direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16×Dic5, C5⋊6(C4×Q16), (C5×Q16)⋊8C4, C2.5(D5×Q16), C40.60(C2×C4), (C2×Q16).7D5, (C2×C8).242D10, C10.127(C4×D4), (C10×Q16).5C2, C10.28(C2×Q16), Q8.1(C2×Dic5), (Q8×Dic5).8C2, (C8×Dic5).5C2, C40⋊5C4.16C2, C2.14(D4×Dic5), C8.10(C2×Dic5), C10.78(C4○D8), (C2×C40).94C22, (C2×Q8).119D10, C22.118(D4×D5), C20.104(C4○D4), C2.5(Q8.D10), C4.34(D4⋊2D5), C4.5(C22×Dic5), (C2×C20).457C23, C20.134(C22×C4), (C2×Dic5).282D4, Q8⋊Dic5.15C2, (Q8×C10).86C22, C4⋊Dic5.180C22, (C4×Dic5).275C22, (C5×Q8).23(C2×C4), (C2×C10).368(C2×D4), (C2×C4).545(C22×D5), (C2×C5⋊2C8).285C22, SmallGroup(320,810)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×Dic5
G = < a,b,c,d | a8=c10=1, b2=a4, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 310 in 110 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, 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, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C4×Q16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C2×C40, C5×Q16, Q8×C10, C8×Dic5, C40⋊5C4, Q8⋊Dic5, Q8×Dic5, C10×Q16, Q16×Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, Q16, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×Q16, C4○D8, C2×Dic5, C22×D5, C4×Q16, D4×D5, D4⋊2D5, C22×Dic5, D5×Q16, Q8.D10, D4×Dic5, Q16×Dic5
►Regular action on 320 points
Generators in S320
(1 141 12 154 39 170 48 139)(2 142 13 155 40 161 49 140)(3 143 14 156 31 162 50 131)(4 144 15 157 32 163 41 132)(5 145 16 158 33 164 42 133)(6 146 17 159 34 165 43 134)(7 147 18 160 35 166 44 135)(8 148 19 151 36 167 45 136)(9 149 20 152 37 168 46 137)(10 150 11 153 38 169 47 138)(21 207 301 186 299 174 316 191)(22 208 302 187 300 175 317 192)(23 209 303 188 291 176 318 193)(24 210 304 189 292 177 319 194)(25 201 305 190 293 178 320 195)(26 202 306 181 294 179 311 196)(27 203 307 182 295 180 312 197)(28 204 308 183 296 171 313 198)(29 205 309 184 297 172 314 199)(30 206 310 185 298 173 315 200)(51 108 85 96 76 130 63 111)(52 109 86 97 77 121 64 112)(53 110 87 98 78 122 65 113)(54 101 88 99 79 123 66 114)(55 102 89 100 80 124 67 115)(56 103 90 91 71 125 68 116)(57 104 81 92 72 126 69 117)(58 105 82 93 73 127 70 118)(59 106 83 94 74 128 61 119)(60 107 84 95 75 129 62 120)(211 273 223 256 244 268 238 289)(212 274 224 257 245 269 239 290)(213 275 225 258 246 270 240 281)(214 276 226 259 247 261 231 282)(215 277 227 260 248 262 232 283)(216 278 228 251 249 263 233 284)(217 279 229 252 250 264 234 285)(218 280 230 253 241 265 235 286)(219 271 221 254 242 266 236 287)(220 272 222 255 243 267 237 288)
(1 74 39 59)(2 75 40 60)(3 76 31 51)(4 77 32 52)(5 78 33 53)(6 79 34 54)(7 80 35 55)(8 71 36 56)(9 72 37 57)(10 73 38 58)(11 82 47 70)(12 83 48 61)(13 84 49 62)(14 85 50 63)(15 86 41 64)(16 87 42 65)(17 88 43 66)(18 89 44 67)(19 90 45 68)(20 81 46 69)(21 271 299 266)(22 272 300 267)(23 273 291 268)(24 274 292 269)(25 275 293 270)(26 276 294 261)(27 277 295 262)(28 278 296 263)(29 279 297 264)(30 280 298 265)(91 167 116 148)(92 168 117 149)(93 169 118 150)(94 170 119 141)(95 161 120 142)(96 162 111 143)(97 163 112 144)(98 164 113 145)(99 165 114 146)(100 166 115 147)(101 134 123 159)(102 135 124 160)(103 136 125 151)(104 137 126 152)(105 138 127 153)(106 139 128 154)(107 140 129 155)(108 131 130 156)(109 132 121 157)(110 133 122 158)(171 249 204 216)(172 250 205 217)(173 241 206 218)(174 242 207 219)(175 243 208 220)(176 244 209 211)(177 245 210 212)(178 246 201 213)(179 247 202 214)(180 248 203 215)(181 231 196 226)(182 232 197 227)(183 233 198 228)(184 234 199 229)(185 235 200 230)(186 236 191 221)(187 237 192 222)(188 238 193 223)(189 239 194 224)(190 240 195 225)(251 308 284 313)(252 309 285 314)(253 310 286 315)(254 301 287 316)(255 302 288 317)(256 303 289 318)(257 304 290 319)(258 305 281 320)(259 306 282 311)(260 307 283 312)
(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)
(1 196 6 191)(2 195 7 200)(3 194 8 199)(4 193 9 198)(5 192 10 197)(11 203 16 208)(12 202 17 207)(13 201 18 206)(14 210 19 205)(15 209 20 204)(21 141 26 146)(22 150 27 145)(23 149 28 144)(24 148 29 143)(25 147 30 142)(31 189 36 184)(32 188 37 183)(33 187 38 182)(34 186 39 181)(35 185 40 190)(41 176 46 171)(42 175 47 180)(43 174 48 179)(44 173 49 178)(45 172 50 177)(51 239 56 234)(52 238 57 233)(53 237 58 232)(54 236 59 231)(55 235 60 240)(61 247 66 242)(62 246 67 241)(63 245 68 250)(64 244 69 249)(65 243 70 248)(71 229 76 224)(72 228 77 223)(73 227 78 222)(74 226 79 221)(75 225 80 230)(81 216 86 211)(82 215 87 220)(83 214 88 219)(84 213 89 218)(85 212 90 217)(91 279 96 274)(92 278 97 273)(93 277 98 272)(94 276 99 271)(95 275 100 280)(101 287 106 282)(102 286 107 281)(103 285 108 290)(104 284 109 289)(105 283 110 288)(111 269 116 264)(112 268 117 263)(113 267 118 262)(114 266 119 261)(115 265 120 270)(121 256 126 251)(122 255 127 260)(123 254 128 259)(124 253 129 258)(125 252 130 257)(131 319 136 314)(132 318 137 313)(133 317 138 312)(134 316 139 311)(135 315 140 320)(151 309 156 304)(152 308 157 303)(153 307 158 302)(154 306 159 301)(155 305 160 310)(161 293 166 298)(162 292 167 297)(163 291 168 296)(164 300 169 295)(165 299 170 294)
G:=sub<Sym(320)| (1,141,12,154,39,170,48,139)(2,142,13,155,40,161,49,140)(3,143,14,156,31,162,50,131)(4,144,15,157,32,163,41,132)(5,145,16,158,33,164,42,133)(6,146,17,159,34,165,43,134)(7,147,18,160,35,166,44,135)(8,148,19,151,36,167,45,136)(9,149,20,152,37,168,46,137)(10,150,11,153,38,169,47,138)(21,207,301,186,299,174,316,191)(22,208,302,187,300,175,317,192)(23,209,303,188,291,176,318,193)(24,210,304,189,292,177,319,194)(25,201,305,190,293,178,320,195)(26,202,306,181,294,179,311,196)(27,203,307,182,295,180,312,197)(28,204,308,183,296,171,313,198)(29,205,309,184,297,172,314,199)(30,206,310,185,298,173,315,200)(51,108,85,96,76,130,63,111)(52,109,86,97,77,121,64,112)(53,110,87,98,78,122,65,113)(54,101,88,99,79,123,66,114)(55,102,89,100,80,124,67,115)(56,103,90,91,71,125,68,116)(57,104,81,92,72,126,69,117)(58,105,82,93,73,127,70,118)(59,106,83,94,74,128,61,119)(60,107,84,95,75,129,62,120)(211,273,223,256,244,268,238,289)(212,274,224,257,245,269,239,290)(213,275,225,258,246,270,240,281)(214,276,226,259,247,261,231,282)(215,277,227,260,248,262,232,283)(216,278,228,251,249,263,233,284)(217,279,229,252,250,264,234,285)(218,280,230,253,241,265,235,286)(219,271,221,254,242,266,236,287)(220,272,222,255,243,267,237,288), (1,74,39,59)(2,75,40,60)(3,76,31,51)(4,77,32,52)(5,78,33,53)(6,79,34,54)(7,80,35,55)(8,71,36,56)(9,72,37,57)(10,73,38,58)(11,82,47,70)(12,83,48,61)(13,84,49,62)(14,85,50,63)(15,86,41,64)(16,87,42,65)(17,88,43,66)(18,89,44,67)(19,90,45,68)(20,81,46,69)(21,271,299,266)(22,272,300,267)(23,273,291,268)(24,274,292,269)(25,275,293,270)(26,276,294,261)(27,277,295,262)(28,278,296,263)(29,279,297,264)(30,280,298,265)(91,167,116,148)(92,168,117,149)(93,169,118,150)(94,170,119,141)(95,161,120,142)(96,162,111,143)(97,163,112,144)(98,164,113,145)(99,165,114,146)(100,166,115,147)(101,134,123,159)(102,135,124,160)(103,136,125,151)(104,137,126,152)(105,138,127,153)(106,139,128,154)(107,140,129,155)(108,131,130,156)(109,132,121,157)(110,133,122,158)(171,249,204,216)(172,250,205,217)(173,241,206,218)(174,242,207,219)(175,243,208,220)(176,244,209,211)(177,245,210,212)(178,246,201,213)(179,247,202,214)(180,248,203,215)(181,231,196,226)(182,232,197,227)(183,233,198,228)(184,234,199,229)(185,235,200,230)(186,236,191,221)(187,237,192,222)(188,238,193,223)(189,239,194,224)(190,240,195,225)(251,308,284,313)(252,309,285,314)(253,310,286,315)(254,301,287,316)(255,302,288,317)(256,303,289,318)(257,304,290,319)(258,305,281,320)(259,306,282,311)(260,307,283,312), (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), (1,196,6,191)(2,195,7,200)(3,194,8,199)(4,193,9,198)(5,192,10,197)(11,203,16,208)(12,202,17,207)(13,201,18,206)(14,210,19,205)(15,209,20,204)(21,141,26,146)(22,150,27,145)(23,149,28,144)(24,148,29,143)(25,147,30,142)(31,189,36,184)(32,188,37,183)(33,187,38,182)(34,186,39,181)(35,185,40,190)(41,176,46,171)(42,175,47,180)(43,174,48,179)(44,173,49,178)(45,172,50,177)(51,239,56,234)(52,238,57,233)(53,237,58,232)(54,236,59,231)(55,235,60,240)(61,247,66,242)(62,246,67,241)(63,245,68,250)(64,244,69,249)(65,243,70,248)(71,229,76,224)(72,228,77,223)(73,227,78,222)(74,226,79,221)(75,225,80,230)(81,216,86,211)(82,215,87,220)(83,214,88,219)(84,213,89,218)(85,212,90,217)(91,279,96,274)(92,278,97,273)(93,277,98,272)(94,276,99,271)(95,275,100,280)(101,287,106,282)(102,286,107,281)(103,285,108,290)(104,284,109,289)(105,283,110,288)(111,269,116,264)(112,268,117,263)(113,267,118,262)(114,266,119,261)(115,265,120,270)(121,256,126,251)(122,255,127,260)(123,254,128,259)(124,253,129,258)(125,252,130,257)(131,319,136,314)(132,318,137,313)(133,317,138,312)(134,316,139,311)(135,315,140,320)(151,309,156,304)(152,308,157,303)(153,307,158,302)(154,306,159,301)(155,305,160,310)(161,293,166,298)(162,292,167,297)(163,291,168,296)(164,300,169,295)(165,299,170,294)>;
G:=Group( (1,141,12,154,39,170,48,139)(2,142,13,155,40,161,49,140)(3,143,14,156,31,162,50,131)(4,144,15,157,32,163,41,132)(5,145,16,158,33,164,42,133)(6,146,17,159,34,165,43,134)(7,147,18,160,35,166,44,135)(8,148,19,151,36,167,45,136)(9,149,20,152,37,168,46,137)(10,150,11,153,38,169,47,138)(21,207,301,186,299,174,316,191)(22,208,302,187,300,175,317,192)(23,209,303,188,291,176,318,193)(24,210,304,189,292,177,319,194)(25,201,305,190,293,178,320,195)(26,202,306,181,294,179,311,196)(27,203,307,182,295,180,312,197)(28,204,308,183,296,171,313,198)(29,205,309,184,297,172,314,199)(30,206,310,185,298,173,315,200)(51,108,85,96,76,130,63,111)(52,109,86,97,77,121,64,112)(53,110,87,98,78,122,65,113)(54,101,88,99,79,123,66,114)(55,102,89,100,80,124,67,115)(56,103,90,91,71,125,68,116)(57,104,81,92,72,126,69,117)(58,105,82,93,73,127,70,118)(59,106,83,94,74,128,61,119)(60,107,84,95,75,129,62,120)(211,273,223,256,244,268,238,289)(212,274,224,257,245,269,239,290)(213,275,225,258,246,270,240,281)(214,276,226,259,247,261,231,282)(215,277,227,260,248,262,232,283)(216,278,228,251,249,263,233,284)(217,279,229,252,250,264,234,285)(218,280,230,253,241,265,235,286)(219,271,221,254,242,266,236,287)(220,272,222,255,243,267,237,288), (1,74,39,59)(2,75,40,60)(3,76,31,51)(4,77,32,52)(5,78,33,53)(6,79,34,54)(7,80,35,55)(8,71,36,56)(9,72,37,57)(10,73,38,58)(11,82,47,70)(12,83,48,61)(13,84,49,62)(14,85,50,63)(15,86,41,64)(16,87,42,65)(17,88,43,66)(18,89,44,67)(19,90,45,68)(20,81,46,69)(21,271,299,266)(22,272,300,267)(23,273,291,268)(24,274,292,269)(25,275,293,270)(26,276,294,261)(27,277,295,262)(28,278,296,263)(29,279,297,264)(30,280,298,265)(91,167,116,148)(92,168,117,149)(93,169,118,150)(94,170,119,141)(95,161,120,142)(96,162,111,143)(97,163,112,144)(98,164,113,145)(99,165,114,146)(100,166,115,147)(101,134,123,159)(102,135,124,160)(103,136,125,151)(104,137,126,152)(105,138,127,153)(106,139,128,154)(107,140,129,155)(108,131,130,156)(109,132,121,157)(110,133,122,158)(171,249,204,216)(172,250,205,217)(173,241,206,218)(174,242,207,219)(175,243,208,220)(176,244,209,211)(177,245,210,212)(178,246,201,213)(179,247,202,214)(180,248,203,215)(181,231,196,226)(182,232,197,227)(183,233,198,228)(184,234,199,229)(185,235,200,230)(186,236,191,221)(187,237,192,222)(188,238,193,223)(189,239,194,224)(190,240,195,225)(251,308,284,313)(252,309,285,314)(253,310,286,315)(254,301,287,316)(255,302,288,317)(256,303,289,318)(257,304,290,319)(258,305,281,320)(259,306,282,311)(260,307,283,312), (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), (1,196,6,191)(2,195,7,200)(3,194,8,199)(4,193,9,198)(5,192,10,197)(11,203,16,208)(12,202,17,207)(13,201,18,206)(14,210,19,205)(15,209,20,204)(21,141,26,146)(22,150,27,145)(23,149,28,144)(24,148,29,143)(25,147,30,142)(31,189,36,184)(32,188,37,183)(33,187,38,182)(34,186,39,181)(35,185,40,190)(41,176,46,171)(42,175,47,180)(43,174,48,179)(44,173,49,178)(45,172,50,177)(51,239,56,234)(52,238,57,233)(53,237,58,232)(54,236,59,231)(55,235,60,240)(61,247,66,242)(62,246,67,241)(63,245,68,250)(64,244,69,249)(65,243,70,248)(71,229,76,224)(72,228,77,223)(73,227,78,222)(74,226,79,221)(75,225,80,230)(81,216,86,211)(82,215,87,220)(83,214,88,219)(84,213,89,218)(85,212,90,217)(91,279,96,274)(92,278,97,273)(93,277,98,272)(94,276,99,271)(95,275,100,280)(101,287,106,282)(102,286,107,281)(103,285,108,290)(104,284,109,289)(105,283,110,288)(111,269,116,264)(112,268,117,263)(113,267,118,262)(114,266,119,261)(115,265,120,270)(121,256,126,251)(122,255,127,260)(123,254,128,259)(124,253,129,258)(125,252,130,257)(131,319,136,314)(132,318,137,313)(133,317,138,312)(134,316,139,311)(135,315,140,320)(151,309,156,304)(152,308,157,303)(153,307,158,302)(154,306,159,301)(155,305,160,310)(161,293,166,298)(162,292,167,297)(163,291,168,296)(164,300,169,295)(165,299,170,294) );
G=PermutationGroup([[(1,141,12,154,39,170,48,139),(2,142,13,155,40,161,49,140),(3,143,14,156,31,162,50,131),(4,144,15,157,32,163,41,132),(5,145,16,158,33,164,42,133),(6,146,17,159,34,165,43,134),(7,147,18,160,35,166,44,135),(8,148,19,151,36,167,45,136),(9,149,20,152,37,168,46,137),(10,150,11,153,38,169,47,138),(21,207,301,186,299,174,316,191),(22,208,302,187,300,175,317,192),(23,209,303,188,291,176,318,193),(24,210,304,189,292,177,319,194),(25,201,305,190,293,178,320,195),(26,202,306,181,294,179,311,196),(27,203,307,182,295,180,312,197),(28,204,308,183,296,171,313,198),(29,205,309,184,297,172,314,199),(30,206,310,185,298,173,315,200),(51,108,85,96,76,130,63,111),(52,109,86,97,77,121,64,112),(53,110,87,98,78,122,65,113),(54,101,88,99,79,123,66,114),(55,102,89,100,80,124,67,115),(56,103,90,91,71,125,68,116),(57,104,81,92,72,126,69,117),(58,105,82,93,73,127,70,118),(59,106,83,94,74,128,61,119),(60,107,84,95,75,129,62,120),(211,273,223,256,244,268,238,289),(212,274,224,257,245,269,239,290),(213,275,225,258,246,270,240,281),(214,276,226,259,247,261,231,282),(215,277,227,260,248,262,232,283),(216,278,228,251,249,263,233,284),(217,279,229,252,250,264,234,285),(218,280,230,253,241,265,235,286),(219,271,221,254,242,266,236,287),(220,272,222,255,243,267,237,288)], [(1,74,39,59),(2,75,40,60),(3,76,31,51),(4,77,32,52),(5,78,33,53),(6,79,34,54),(7,80,35,55),(8,71,36,56),(9,72,37,57),(10,73,38,58),(11,82,47,70),(12,83,48,61),(13,84,49,62),(14,85,50,63),(15,86,41,64),(16,87,42,65),(17,88,43,66),(18,89,44,67),(19,90,45,68),(20,81,46,69),(21,271,299,266),(22,272,300,267),(23,273,291,268),(24,274,292,269),(25,275,293,270),(26,276,294,261),(27,277,295,262),(28,278,296,263),(29,279,297,264),(30,280,298,265),(91,167,116,148),(92,168,117,149),(93,169,118,150),(94,170,119,141),(95,161,120,142),(96,162,111,143),(97,163,112,144),(98,164,113,145),(99,165,114,146),(100,166,115,147),(101,134,123,159),(102,135,124,160),(103,136,125,151),(104,137,126,152),(105,138,127,153),(106,139,128,154),(107,140,129,155),(108,131,130,156),(109,132,121,157),(110,133,122,158),(171,249,204,216),(172,250,205,217),(173,241,206,218),(174,242,207,219),(175,243,208,220),(176,244,209,211),(177,245,210,212),(178,246,201,213),(179,247,202,214),(180,248,203,215),(181,231,196,226),(182,232,197,227),(183,233,198,228),(184,234,199,229),(185,235,200,230),(186,236,191,221),(187,237,192,222),(188,238,193,223),(189,239,194,224),(190,240,195,225),(251,308,284,313),(252,309,285,314),(253,310,286,315),(254,301,287,316),(255,302,288,317),(256,303,289,318),(257,304,290,319),(258,305,281,320),(259,306,282,311),(260,307,283,312)], [(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)], [(1,196,6,191),(2,195,7,200),(3,194,8,199),(4,193,9,198),(5,192,10,197),(11,203,16,208),(12,202,17,207),(13,201,18,206),(14,210,19,205),(15,209,20,204),(21,141,26,146),(22,150,27,145),(23,149,28,144),(24,148,29,143),(25,147,30,142),(31,189,36,184),(32,188,37,183),(33,187,38,182),(34,186,39,181),(35,185,40,190),(41,176,46,171),(42,175,47,180),(43,174,48,179),(44,173,49,178),(45,172,50,177),(51,239,56,234),(52,238,57,233),(53,237,58,232),(54,236,59,231),(55,235,60,240),(61,247,66,242),(62,246,67,241),(63,245,68,250),(64,244,69,249),(65,243,70,248),(71,229,76,224),(72,228,77,223),(73,227,78,222),(74,226,79,221),(75,225,80,230),(81,216,86,211),(82,215,87,220),(83,214,88,219),(84,213,89,218),(85,212,90,217),(91,279,96,274),(92,278,97,273),(93,277,98,272),(94,276,99,271),(95,275,100,280),(101,287,106,282),(102,286,107,281),(103,285,108,290),(104,284,109,289),(105,283,110,288),(111,269,116,264),(112,268,117,263),(113,267,118,262),(114,266,119,261),(115,265,120,270),(121,256,126,251),(122,255,127,260),(123,254,128,259),(124,253,129,258),(125,252,130,257),(131,319,136,314),(132,318,137,313),(133,317,138,312),(134,316,139,311),(135,315,140,320),(151,309,156,304),(152,308,157,303),(153,307,158,302),(154,306,159,301),(155,305,160,310),(161,293,166,298),(162,292,167,297),(163,291,168,296),(164,300,169,295),(165,299,170,294)]])
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 | Dic5 | D10 | C4○D8 | D4⋊2D5 | D4×D5 | D5×Q16 | Q8.D10 |
| kernel | Q16×Dic5 | C8×Dic5 | C40⋊5C4 | Q8⋊Dic5 | Q8×Dic5 | C10×Q16 | C5×Q16 | C2×Dic5 | C2×Q16 | Dic5 | C20 | C2×C8 | Q16 | C2×Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 8 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of Q16×Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 20 |
| 0 | 0 | 20 | 38 |
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 9 | 0 | 0 |
| 19 | 19 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,3,20,0,0,20,38],[0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[22,19,0,0,9,19,0,0,0,0,1,0,0,0,0,1] >;
Q16×Dic5 in GAP, Magma, Sage, TeX
Q_{16}\times {\rm Dic}_5 % in TeX
G:=Group("Q16xDic5"); // GroupNames label
G:=SmallGroup(320,810);
// by ID
G=gap.SmallGroup(320,810);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^10=1,b^2=a^4,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations