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G = Q16×Dic5order 320 = 26·5

Direct product of Q16 and Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q16×Dic5, C56(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, C405C4.16C2, 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(D42D5), 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×C52C8).285C22, SmallGroup(320,810)

Series: Derived Chief Lower central Upper central

C1C20 — Q16×Dic5
C1C5C10C2×C10C2×C20C4×Dic5Q8×Dic5 — Q16×Dic5
C5C10C20 — Q16×Dic5
C1C22C2×C4C2×Q16

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 [×3], C4 [×2], C4 [×9], C22, C5, C8 [×2], C8, C2×C4, C2×C4 [×6], Q8 [×4], Q8 [×2], C10 [×3], C42 [×3], C4⋊C4 [×4], C2×C8, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×4], C2×C10, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C52C8, C40 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C4×Q16, C2×C52C8, C4×Dic5, C4×Dic5 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C2×C40, C5×Q16 [×4], Q8×C10 [×2], C8×Dic5, C405C4, Q8⋊Dic5 [×2], Q8×Dic5 [×2], C10×Q16, Q16×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, Q16 [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×Q16, C4○D8, C2×Dic5 [×6], C22×D5, C4×Q16, D4×D5, D42D5, C22×Dic5, D5×Q16, Q8.D10, D4×Dic5, Q16×Dic5

Smallest permutation representation of Q16×Dic5
Regular action on 320 points
Generators in S320
(1 147 16 154 39 170 48 131)(2 148 17 155 40 161 49 132)(3 149 18 156 31 162 50 133)(4 150 19 157 32 163 41 134)(5 141 20 158 33 164 42 135)(6 142 11 159 34 165 43 136)(7 143 12 160 35 166 44 137)(8 144 13 151 36 167 45 138)(9 145 14 152 37 168 46 139)(10 146 15 153 38 169 47 140)(21 207 294 179 303 188 316 191)(22 208 295 180 304 189 317 192)(23 209 296 171 305 190 318 193)(24 210 297 172 306 181 319 194)(25 201 298 173 307 182 320 195)(26 202 299 174 308 183 311 196)(27 203 300 175 309 184 312 197)(28 204 291 176 310 185 313 198)(29 205 292 177 301 186 314 199)(30 206 293 178 302 187 315 200)(51 102 83 96 74 128 67 119)(52 103 84 97 75 129 68 120)(53 104 85 98 76 130 69 111)(54 105 86 99 77 121 70 112)(55 106 87 100 78 122 61 113)(56 107 88 91 79 123 62 114)(57 108 89 92 80 124 63 115)(58 109 90 93 71 125 64 116)(59 110 81 94 72 126 65 117)(60 101 82 95 73 127 66 118)(211 265 249 256 233 284 230 278)(212 266 250 257 234 285 221 279)(213 267 241 258 235 286 222 280)(214 268 242 259 236 287 223 271)(215 269 243 260 237 288 224 272)(216 270 244 251 238 289 225 273)(217 261 245 252 239 290 226 274)(218 262 246 253 240 281 227 275)(219 263 247 254 231 282 228 276)(220 264 248 255 232 283 229 277)
(1 74 39 51)(2 75 40 52)(3 76 31 53)(4 77 32 54)(5 78 33 55)(6 79 34 56)(7 80 35 57)(8 71 36 58)(9 72 37 59)(10 73 38 60)(11 88 43 62)(12 89 44 63)(13 90 45 64)(14 81 46 65)(15 82 47 66)(16 83 48 67)(17 84 49 68)(18 85 50 69)(19 86 41 70)(20 87 42 61)(21 271 303 259)(22 272 304 260)(23 273 305 251)(24 274 306 252)(25 275 307 253)(26 276 308 254)(27 277 309 255)(28 278 310 256)(29 279 301 257)(30 280 302 258)(91 165 114 142)(92 166 115 143)(93 167 116 144)(94 168 117 145)(95 169 118 146)(96 170 119 147)(97 161 120 148)(98 162 111 149)(99 163 112 150)(100 164 113 141)(101 140 127 153)(102 131 128 154)(103 132 129 155)(104 133 130 156)(105 134 121 157)(106 135 122 158)(107 136 123 159)(108 137 124 160)(109 138 125 151)(110 139 126 152)(171 238 193 216)(172 239 194 217)(173 240 195 218)(174 231 196 219)(175 232 197 220)(176 233 198 211)(177 234 199 212)(178 235 200 213)(179 236 191 214)(180 237 192 215)(181 245 210 226)(182 246 201 227)(183 247 202 228)(184 248 203 229)(185 249 204 230)(186 250 205 221)(187 241 206 222)(188 242 207 223)(189 243 208 224)(190 244 209 225)(261 297 290 319)(262 298 281 320)(263 299 282 311)(264 300 283 312)(265 291 284 313)(266 292 285 314)(267 293 286 315)(268 294 287 316)(269 295 288 317)(270 296 289 318)
(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)(11 207 16 202)(12 206 17 201)(13 205 18 210)(14 204 19 209)(15 203 20 208)(21 147 26 142)(22 146 27 141)(23 145 28 150)(24 144 29 149)(25 143 30 148)(31 172 36 177)(32 171 37 176)(33 180 38 175)(34 179 39 174)(35 178 40 173)(41 190 46 185)(42 189 47 184)(43 188 48 183)(44 187 49 182)(45 186 50 181)(51 231 56 236)(52 240 57 235)(53 239 58 234)(54 238 59 233)(55 237 60 232)(61 243 66 248)(62 242 67 247)(63 241 68 246)(64 250 69 245)(65 249 70 244)(71 212 76 217)(72 211 77 216)(73 220 78 215)(74 219 79 214)(75 218 80 213)(81 230 86 225)(82 229 87 224)(83 228 88 223)(84 227 89 222)(85 226 90 221)(91 271 96 276)(92 280 97 275)(93 279 98 274)(94 278 99 273)(95 277 100 272)(101 283 106 288)(102 282 107 287)(103 281 108 286)(104 290 109 285)(105 289 110 284)(111 252 116 257)(112 251 117 256)(113 260 118 255)(114 259 119 254)(115 258 120 253)(121 270 126 265)(122 269 127 264)(123 268 128 263)(124 267 129 262)(125 266 130 261)(131 311 136 316)(132 320 137 315)(133 319 138 314)(134 318 139 313)(135 317 140 312)(151 292 156 297)(152 291 157 296)(153 300 158 295)(154 299 159 294)(155 298 160 293)(161 307 166 302)(162 306 167 301)(163 305 168 310)(164 304 169 309)(165 303 170 308)

G:=sub<Sym(320)| (1,147,16,154,39,170,48,131)(2,148,17,155,40,161,49,132)(3,149,18,156,31,162,50,133)(4,150,19,157,32,163,41,134)(5,141,20,158,33,164,42,135)(6,142,11,159,34,165,43,136)(7,143,12,160,35,166,44,137)(8,144,13,151,36,167,45,138)(9,145,14,152,37,168,46,139)(10,146,15,153,38,169,47,140)(21,207,294,179,303,188,316,191)(22,208,295,180,304,189,317,192)(23,209,296,171,305,190,318,193)(24,210,297,172,306,181,319,194)(25,201,298,173,307,182,320,195)(26,202,299,174,308,183,311,196)(27,203,300,175,309,184,312,197)(28,204,291,176,310,185,313,198)(29,205,292,177,301,186,314,199)(30,206,293,178,302,187,315,200)(51,102,83,96,74,128,67,119)(52,103,84,97,75,129,68,120)(53,104,85,98,76,130,69,111)(54,105,86,99,77,121,70,112)(55,106,87,100,78,122,61,113)(56,107,88,91,79,123,62,114)(57,108,89,92,80,124,63,115)(58,109,90,93,71,125,64,116)(59,110,81,94,72,126,65,117)(60,101,82,95,73,127,66,118)(211,265,249,256,233,284,230,278)(212,266,250,257,234,285,221,279)(213,267,241,258,235,286,222,280)(214,268,242,259,236,287,223,271)(215,269,243,260,237,288,224,272)(216,270,244,251,238,289,225,273)(217,261,245,252,239,290,226,274)(218,262,246,253,240,281,227,275)(219,263,247,254,231,282,228,276)(220,264,248,255,232,283,229,277), (1,74,39,51)(2,75,40,52)(3,76,31,53)(4,77,32,54)(5,78,33,55)(6,79,34,56)(7,80,35,57)(8,71,36,58)(9,72,37,59)(10,73,38,60)(11,88,43,62)(12,89,44,63)(13,90,45,64)(14,81,46,65)(15,82,47,66)(16,83,48,67)(17,84,49,68)(18,85,50,69)(19,86,41,70)(20,87,42,61)(21,271,303,259)(22,272,304,260)(23,273,305,251)(24,274,306,252)(25,275,307,253)(26,276,308,254)(27,277,309,255)(28,278,310,256)(29,279,301,257)(30,280,302,258)(91,165,114,142)(92,166,115,143)(93,167,116,144)(94,168,117,145)(95,169,118,146)(96,170,119,147)(97,161,120,148)(98,162,111,149)(99,163,112,150)(100,164,113,141)(101,140,127,153)(102,131,128,154)(103,132,129,155)(104,133,130,156)(105,134,121,157)(106,135,122,158)(107,136,123,159)(108,137,124,160)(109,138,125,151)(110,139,126,152)(171,238,193,216)(172,239,194,217)(173,240,195,218)(174,231,196,219)(175,232,197,220)(176,233,198,211)(177,234,199,212)(178,235,200,213)(179,236,191,214)(180,237,192,215)(181,245,210,226)(182,246,201,227)(183,247,202,228)(184,248,203,229)(185,249,204,230)(186,250,205,221)(187,241,206,222)(188,242,207,223)(189,243,208,224)(190,244,209,225)(261,297,290,319)(262,298,281,320)(263,299,282,311)(264,300,283,312)(265,291,284,313)(266,292,285,314)(267,293,286,315)(268,294,287,316)(269,295,288,317)(270,296,289,318), (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)(11,207,16,202)(12,206,17,201)(13,205,18,210)(14,204,19,209)(15,203,20,208)(21,147,26,142)(22,146,27,141)(23,145,28,150)(24,144,29,149)(25,143,30,148)(31,172,36,177)(32,171,37,176)(33,180,38,175)(34,179,39,174)(35,178,40,173)(41,190,46,185)(42,189,47,184)(43,188,48,183)(44,187,49,182)(45,186,50,181)(51,231,56,236)(52,240,57,235)(53,239,58,234)(54,238,59,233)(55,237,60,232)(61,243,66,248)(62,242,67,247)(63,241,68,246)(64,250,69,245)(65,249,70,244)(71,212,76,217)(72,211,77,216)(73,220,78,215)(74,219,79,214)(75,218,80,213)(81,230,86,225)(82,229,87,224)(83,228,88,223)(84,227,89,222)(85,226,90,221)(91,271,96,276)(92,280,97,275)(93,279,98,274)(94,278,99,273)(95,277,100,272)(101,283,106,288)(102,282,107,287)(103,281,108,286)(104,290,109,285)(105,289,110,284)(111,252,116,257)(112,251,117,256)(113,260,118,255)(114,259,119,254)(115,258,120,253)(121,270,126,265)(122,269,127,264)(123,268,128,263)(124,267,129,262)(125,266,130,261)(131,311,136,316)(132,320,137,315)(133,319,138,314)(134,318,139,313)(135,317,140,312)(151,292,156,297)(152,291,157,296)(153,300,158,295)(154,299,159,294)(155,298,160,293)(161,307,166,302)(162,306,167,301)(163,305,168,310)(164,304,169,309)(165,303,170,308)>;

G:=Group( (1,147,16,154,39,170,48,131)(2,148,17,155,40,161,49,132)(3,149,18,156,31,162,50,133)(4,150,19,157,32,163,41,134)(5,141,20,158,33,164,42,135)(6,142,11,159,34,165,43,136)(7,143,12,160,35,166,44,137)(8,144,13,151,36,167,45,138)(9,145,14,152,37,168,46,139)(10,146,15,153,38,169,47,140)(21,207,294,179,303,188,316,191)(22,208,295,180,304,189,317,192)(23,209,296,171,305,190,318,193)(24,210,297,172,306,181,319,194)(25,201,298,173,307,182,320,195)(26,202,299,174,308,183,311,196)(27,203,300,175,309,184,312,197)(28,204,291,176,310,185,313,198)(29,205,292,177,301,186,314,199)(30,206,293,178,302,187,315,200)(51,102,83,96,74,128,67,119)(52,103,84,97,75,129,68,120)(53,104,85,98,76,130,69,111)(54,105,86,99,77,121,70,112)(55,106,87,100,78,122,61,113)(56,107,88,91,79,123,62,114)(57,108,89,92,80,124,63,115)(58,109,90,93,71,125,64,116)(59,110,81,94,72,126,65,117)(60,101,82,95,73,127,66,118)(211,265,249,256,233,284,230,278)(212,266,250,257,234,285,221,279)(213,267,241,258,235,286,222,280)(214,268,242,259,236,287,223,271)(215,269,243,260,237,288,224,272)(216,270,244,251,238,289,225,273)(217,261,245,252,239,290,226,274)(218,262,246,253,240,281,227,275)(219,263,247,254,231,282,228,276)(220,264,248,255,232,283,229,277), (1,74,39,51)(2,75,40,52)(3,76,31,53)(4,77,32,54)(5,78,33,55)(6,79,34,56)(7,80,35,57)(8,71,36,58)(9,72,37,59)(10,73,38,60)(11,88,43,62)(12,89,44,63)(13,90,45,64)(14,81,46,65)(15,82,47,66)(16,83,48,67)(17,84,49,68)(18,85,50,69)(19,86,41,70)(20,87,42,61)(21,271,303,259)(22,272,304,260)(23,273,305,251)(24,274,306,252)(25,275,307,253)(26,276,308,254)(27,277,309,255)(28,278,310,256)(29,279,301,257)(30,280,302,258)(91,165,114,142)(92,166,115,143)(93,167,116,144)(94,168,117,145)(95,169,118,146)(96,170,119,147)(97,161,120,148)(98,162,111,149)(99,163,112,150)(100,164,113,141)(101,140,127,153)(102,131,128,154)(103,132,129,155)(104,133,130,156)(105,134,121,157)(106,135,122,158)(107,136,123,159)(108,137,124,160)(109,138,125,151)(110,139,126,152)(171,238,193,216)(172,239,194,217)(173,240,195,218)(174,231,196,219)(175,232,197,220)(176,233,198,211)(177,234,199,212)(178,235,200,213)(179,236,191,214)(180,237,192,215)(181,245,210,226)(182,246,201,227)(183,247,202,228)(184,248,203,229)(185,249,204,230)(186,250,205,221)(187,241,206,222)(188,242,207,223)(189,243,208,224)(190,244,209,225)(261,297,290,319)(262,298,281,320)(263,299,282,311)(264,300,283,312)(265,291,284,313)(266,292,285,314)(267,293,286,315)(268,294,287,316)(269,295,288,317)(270,296,289,318), (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)(11,207,16,202)(12,206,17,201)(13,205,18,210)(14,204,19,209)(15,203,20,208)(21,147,26,142)(22,146,27,141)(23,145,28,150)(24,144,29,149)(25,143,30,148)(31,172,36,177)(32,171,37,176)(33,180,38,175)(34,179,39,174)(35,178,40,173)(41,190,46,185)(42,189,47,184)(43,188,48,183)(44,187,49,182)(45,186,50,181)(51,231,56,236)(52,240,57,235)(53,239,58,234)(54,238,59,233)(55,237,60,232)(61,243,66,248)(62,242,67,247)(63,241,68,246)(64,250,69,245)(65,249,70,244)(71,212,76,217)(72,211,77,216)(73,220,78,215)(74,219,79,214)(75,218,80,213)(81,230,86,225)(82,229,87,224)(83,228,88,223)(84,227,89,222)(85,226,90,221)(91,271,96,276)(92,280,97,275)(93,279,98,274)(94,278,99,273)(95,277,100,272)(101,283,106,288)(102,282,107,287)(103,281,108,286)(104,290,109,285)(105,289,110,284)(111,252,116,257)(112,251,117,256)(113,260,118,255)(114,259,119,254)(115,258,120,253)(121,270,126,265)(122,269,127,264)(123,268,128,263)(124,267,129,262)(125,266,130,261)(131,311,136,316)(132,320,137,315)(133,319,138,314)(134,318,139,313)(135,317,140,312)(151,292,156,297)(152,291,157,296)(153,300,158,295)(154,299,159,294)(155,298,160,293)(161,307,166,302)(162,306,167,301)(163,305,168,310)(164,304,169,309)(165,303,170,308) );

G=PermutationGroup([(1,147,16,154,39,170,48,131),(2,148,17,155,40,161,49,132),(3,149,18,156,31,162,50,133),(4,150,19,157,32,163,41,134),(5,141,20,158,33,164,42,135),(6,142,11,159,34,165,43,136),(7,143,12,160,35,166,44,137),(8,144,13,151,36,167,45,138),(9,145,14,152,37,168,46,139),(10,146,15,153,38,169,47,140),(21,207,294,179,303,188,316,191),(22,208,295,180,304,189,317,192),(23,209,296,171,305,190,318,193),(24,210,297,172,306,181,319,194),(25,201,298,173,307,182,320,195),(26,202,299,174,308,183,311,196),(27,203,300,175,309,184,312,197),(28,204,291,176,310,185,313,198),(29,205,292,177,301,186,314,199),(30,206,293,178,302,187,315,200),(51,102,83,96,74,128,67,119),(52,103,84,97,75,129,68,120),(53,104,85,98,76,130,69,111),(54,105,86,99,77,121,70,112),(55,106,87,100,78,122,61,113),(56,107,88,91,79,123,62,114),(57,108,89,92,80,124,63,115),(58,109,90,93,71,125,64,116),(59,110,81,94,72,126,65,117),(60,101,82,95,73,127,66,118),(211,265,249,256,233,284,230,278),(212,266,250,257,234,285,221,279),(213,267,241,258,235,286,222,280),(214,268,242,259,236,287,223,271),(215,269,243,260,237,288,224,272),(216,270,244,251,238,289,225,273),(217,261,245,252,239,290,226,274),(218,262,246,253,240,281,227,275),(219,263,247,254,231,282,228,276),(220,264,248,255,232,283,229,277)], [(1,74,39,51),(2,75,40,52),(3,76,31,53),(4,77,32,54),(5,78,33,55),(6,79,34,56),(7,80,35,57),(8,71,36,58),(9,72,37,59),(10,73,38,60),(11,88,43,62),(12,89,44,63),(13,90,45,64),(14,81,46,65),(15,82,47,66),(16,83,48,67),(17,84,49,68),(18,85,50,69),(19,86,41,70),(20,87,42,61),(21,271,303,259),(22,272,304,260),(23,273,305,251),(24,274,306,252),(25,275,307,253),(26,276,308,254),(27,277,309,255),(28,278,310,256),(29,279,301,257),(30,280,302,258),(91,165,114,142),(92,166,115,143),(93,167,116,144),(94,168,117,145),(95,169,118,146),(96,170,119,147),(97,161,120,148),(98,162,111,149),(99,163,112,150),(100,164,113,141),(101,140,127,153),(102,131,128,154),(103,132,129,155),(104,133,130,156),(105,134,121,157),(106,135,122,158),(107,136,123,159),(108,137,124,160),(109,138,125,151),(110,139,126,152),(171,238,193,216),(172,239,194,217),(173,240,195,218),(174,231,196,219),(175,232,197,220),(176,233,198,211),(177,234,199,212),(178,235,200,213),(179,236,191,214),(180,237,192,215),(181,245,210,226),(182,246,201,227),(183,247,202,228),(184,248,203,229),(185,249,204,230),(186,250,205,221),(187,241,206,222),(188,242,207,223),(189,243,208,224),(190,244,209,225),(261,297,290,319),(262,298,281,320),(263,299,282,311),(264,300,283,312),(265,291,284,313),(266,292,285,314),(267,293,286,315),(268,294,287,316),(269,295,288,317),(270,296,289,318)], [(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),(11,207,16,202),(12,206,17,201),(13,205,18,210),(14,204,19,209),(15,203,20,208),(21,147,26,142),(22,146,27,141),(23,145,28,150),(24,144,29,149),(25,143,30,148),(31,172,36,177),(32,171,37,176),(33,180,38,175),(34,179,39,174),(35,178,40,173),(41,190,46,185),(42,189,47,184),(43,188,48,183),(44,187,49,182),(45,186,50,181),(51,231,56,236),(52,240,57,235),(53,239,58,234),(54,238,59,233),(55,237,60,232),(61,243,66,248),(62,242,67,247),(63,241,68,246),(64,250,69,245),(65,249,70,244),(71,212,76,217),(72,211,77,216),(73,220,78,215),(74,219,79,214),(75,218,80,213),(81,230,86,225),(82,229,87,224),(83,228,88,223),(84,227,89,222),(85,226,90,221),(91,271,96,276),(92,280,97,275),(93,279,98,274),(94,278,99,273),(95,277,100,272),(101,283,106,288),(102,282,107,287),(103,281,108,286),(104,290,109,285),(105,289,110,284),(111,252,116,257),(112,251,117,256),(113,260,118,255),(114,259,119,254),(115,258,120,253),(121,270,126,265),(122,269,127,264),(123,268,128,263),(124,267,129,262),(125,266,130,261),(131,311,136,316),(132,320,137,315),(133,319,138,314),(134,318,139,313),(135,317,140,312),(151,292,156,297),(152,291,157,296),(153,300,158,295),(154,299,159,294),(155,298,160,293),(161,307,166,302),(162,306,167,301),(163,305,168,310),(164,304,169,309),(165,303,170,308)])

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444444444558888888810···102020202020···2040···40
size11112244445555101020202020222222101010102···244448···84···4

56 irreducible representations

dim1111111222222224444
type++++++++-+-+-+-+
imageC1C2C2C2C2C2C4D4D5Q16C4○D4D10Dic5D10C4○D8D42D5D4×D5D5×Q16Q8.D10
kernelQ16×Dic5C8×Dic5C405C4Q8⋊Dic5Q8×Dic5C10×Q16C5×Q16C2×Dic5C2×Q16Dic5C20C2×C8Q16C2×Q8C10C4C22C2C2
# reps1112218224228442244

Matrix representation of Q16×Dic5 in GL4(𝔽41) generated by

40000
04000
001229
001212
,
1000
0100
00320
002038
,
04000
1700
0010
0001
,
22900
191900
0010
0001
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

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