142,293,147,298)(143,292,148,297)(144,291,149,296)(145,300,150,295)(161,311,166,316)(162,320,167,315)(163,319,168,314)(164,318,169,313)(165,317,170,312), (1,158,38,148,28,165,48,138)(2,159,39,149,29,166,49,139)(3,160,40,150,30,167,50,140)(4,151,31,141,21,168,41,131)(5,152,32,142,22,169,42,132)(6,153,33,143,23,170,43,133)(7,154,34,144,24,161,44,134)(8,155,35,145,25,162,45,135)(9,156,36,146,26,163,46,136)(10,157,37,147,27,164,47,137)(11,202,297,177,317,197,302,182)(12,203,298,178,318,198,303,183)(13,204,299,179,319,199,304,184)(14,205,300,180,320,200,305,185)(15,206,291,171,311,191,306,186)(16,207,292,172,312,192,307,187)(17,208,293,173,313,193,308,188)(18,209,294,174,314,194,309,189)(19,210,295,175,315,195,310,190)(20,201,296,176,316,196,301,181)(51,116,81,96,61,126,71,106)(52,117,82,97,62,127,72,107)(53,118,83,98,63,128,73,108)(54,119,84,99,64,129,74,109)(55,120,85,100,65,130,75,110)(56,111,86,91,66,121,76,101)(57,112,87,92,67,122,77,102)(58,113,88,93,68,123,78,103)(59,114,89,94,69,124,79,104)(60,115,90,95,70,125,80,105)(211,276,246,256,226,281,231,261)(212,277,247,257,227,282,232,262)(213,278,248,258,228,283,233,263)(214,279,249,259,229,284,234,264)(215,280,250,260,230,285,235,265)(216,271,241,251,221,286,236,266)(217,272,242,252,222,287,237,267)(218,273,243,253,223,288,238,268)(219,274,244,254,224,289,239,269)(220,275,245,255,225,290,240,270), (1,217,28,222)(2,216,29,221)(3,215,30,230)(4,214,21,229)(5,213,22,228)(6,212,23,227)(7,211,24,226)(8,220,25,225)(9,219,26,224)(10,218,27,223)(11,98,317,108)(12,97,318,107)(13,96,319,106)(14,95,320,105)(15,94,311,104)(16,93,312,103)(17,92,313,102)(18,91,314,101)(19,100,315,110)(20,99,316,109)(31,234,41,249)(32,233,42,248)(33,232,43,247)(34,231,44,246)(35,240,45,245)(36,239,46,244)(37,238,47,243)(38,237,48,242)(39,236,49,241)(40,235,50,250)(51,184,61,179)(52,183,62,178)(53,182,63,177)(54,181,64,176)(55,190,65,175)(56,189,66,174)(57,188,67,173)(58,187,68,172)(59,186,69,171)(60,185,70,180)(71,204,81,199)(72,203,82,198)(73,202,83,197)(74,201,84,196)(75,210,85,195)(76,209,86,194)(77,208,87,193)(78,207,88,192)(79,206,89,191)(80,205,90,200)(111,309,121,294)(112,308,122,293)(113,307,123,292)(114,306,124,291)(115,305,125,300)(116,304,126,299)(117,303,127,298)(118,302,128,297)(119,301,129,296)(120,310,130,295)(131,279,141,284)(132,278,142,283)(133,277,143,282)(134,276,144,281)(135,275,145,290)(136,274,146,289)(137,273,147,288)(138,272,148,287)(139,271,149,286)(140,280,150,285)(151,264,168,259)(152,263,169,258)(153,262,170,257)(154,261,161,256)(155,270,162,255)(156,269,163,254)(157,268,164,253)(158,267,165,252)(159,266,166,251)(160,265,167,260) );
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),(11,153,16,158),(12,152,17,157),(13,151,18,156),(14,160,19,155),(15,159,20,154),(21,174,26,179),(22,173,27,178),(23,172,28,177),(24,171,29,176),(25,180,30,175),(31,209,36,204),(32,208,37,203),(33,207,38,202),(34,206,39,201),(35,205,40,210),(41,194,46,199),(42,193,47,198),(43,192,48,197),(44,191,49,196),(45,200,50,195),(51,229,56,224),(52,228,57,223),(53,227,58,222),(54,226,59,221),(55,225,60,230),(61,214,66,219),(62,213,67,218),(63,212,68,217),(64,211,69,216),(65,220,70,215),(71,249,76,244),(72,248,77,243),(73,247,78,242),(74,246,79,241),(75,245,80,250),(81,234,86,239),(82,233,87,238),(83,232,88,237),(84,231,89,236),(85,240,90,235),(91,269,96,264),(92,268,97,263),(93,267,98,262),(94,266,99,261),(95,265,100,270),(101,254,106,259),(102,253,107,258),(103,252,108,257),(104,251,109,256),(105,260,110,255),(111,289,116,284),(112,288,117,283),(113,287,118,282),(114,286,119,281),(115,285,120,290),(121,274,126,279),(122,273,127,278),(123,272,128,277),(124,271,129,276),(125,280,130,275),(131,309,136,304),(132,308,137,303),(133,307,138,302),(134,306,139,301),(135,305,140,310),(141,294,146,299),(142,293,147,298),(143,292,148,297),(144,291,149,296),(145,300,150,295),(161,311,166,316),(162,320,167,315),(163,319,168,314),(164,318,169,313),(165,317,170,312)], [(1,158,38,148,28,165,48,138),(2,159,39,149,29,166,49,139),(3,160,40,150,30,167,50,140),(4,151,31,141,21,168,41,131),(5,152,32,142,22,169,42,132),(6,153,33,143,23,170,43,133),(7,154,34,144,24,161,44,134),(8,155,35,145,25,162,45,135),(9,156,36,146,26,163,46,136),(10,157,37,147,27,164,47,137),(11,202,297,177,317,197,302,182),(12,203,298,178,318,198,303,183),(13,204,299,179,319,199,304,184),(14,205,300,180,320,200,305,185),(15,206,291,171,311,191,306,186),(16,207,292,172,312,192,307,187),(17,208,293,173,313,193,308,188),(18,209,294,174,314,194,309,189),(19,210,295,175,315,195,310,190),(20,201,296,176,316,196,301,181),(51,116,81,96,61,126,71,106),(52,117,82,97,62,127,72,107),(53,118,83,98,63,128,73,108),(54,119,84,99,64,129,74,109),(55,120,85,100,65,130,75,110),(56,111,86,91,66,121,76,101),(57,112,87,92,67,122,77,102),(58,113,88,93,68,123,78,103),(59,114,89,94,69,124,79,104),(60,115,90,95,70,125,80,105),(211,276,246,256,226,281,231,261),(212,277,247,257,227,282,232,262),(213,278,248,258,228,283,233,263),(214,279,249,259,229,284,234,264),(215,280,250,260,230,285,235,265),(216,271,241,251,221,286,236,266),(217,272,242,252,222,287,237,267),(218,273,243,253,223,288,238,268),(219,274,244,254,224,289,239,269),(220,275,245,255,225,290,240,270)], [(1,217,28,222),(2,216,29,221),(3,215,30,230),(4,214,21,229),(5,213,22,228),(6,212,23,227),(7,211,24,226),(8,220,25,225),(9,219,26,224),(10,218,27,223),(11,98,317,108),(12,97,318,107),(13,96,319,106),(14,95,320,105),(15,94,311,104),(16,93,312,103),(17,92,313,102),(18,91,314,101),(19,100,315,110),(20,99,316,109),(31,234,41,249),(32,233,42,248),(33,232,43,247),(34,231,44,246),(35,240,45,245),(36,239,46,244),(37,238,47,243),(38,237,48,242),(39,236,49,241),(40,235,50,250),(51,184,61,179),(52,183,62,178),(53,182,63,177),(54,181,64,176),(55,190,65,175),(56,189,66,174),(57,188,67,173),(58,187,68,172),(59,186,69,171),(60,185,70,180),(71,204,81,199),(72,203,82,198),(73,202,83,197),(74,201,84,196),(75,210,85,195),(76,209,86,194),(77,208,87,193),(78,207,88,192),(79,206,89,191),(80,205,90,200),(111,309,121,294),(112,308,122,293),(113,307,123,292),(114,306,124,291),(115,305,125,300),(116,304,126,299),(117,303,127,298),(118,302,128,297),(119,301,129,296),(120,310,130,295),(131,279,141,284),(132,278,142,283),(133,277,143,282),(134,276,144,281),(135,275,145,290),(136,274,146,289),(137,273,147,288),(138,272,148,287),(139,271,149,286),(140,280,150,285),(151,264,168,259),(152,263,169,258),(153,262,170,257),(154,261,161,256),(155,270,162,255),(156,269,163,254),(157,268,164,253),(158,267,165,252),(159,266,166,251),(160,265,167,260)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | Q16 | C4○D4 | D10 | D10 | C4○D8 | C4×D5 | Q8⋊2D5 | D4×D5 | D8⋊3D5 | D5×Q16 |
| kernel | Dic5⋊5Q16 | C10.Q16 | C8×Dic5 | C5×C2.D8 | Dic5⋊3Q8 | C2×Dic20 | Dic20 | C2×Dic5 | C2.D8 | Dic5 | C20 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊5Q16 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 6 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 36 | 33 |
| 0 | 0 | 0 | 3 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 0 | 0 |
| 0 | 29 | 24 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 12 | 39 | 0 | 0 |
| 0 | 11 | 29 | 0 | 0 |
| 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 38 | 36 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,1,6],[32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,36,3,0,0,0,33,5],[1,0,0,0,0,0,0,29,0,0,0,24,24,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,12,11,0,0,0,39,29,0,0,0,0,0,5,38,0,0,0,8,36] >;
Dic5⋊5Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_5Q_{16} % in TeX
G:=Group("Dic5:5Q16"); // GroupNames label
G:=SmallGroup(320,500);
// by ID
G=gap.SmallGroup(320,500);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=c^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations