metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.13D8, Dic5.8Q16, C20.27M4(2), C5⋊2C8⋊5C8, C5⋊2(C8⋊1C8), (C2×C8).6F5, (C2×C40).3C4, C10.6(C4⋊C8), C20.13(C2×C8), C4.8(D5⋊C8), C2.2(D5.D8), C20⋊C8.10C2, C10.6(C2.D8), (C2×Dic5).30Q8, (C8×Dic5).12C2, C22.18(C4⋊F5), C2.2(C40.C4), C10.4(C8.C4), C4.5(C22.F5), (C2×Dic5).173D4, C2.5(Dic5⋊C8), (C4×Dic5).343C22, (C2×C5⋊2C8).22C4, (C2×C4).120(C2×F5), (C2×C10).11(C4⋊C4), (C2×C20).117(C2×C4), SmallGroup(320,222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.13D8
G = < a,b,c,d | a10=c8=1, b2=a5, d2=a5b, bab-1=a-1, ac=ca, dad-1=a3, bc=cb, bd=db, dcd-1=a5c-1 >
Smallest permutation representation of Dic5.13D8
►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 67 6 62)(2 66 7 61)(3 65 8 70)(4 64 9 69)(5 63 10 68)(11 281 16 286)(12 290 17 285)(13 289 18 284)(14 288 19 283)(15 287 20 282)(21 53 26 58)(22 52 27 57)(23 51 28 56)(24 60 29 55)(25 59 30 54)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)(41 89 46 84)(42 88 47 83)(43 87 48 82)(44 86 49 81)(45 85 50 90)(91 134 96 139)(92 133 97 138)(93 132 98 137)(94 131 99 136)(95 140 100 135)(101 149 106 144)(102 148 107 143)(103 147 108 142)(104 146 109 141)(105 145 110 150)(111 154 116 159)(112 153 117 158)(113 152 118 157)(114 151 119 156)(115 160 120 155)(121 168 126 163)(122 167 127 162)(123 166 128 161)(124 165 129 170)(125 164 130 169)(171 214 176 219)(172 213 177 218)(173 212 178 217)(174 211 179 216)(175 220 180 215)(181 229 186 224)(182 228 187 223)(183 227 188 222)(184 226 189 221)(185 225 190 230)(191 234 196 239)(192 233 197 238)(193 232 198 237)(194 231 199 236)(195 240 200 235)(201 249 206 244)(202 248 207 243)(203 247 208 242)(204 246 209 241)(205 245 210 250)(251 294 256 299)(252 293 257 298)(253 292 258 297)(254 291 259 296)(255 300 260 295)(261 309 266 304)(262 308 267 303)(263 307 268 302)(264 306 269 301)(265 305 270 310)(271 314 276 319)(272 313 277 318)(273 312 278 317)(274 311 279 316)(275 320 280 315)
(1 125 45 94 23 119 39 103)(2 126 46 95 24 120 40 104)(3 127 47 96 25 111 31 105)(4 128 48 97 26 112 32 106)(5 129 49 98 27 113 33 107)(6 130 50 99 28 114 34 108)(7 121 41 100 29 115 35 109)(8 122 42 91 30 116 36 110)(9 123 43 92 21 117 37 101)(10 124 44 93 22 118 38 102)(11 223 308 231 316 216 291 249)(12 224 309 232 317 217 292 250)(13 225 310 233 318 218 293 241)(14 226 301 234 319 219 294 242)(15 227 302 235 320 220 295 243)(16 228 303 236 311 211 296 244)(17 229 304 237 312 212 297 245)(18 230 305 238 313 213 298 246)(19 221 306 239 314 214 299 247)(20 222 307 240 315 215 300 248)(51 156 76 147 67 164 85 131)(52 157 77 148 68 165 86 132)(53 158 78 149 69 166 87 133)(54 159 79 150 70 167 88 134)(55 160 80 141 61 168 89 135)(56 151 71 142 62 169 90 136)(57 152 72 143 63 170 81 137)(58 153 73 144 64 161 82 138)(59 154 74 145 65 162 83 139)(60 155 75 146 66 163 84 140)(171 256 203 288 189 264 196 271)(172 257 204 289 190 265 197 272)(173 258 205 290 181 266 198 273)(174 259 206 281 182 267 199 274)(175 260 207 282 183 268 200 275)(176 251 208 283 184 269 191 276)(177 252 209 284 185 270 192 277)(178 253 210 285 186 261 193 278)(179 254 201 286 187 262 194 279)(180 255 202 287 188 263 195 280)
(1 13 62 284 6 18 67 289)(2 20 61 287 7 15 66 282)(3 17 70 290 8 12 65 285)(4 14 69 283 9 19 64 288)(5 11 68 286 10 16 63 281)(21 314 58 271 26 319 53 276)(22 311 57 274 27 316 52 279)(23 318 56 277 28 313 51 272)(24 315 55 280 29 320 60 275)(25 312 54 273 30 317 59 278)(31 304 79 266 36 309 74 261)(32 301 78 269 37 306 73 264)(33 308 77 262 38 303 72 267)(34 305 76 265 39 310 71 270)(35 302 75 268 40 307 80 263)(41 295 84 260 46 300 89 255)(42 292 83 253 47 297 88 258)(43 299 82 256 48 294 87 251)(44 296 81 259 49 291 86 254)(45 293 90 252 50 298 85 257)(91 212 139 173 96 217 134 178)(92 219 138 176 97 214 133 171)(93 216 137 179 98 211 132 174)(94 213 136 172 99 218 131 177)(95 220 135 175 100 215 140 180)(101 226 144 184 106 221 149 189)(102 223 143 187 107 228 148 182)(103 230 142 190 108 225 147 185)(104 227 141 183 109 222 146 188)(105 224 150 186 110 229 145 181)(111 232 159 193 116 237 154 198)(112 239 158 196 117 234 153 191)(113 236 157 199 118 231 152 194)(114 233 156 192 119 238 151 197)(115 240 155 195 120 235 160 200)(121 248 163 202 126 243 168 207)(122 245 162 205 127 250 167 210)(123 242 161 208 128 247 166 203)(124 249 170 201 129 244 165 206)(125 246 169 204 130 241 164 209)
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), (1,67,6,62)(2,66,7,61)(3,65,8,70)(4,64,9,69)(5,63,10,68)(11,281,16,286)(12,290,17,285)(13,289,18,284)(14,288,19,283)(15,287,20,282)(21,53,26,58)(22,52,27,57)(23,51,28,56)(24,60,29,55)(25,59,30,54)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75)(41,89,46,84)(42,88,47,83)(43,87,48,82)(44,86,49,81)(45,85,50,90)(91,134,96,139)(92,133,97,138)(93,132,98,137)(94,131,99,136)(95,140,100,135)(101,149,106,144)(102,148,107,143)(103,147,108,142)(104,146,109,141)(105,145,110,150)(111,154,116,159)(112,153,117,158)(113,152,118,157)(114,151,119,156)(115,160,120,155)(121,168,126,163)(122,167,127,162)(123,166,128,161)(124,165,129,170)(125,164,130,169)(171,214,176,219)(172,213,177,218)(173,212,178,217)(174,211,179,216)(175,220,180,215)(181,229,186,224)(182,228,187,223)(183,227,188,222)(184,226,189,221)(185,225,190,230)(191,234,196,239)(192,233,197,238)(193,232,198,237)(194,231,199,236)(195,240,200,235)(201,249,206,244)(202,248,207,243)(203,247,208,242)(204,246,209,241)(205,245,210,250)(251,294,256,299)(252,293,257,298)(253,292,258,297)(254,291,259,296)(255,300,260,295)(261,309,266,304)(262,308,267,303)(263,307,268,302)(264,306,269,301)(265,305,270,310)(271,314,276,319)(272,313,277,318)(273,312,278,317)(274,311,279,316)(275,320,280,315), (1,125,45,94,23,119,39,103)(2,126,46,95,24,120,40,104)(3,127,47,96,25,111,31,105)(4,128,48,97,26,112,32,106)(5,129,49,98,27,113,33,107)(6,130,50,99,28,114,34,108)(7,121,41,100,29,115,35,109)(8,122,42,91,30,116,36,110)(9,123,43,92,21,117,37,101)(10,124,44,93,22,118,38,102)(11,223,308,231,316,216,291,249)(12,224,309,232,317,217,292,250)(13,225,310,233,318,218,293,241)(14,226,301,234,319,219,294,242)(15,227,302,235,320,220,295,243)(16,228,303,236,311,211,296,244)(17,229,304,237,312,212,297,245)(18,230,305,238,313,213,298,246)(19,221,306,239,314,214,299,247)(20,222,307,240,315,215,300,248)(51,156,76,147,67,164,85,131)(52,157,77,148,68,165,86,132)(53,158,78,149,69,166,87,133)(54,159,79,150,70,167,88,134)(55,160,80,141,61,168,89,135)(56,151,71,142,62,169,90,136)(57,152,72,143,63,170,81,137)(58,153,73,144,64,161,82,138)(59,154,74,145,65,162,83,139)(60,155,75,146,66,163,84,140)(171,256,203,288,189,264,196,271)(172,257,204,289,190,265,197,272)(173,258,205,290,181,266,198,273)(174,259,206,281,182,267,199,274)(175,260,207,282,183,268,200,275)(176,251,208,283,184,269,191,276)(177,252,209,284,185,270,192,277)(178,253,210,285,186,261,193,278)(179,254,201,286,187,262,194,279)(180,255,202,287,188,263,195,280), (1,13,62,284,6,18,67,289)(2,20,61,287,7,15,66,282)(3,17,70,290,8,12,65,285)(4,14,69,283,9,19,64,288)(5,11,68,286,10,16,63,281)(21,314,58,271,26,319,53,276)(22,311,57,274,27,316,52,279)(23,318,56,277,28,313,51,272)(24,315,55,280,29,320,60,275)(25,312,54,273,30,317,59,278)(31,304,79,266,36,309,74,261)(32,301,78,269,37,306,73,264)(33,308,77,262,38,303,72,267)(34,305,76,265,39,310,71,270)(35,302,75,268,40,307,80,263)(41,295,84,260,46,300,89,255)(42,292,83,253,47,297,88,258)(43,299,82,256,48,294,87,251)(44,296,81,259,49,291,86,254)(45,293,90,252,50,298,85,257)(91,212,139,173,96,217,134,178)(92,219,138,176,97,214,133,171)(93,216,137,179,98,211,132,174)(94,213,136,172,99,218,131,177)(95,220,135,175,100,215,140,180)(101,226,144,184,106,221,149,189)(102,223,143,187,107,228,148,182)(103,230,142,190,108,225,147,185)(104,227,141,183,109,222,146,188)(105,224,150,186,110,229,145,181)(111,232,159,193,116,237,154,198)(112,239,158,196,117,234,153,191)(113,236,157,199,118,231,152,194)(114,233,156,192,119,238,151,197)(115,240,155,195,120,235,160,200)(121,248,163,202,126,243,168,207)(122,245,162,205,127,250,167,210)(123,242,161,208,128,247,166,203)(124,249,170,201,129,244,165,206)(125,246,169,204,130,241,164,209)>;
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), (1,67,6,62)(2,66,7,61)(3,65,8,70)(4,64,9,69)(5,63,10,68)(11,281,16,286)(12,290,17,285)(13,289,18,284)(14,288,19,283)(15,287,20,282)(21,53,26,58)(22,52,27,57)(23,51,28,56)(24,60,29,55)(25,59,30,54)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75)(41,89,46,84)(42,88,47,83)(43,87,48,82)(44,86,49,81)(45,85,50,90)(91,134,96,139)(92,133,97,138)(93,132,98,137)(94,131,99,136)(95,140,100,135)(101,149,106,144)(102,148,107,143)(103,147,108,142)(104,146,109,141)(105,145,110,150)(111,154,116,159)(112,153,117,158)(113,152,118,157)(114,151,119,156)(115,160,120,155)(121,168,126,163)(122,167,127,162)(123,166,128,161)(124,165,129,170)(125,164,130,169)(171,214,176,219)(172,213,177,218)(173,212,178,217)(174,211,179,216)(175,220,180,215)(181,229,186,224)(182,228,187,223)(183,227,188,222)(184,226,189,221)(185,225,190,230)(191,234,196,239)(192,233,197,238)(193,232,198,237)(194,231,199,236)(195,240,200,235)(201,249,206,244)(202,248,207,243)(203,247,208,242)(204,246,209,241)(205,245,210,250)(251,294,256,299)(252,293,257,298)(253,292,258,297)(254,291,259,296)(255,300,260,295)(261,309,266,304)(262,308,267,303)(263,307,268,302)(264,306,269,301)(265,305,270,310)(271,314,276,319)(272,313,277,318)(273,312,278,317)(274,311,279,316)(275,320,280,315), (1,125,45,94,23,119,39,103)(2,126,46,95,24,120,40,104)(3,127,47,96,25,111,31,105)(4,128,48,97,26,112,32,106)(5,129,49,98,27,113,33,107)(6,130,50,99,28,114,34,108)(7,121,41,100,29,115,35,109)(8,122,42,91,30,116,36,110)(9,123,43,92,21,117,37,101)(10,124,44,93,22,118,38,102)(11,223,308,231,316,216,291,249)(12,224,309,232,317,217,292,250)(13,225,310,233,318,218,293,241)(14,226,301,234,319,219,294,242)(15,227,302,235,320,220,295,243)(16,228,303,236,311,211,296,244)(17,229,304,237,312,212,297,245)(18,230,305,238,313,213,298,246)(19,221,306,239,314,214,299,247)(20,222,307,240,315,215,300,248)(51,156,76,147,67,164,85,131)(52,157,77,148,68,165,86,132)(53,158,78,149,69,166,87,133)(54,159,79,150,70,167,88,134)(55,160,80,141,61,168,89,135)(56,151,71,142,62,169,90,136)(57,152,72,143,63,170,81,137)(58,153,73,144,64,161,82,138)(59,154,74,145,65,162,83,139)(60,155,75,146,66,163,84,140)(171,256,203,288,189,264,196,271)(172,257,204,289,190,265,197,272)(173,258,205,290,181,266,198,273)(174,259,206,281,182,267,199,274)(175,260,207,282,183,268,200,275)(176,251,208,283,184,269,191,276)(177,252,209,284,185,270,192,277)(178,253,210,285,186,261,193,278)(179,254,201,286,187,262,194,279)(180,255,202,287,188,263,195,280), (1,13,62,284,6,18,67,289)(2,20,61,287,7,15,66,282)(3,17,70,290,8,12,65,285)(4,14,69,283,9,19,64,288)(5,11,68,286,10,16,63,281)(21,314,58,271,26,319,53,276)(22,311,57,274,27,316,52,279)(23,318,56,277,28,313,51,272)(24,315,55,280,29,320,60,275)(25,312,54,273,30,317,59,278)(31,304,79,266,36,309,74,261)(32,301,78,269,37,306,73,264)(33,308,77,262,38,303,72,267)(34,305,76,265,39,310,71,270)(35,302,75,268,40,307,80,263)(41,295,84,260,46,300,89,255)(42,292,83,253,47,297,88,258)(43,299,82,256,48,294,87,251)(44,296,81,259,49,291,86,254)(45,293,90,252,50,298,85,257)(91,212,139,173,96,217,134,178)(92,219,138,176,97,214,133,171)(93,216,137,179,98,211,132,174)(94,213,136,172,99,218,131,177)(95,220,135,175,100,215,140,180)(101,226,144,184,106,221,149,189)(102,223,143,187,107,228,148,182)(103,230,142,190,108,225,147,185)(104,227,141,183,109,222,146,188)(105,224,150,186,110,229,145,181)(111,232,159,193,116,237,154,198)(112,239,158,196,117,234,153,191)(113,236,157,199,118,231,152,194)(114,233,156,192,119,238,151,197)(115,240,155,195,120,235,160,200)(121,248,163,202,126,243,168,207)(122,245,162,205,127,250,167,210)(123,242,161,208,128,247,166,203)(124,249,170,201,129,244,165,206)(125,246,169,204,130,241,164,209) );
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)], [(1,67,6,62),(2,66,7,61),(3,65,8,70),(4,64,9,69),(5,63,10,68),(11,281,16,286),(12,290,17,285),(13,289,18,284),(14,288,19,283),(15,287,20,282),(21,53,26,58),(22,52,27,57),(23,51,28,56),(24,60,29,55),(25,59,30,54),(31,74,36,79),(32,73,37,78),(33,72,38,77),(34,71,39,76),(35,80,40,75),(41,89,46,84),(42,88,47,83),(43,87,48,82),(44,86,49,81),(45,85,50,90),(91,134,96,139),(92,133,97,138),(93,132,98,137),(94,131,99,136),(95,140,100,135),(101,149,106,144),(102,148,107,143),(103,147,108,142),(104,146,109,141),(105,145,110,150),(111,154,116,159),(112,153,117,158),(113,152,118,157),(114,151,119,156),(115,160,120,155),(121,168,126,163),(122,167,127,162),(123,166,128,161),(124,165,129,170),(125,164,130,169),(171,214,176,219),(172,213,177,218),(173,212,178,217),(174,211,179,216),(175,220,180,215),(181,229,186,224),(182,228,187,223),(183,227,188,222),(184,226,189,221),(185,225,190,230),(191,234,196,239),(192,233,197,238),(193,232,198,237),(194,231,199,236),(195,240,200,235),(201,249,206,244),(202,248,207,243),(203,247,208,242),(204,246,209,241),(205,245,210,250),(251,294,256,299),(252,293,257,298),(253,292,258,297),(254,291,259,296),(255,300,260,295),(261,309,266,304),(262,308,267,303),(263,307,268,302),(264,306,269,301),(265,305,270,310),(271,314,276,319),(272,313,277,318),(273,312,278,317),(274,311,279,316),(275,320,280,315)], [(1,125,45,94,23,119,39,103),(2,126,46,95,24,120,40,104),(3,127,47,96,25,111,31,105),(4,128,48,97,26,112,32,106),(5,129,49,98,27,113,33,107),(6,130,50,99,28,114,34,108),(7,121,41,100,29,115,35,109),(8,122,42,91,30,116,36,110),(9,123,43,92,21,117,37,101),(10,124,44,93,22,118,38,102),(11,223,308,231,316,216,291,249),(12,224,309,232,317,217,292,250),(13,225,310,233,318,218,293,241),(14,226,301,234,319,219,294,242),(15,227,302,235,320,220,295,243),(16,228,303,236,311,211,296,244),(17,229,304,237,312,212,297,245),(18,230,305,238,313,213,298,246),(19,221,306,239,314,214,299,247),(20,222,307,240,315,215,300,248),(51,156,76,147,67,164,85,131),(52,157,77,148,68,165,86,132),(53,158,78,149,69,166,87,133),(54,159,79,150,70,167,88,134),(55,160,80,141,61,168,89,135),(56,151,71,142,62,169,90,136),(57,152,72,143,63,170,81,137),(58,153,73,144,64,161,82,138),(59,154,74,145,65,162,83,139),(60,155,75,146,66,163,84,140),(171,256,203,288,189,264,196,271),(172,257,204,289,190,265,197,272),(173,258,205,290,181,266,198,273),(174,259,206,281,182,267,199,274),(175,260,207,282,183,268,200,275),(176,251,208,283,184,269,191,276),(177,252,209,284,185,270,192,277),(178,253,210,285,186,261,193,278),(179,254,201,286,187,262,194,279),(180,255,202,287,188,263,195,280)], [(1,13,62,284,6,18,67,289),(2,20,61,287,7,15,66,282),(3,17,70,290,8,12,65,285),(4,14,69,283,9,19,64,288),(5,11,68,286,10,16,63,281),(21,314,58,271,26,319,53,276),(22,311,57,274,27,316,52,279),(23,318,56,277,28,313,51,272),(24,315,55,280,29,320,60,275),(25,312,54,273,30,317,59,278),(31,304,79,266,36,309,74,261),(32,301,78,269,37,306,73,264),(33,308,77,262,38,303,72,267),(34,305,76,265,39,310,71,270),(35,302,75,268,40,307,80,263),(41,295,84,260,46,300,89,255),(42,292,83,253,47,297,88,258),(43,299,82,256,48,294,87,251),(44,296,81,259,49,291,86,254),(45,293,90,252,50,298,85,257),(91,212,139,173,96,217,134,178),(92,219,138,176,97,214,133,171),(93,216,137,179,98,211,132,174),(94,213,136,172,99,218,131,177),(95,220,135,175,100,215,140,180),(101,226,144,184,106,221,149,189),(102,223,143,187,107,228,148,182),(103,230,142,190,108,225,147,185),(104,227,141,183,109,222,146,188),(105,224,150,186,110,229,145,181),(111,232,159,193,116,237,154,198),(112,239,158,196,117,234,153,191),(113,236,157,199,118,231,152,194),(114,233,156,192,119,238,151,197),(115,240,155,195,120,235,160,200),(121,248,163,202,126,243,168,207),(122,245,162,205,127,250,167,210),(123,242,161,208,128,247,166,203),(124,249,170,201,129,244,165,206),(125,246,169,204,130,241,164,209)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D8 | Q16 | M4(2) | C8.C4 | F5 | C2×F5 | D5⋊C8 | C22.F5 | C4⋊F5 | D5.D8 | C40.C4 |
| kernel | Dic5.13D8 | C8×Dic5 | C20⋊C8 | C2×C5⋊2C8 | C2×C40 | C5⋊2C8 | C2×Dic5 | C2×Dic5 | Dic5 | Dic5 | C20 | C10 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5.13D8 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 21 | 8 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 35 | 35 |
| 0 | 0 | 0 | 0 | 28 | 20 | 20 | 26 |
| 0 | 0 | 0 | 0 | 33 | 33 | 39 | 13 |
| 9 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 27 | 27 |
| 0 | 0 | 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 7 | 14 |
| 0 | 0 | 0 | 0 | 27 | 27 | 0 | 34 |
| 16 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 33 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 30 | 39 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 | 11 | 19 |
| 0 | 0 | 0 | 0 | 0 | 8 | 30 | 39 |
| 0 | 0 | 0 | 0 | 22 | 31 | 33 | 33 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,15,28,33,0,0,0,0,21,2,20,33,0,0,0,0,8,35,20,39,0,0,0,0,0,35,26,13],[9,25,0,0,0,0,0,0,5,32,0,0,0,0,0,0,0,0,24,29,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,0,34,14,0,27,0,0,0,0,0,7,14,27,0,0,0,0,27,14,7,0,0,0,0,0,27,0,14,34],[16,33,0,0,0,0,0,0,36,25,0,0,0,0,0,0,0,0,39,27,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,8,9,0,22,0,0,0,0,30,11,8,31,0,0,0,0,39,11,30,33,0,0,0,0,0,19,39,33] >;
Dic5.13D8 in GAP, Magma, Sage, TeX
{\rm Dic}_5._{13}D_8 % in TeX
G:=Group("Dic5.13D8"); // GroupNames label
G:=SmallGroup(320,222);
// by ID
G=gap.SmallGroup(320,222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,268,1123,136,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=a^5*b,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations
Export