Copied to
clipboard

G = Dic5.9Q16order 320 = 26·5

1st non-split extension by Dic5 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82Dic10, Dic5.9Q16, (C5×Q8)⋊2Q8, C20⋊Q8.4C2, C20.6(C2×Q8), C2.8(D5×Q16), C4⋊C4.17D10, C51(C4.Q16), (C2×C8).14D10, C405C4.6C2, Q8⋊C4.3D5, C10.15(C2×Q16), (Q8×Dic5).4C2, C4.6(C2×Dic10), (C2×C40).14C22, (C2×Q8).100D10, Q8⋊Dic5.3C2, C10.D8.2C2, C22.188(D4×D5), C20.8Q8.3C2, C20.158(C4○D4), C4.83(D42D5), C2.14(D40⋊C2), C10.59(C8⋊C22), (C2×C20).234C23, (C2×Dic5).205D4, C10.12(C22⋊Q8), C4⋊Dic5.83C22, (Q8×C10).17C22, (C4×Dic5).24C22, C2.17(Dic5.14D4), (C2×C10).247(C2×D4), (C5×C4⋊C4).35C22, (C5×Q8⋊C4).3C2, (C2×C52C8).29C22, (C2×C4).341(C22×D5), SmallGroup(320,421)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic5.9Q16
C1C5C10C2×C10C2×C20C4×Dic5Q8×Dic5 — Dic5.9Q16
C5C10C2×C20 — Dic5.9Q16
C1C22C2×C4Q8⋊C4

Generators and relations for Dic5.9Q16
 G = < a,b,c,d | a10=c8=1, b2=a5, d2=a5c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c-1 >

Subgroups: 342 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×2], Q8 [×3], C10 [×3], C42 [×2], C4⋊C4, C4⋊C4 [×4], C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C4.Q16, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5 [×2], C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.D8, C20.8Q8, C405C4, Q8⋊Dic5, C5×Q8⋊C4, C20⋊Q8, Q8×Dic5, Dic5.9Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, Q16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×Q16, C8⋊C22, Dic10 [×2], C22×D5, C4.Q16, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D40⋊C2, D5×Q16, Dic5.9Q16

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

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444455888810···102020202020···2040···40
size111122448101020202040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type+++++++++-+-+++-+-++-
imageC1C2C2C2C2C2C2C2D4Q8D5Q16C4○D4D10D10D10Dic10C8⋊C22D42D5D4×D5D40⋊C2D5×Q16
kernelDic5.9Q16C10.D8C20.8Q8C405C4Q8⋊Dic5C5×Q8⋊C4C20⋊Q8Q8×Dic5C2×Dic5C5×Q8Q8⋊C4Dic5C20C4⋊C4C2×C8C2×Q8Q8C10C4C22C2C2
# reps1111111122242222812244

Matrix representation of Dic5.9Q16 in GL6(𝔽41)

100000
010000
0037000
0021000
0000400
0000040
,
100000
010000
0032100
00253800
000026
0000639
,
17170000
1200000
0040000
0004000
00001323
00002328
,
12220000
40290000
001000
000100
000001
0000400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,2,0,0,0,0,0,10,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,25,0,0,0,0,21,38,0,0,0,0,0,0,2,6,0,0,0,0,6,39],[17,12,0,0,0,0,17,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,13,23,0,0,0,0,23,28],[12,40,0,0,0,0,22,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;

Dic5.9Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_5._9Q_{16}
% in TeX

G:=Group("Dic5.9Q16");
// GroupNames label

G:=SmallGroup(320,421);
// by ID

G=gap.SmallGroup(320,421);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,254,219,226,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=a^5*c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

׿
×
𝔽