metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊9Q8, C42.170D10, C10.812+ 1+4, C5⋊2Q82, C4⋊Q8.15D5, C4.18(Q8×D5), C20⋊Q8.14C2, C20.53(C2×Q8), C4⋊C4.215D10, (C2×Q8).85D10, (C2×C20).99C23, Dic5.27(C2×Q8), C10.45(C22×Q8), (C4×C20).207C22, (C2×C10).266C24, (C4×Dic10).26C2, C2.85(D4⋊6D10), Dic5⋊Q8.11C2, Dic5⋊3Q8.13C2, C4⋊Dic5.383C22, (Q8×C10).133C22, C22.287(C23×D5), (C2×Dic5).281C23, (C4×Dic5).166C22, C10.D4.58C22, (C2×Dic10).192C22, C2.28(C2×Q8×D5), (C5×C4⋊Q8).15C2, (C2×C4).91(C22×D5), (C5×C4⋊C4).209C22, SmallGroup(320,1394)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊9Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 630 in 212 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, C4⋊Q8, C4⋊Q8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Q82, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×Dic10, Dic5⋊3Q8, C20⋊Q8, Dic5⋊Q8, C5×C4⋊Q8, Dic10⋊9Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, C22×D5, Q82, Q8×D5, C23×D5, D4⋊6D10, C2×Q8×D5, Dic10⋊9Q8
►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)
(1 133 11 123)(2 132 12 122)(3 131 13 121)(4 130 14 140)(5 129 15 139)(6 128 16 138)(7 127 17 137)(8 126 18 136)(9 125 19 135)(10 124 20 134)(21 114 31 104)(22 113 32 103)(23 112 33 102)(24 111 34 101)(25 110 35 120)(26 109 36 119)(27 108 37 118)(28 107 38 117)(29 106 39 116)(30 105 40 115)(41 294 51 284)(42 293 52 283)(43 292 53 282)(44 291 54 281)(45 290 55 300)(46 289 56 299)(47 288 57 298)(48 287 58 297)(49 286 59 296)(50 285 60 295)(61 143 71 153)(62 142 72 152)(63 141 73 151)(64 160 74 150)(65 159 75 149)(66 158 76 148)(67 157 77 147)(68 156 78 146)(69 155 79 145)(70 154 80 144)(81 266 91 276)(82 265 92 275)(83 264 93 274)(84 263 94 273)(85 262 95 272)(86 261 96 271)(87 280 97 270)(88 279 98 269)(89 278 99 268)(90 277 100 267)(161 242 171 252)(162 241 172 251)(163 260 173 250)(164 259 174 249)(165 258 175 248)(166 257 176 247)(167 256 177 246)(168 255 178 245)(169 254 179 244)(170 253 180 243)(181 309 191 319)(182 308 192 318)(183 307 193 317)(184 306 194 316)(185 305 195 315)(186 304 196 314)(187 303 197 313)(188 302 198 312)(189 301 199 311)(190 320 200 310)(201 225 211 235)(202 224 212 234)(203 223 213 233)(204 222 214 232)(205 221 215 231)(206 240 216 230)(207 239 217 229)(208 238 218 228)(209 237 219 227)(210 236 220 226)
(1 63 301 84)(2 74 302 95)(3 65 303 86)(4 76 304 97)(5 67 305 88)(6 78 306 99)(7 69 307 90)(8 80 308 81)(9 71 309 92)(10 62 310 83)(11 73 311 94)(12 64 312 85)(13 75 313 96)(14 66 314 87)(15 77 315 98)(16 68 316 89)(17 79 317 100)(18 70 318 91)(19 61 319 82)(20 72 320 93)(21 220 251 284)(22 211 252 295)(23 202 253 286)(24 213 254 297)(25 204 255 288)(26 215 256 299)(27 206 257 290)(28 217 258 281)(29 208 259 292)(30 219 260 283)(31 210 241 294)(32 201 242 285)(33 212 243 296)(34 203 244 287)(35 214 245 298)(36 205 246 289)(37 216 247 300)(38 207 248 291)(39 218 249 282)(40 209 250 293)(41 114 226 162)(42 105 227 173)(43 116 228 164)(44 107 229 175)(45 118 230 166)(46 109 231 177)(47 120 232 168)(48 111 233 179)(49 102 234 170)(50 113 235 161)(51 104 236 172)(52 115 237 163)(53 106 238 174)(54 117 239 165)(55 108 240 176)(56 119 221 167)(57 110 222 178)(58 101 223 169)(59 112 224 180)(60 103 225 171)(121 149 187 271)(122 160 188 262)(123 151 189 273)(124 142 190 264)(125 153 191 275)(126 144 192 266)(127 155 193 277)(128 146 194 268)(129 157 195 279)(130 148 196 270)(131 159 197 261)(132 150 198 272)(133 141 199 263)(134 152 200 274)(135 143 181 265)(136 154 182 276)(137 145 183 267)(138 156 184 278)(139 147 185 269)(140 158 186 280)
(1 287 301 203)(2 288 302 204)(3 289 303 205)(4 290 304 206)(5 291 305 207)(6 292 306 208)(7 293 307 209)(8 294 308 210)(9 295 309 211)(10 296 310 212)(11 297 311 213)(12 298 312 214)(13 299 313 215)(14 300 314 216)(15 281 315 217)(16 282 316 218)(17 283 317 219)(18 284 318 220)(19 285 319 201)(20 286 320 202)(21 70 251 91)(22 71 252 92)(23 72 253 93)(24 73 254 94)(25 74 255 95)(26 75 256 96)(27 76 257 97)(28 77 258 98)(29 78 259 99)(30 79 260 100)(31 80 241 81)(32 61 242 82)(33 62 243 83)(34 63 244 84)(35 64 245 85)(36 65 246 86)(37 66 247 87)(38 67 248 88)(39 68 249 89)(40 69 250 90)(41 192 226 126)(42 193 227 127)(43 194 228 128)(44 195 229 129)(45 196 230 130)(46 197 231 131)(47 198 232 132)(48 199 233 133)(49 200 234 134)(50 181 235 135)(51 182 236 136)(52 183 237 137)(53 184 238 138)(54 185 239 139)(55 186 240 140)(56 187 221 121)(57 188 222 122)(58 189 223 123)(59 190 224 124)(60 191 225 125)(101 151 169 273)(102 152 170 274)(103 153 171 275)(104 154 172 276)(105 155 173 277)(106 156 174 278)(107 157 175 279)(108 158 176 280)(109 159 177 261)(110 160 178 262)(111 141 179 263)(112 142 180 264)(113 143 161 265)(114 144 162 266)(115 145 163 267)(116 146 164 268)(117 147 165 269)(118 148 166 270)(119 149 167 271)(120 150 168 272)
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), (1,133,11,123)(2,132,12,122)(3,131,13,121)(4,130,14,140)(5,129,15,139)(6,128,16,138)(7,127,17,137)(8,126,18,136)(9,125,19,135)(10,124,20,134)(21,114,31,104)(22,113,32,103)(23,112,33,102)(24,111,34,101)(25,110,35,120)(26,109,36,119)(27,108,37,118)(28,107,38,117)(29,106,39,116)(30,105,40,115)(41,294,51,284)(42,293,52,283)(43,292,53,282)(44,291,54,281)(45,290,55,300)(46,289,56,299)(47,288,57,298)(48,287,58,297)(49,286,59,296)(50,285,60,295)(61,143,71,153)(62,142,72,152)(63,141,73,151)(64,160,74,150)(65,159,75,149)(66,158,76,148)(67,157,77,147)(68,156,78,146)(69,155,79,145)(70,154,80,144)(81,266,91,276)(82,265,92,275)(83,264,93,274)(84,263,94,273)(85,262,95,272)(86,261,96,271)(87,280,97,270)(88,279,98,269)(89,278,99,268)(90,277,100,267)(161,242,171,252)(162,241,172,251)(163,260,173,250)(164,259,174,249)(165,258,175,248)(166,257,176,247)(167,256,177,246)(168,255,178,245)(169,254,179,244)(170,253,180,243)(181,309,191,319)(182,308,192,318)(183,307,193,317)(184,306,194,316)(185,305,195,315)(186,304,196,314)(187,303,197,313)(188,302,198,312)(189,301,199,311)(190,320,200,310)(201,225,211,235)(202,224,212,234)(203,223,213,233)(204,222,214,232)(205,221,215,231)(206,240,216,230)(207,239,217,229)(208,238,218,228)(209,237,219,227)(210,236,220,226), (1,63,301,84)(2,74,302,95)(3,65,303,86)(4,76,304,97)(5,67,305,88)(6,78,306,99)(7,69,307,90)(8,80,308,81)(9,71,309,92)(10,62,310,83)(11,73,311,94)(12,64,312,85)(13,75,313,96)(14,66,314,87)(15,77,315,98)(16,68,316,89)(17,79,317,100)(18,70,318,91)(19,61,319,82)(20,72,320,93)(21,220,251,284)(22,211,252,295)(23,202,253,286)(24,213,254,297)(25,204,255,288)(26,215,256,299)(27,206,257,290)(28,217,258,281)(29,208,259,292)(30,219,260,283)(31,210,241,294)(32,201,242,285)(33,212,243,296)(34,203,244,287)(35,214,245,298)(36,205,246,289)(37,216,247,300)(38,207,248,291)(39,218,249,282)(40,209,250,293)(41,114,226,162)(42,105,227,173)(43,116,228,164)(44,107,229,175)(45,118,230,166)(46,109,231,177)(47,120,232,168)(48,111,233,179)(49,102,234,170)(50,113,235,161)(51,104,236,172)(52,115,237,163)(53,106,238,174)(54,117,239,165)(55,108,240,176)(56,119,221,167)(57,110,222,178)(58,101,223,169)(59,112,224,180)(60,103,225,171)(121,149,187,271)(122,160,188,262)(123,151,189,273)(124,142,190,264)(125,153,191,275)(126,144,192,266)(127,155,193,277)(128,146,194,268)(129,157,195,279)(130,148,196,270)(131,159,197,261)(132,150,198,272)(133,141,199,263)(134,152,200,274)(135,143,181,265)(136,154,182,276)(137,145,183,267)(138,156,184,278)(139,147,185,269)(140,158,186,280), (1,287,301,203)(2,288,302,204)(3,289,303,205)(4,290,304,206)(5,291,305,207)(6,292,306,208)(7,293,307,209)(8,294,308,210)(9,295,309,211)(10,296,310,212)(11,297,311,213)(12,298,312,214)(13,299,313,215)(14,300,314,216)(15,281,315,217)(16,282,316,218)(17,283,317,219)(18,284,318,220)(19,285,319,201)(20,286,320,202)(21,70,251,91)(22,71,252,92)(23,72,253,93)(24,73,254,94)(25,74,255,95)(26,75,256,96)(27,76,257,97)(28,77,258,98)(29,78,259,99)(30,79,260,100)(31,80,241,81)(32,61,242,82)(33,62,243,83)(34,63,244,84)(35,64,245,85)(36,65,246,86)(37,66,247,87)(38,67,248,88)(39,68,249,89)(40,69,250,90)(41,192,226,126)(42,193,227,127)(43,194,228,128)(44,195,229,129)(45,196,230,130)(46,197,231,131)(47,198,232,132)(48,199,233,133)(49,200,234,134)(50,181,235,135)(51,182,236,136)(52,183,237,137)(53,184,238,138)(54,185,239,139)(55,186,240,140)(56,187,221,121)(57,188,222,122)(58,189,223,123)(59,190,224,124)(60,191,225,125)(101,151,169,273)(102,152,170,274)(103,153,171,275)(104,154,172,276)(105,155,173,277)(106,156,174,278)(107,157,175,279)(108,158,176,280)(109,159,177,261)(110,160,178,262)(111,141,179,263)(112,142,180,264)(113,143,161,265)(114,144,162,266)(115,145,163,267)(116,146,164,268)(117,147,165,269)(118,148,166,270)(119,149,167,271)(120,150,168,272)>;
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), (1,133,11,123)(2,132,12,122)(3,131,13,121)(4,130,14,140)(5,129,15,139)(6,128,16,138)(7,127,17,137)(8,126,18,136)(9,125,19,135)(10,124,20,134)(21,114,31,104)(22,113,32,103)(23,112,33,102)(24,111,34,101)(25,110,35,120)(26,109,36,119)(27,108,37,118)(28,107,38,117)(29,106,39,116)(30,105,40,115)(41,294,51,284)(42,293,52,283)(43,292,53,282)(44,291,54,281)(45,290,55,300)(46,289,56,299)(47,288,57,298)(48,287,58,297)(49,286,59,296)(50,285,60,295)(61,143,71,153)(62,142,72,152)(63,141,73,151)(64,160,74,150)(65,159,75,149)(66,158,76,148)(67,157,77,147)(68,156,78,146)(69,155,79,145)(70,154,80,144)(81,266,91,276)(82,265,92,275)(83,264,93,274)(84,263,94,273)(85,262,95,272)(86,261,96,271)(87,280,97,270)(88,279,98,269)(89,278,99,268)(90,277,100,267)(161,242,171,252)(162,241,172,251)(163,260,173,250)(164,259,174,249)(165,258,175,248)(166,257,176,247)(167,256,177,246)(168,255,178,245)(169,254,179,244)(170,253,180,243)(181,309,191,319)(182,308,192,318)(183,307,193,317)(184,306,194,316)(185,305,195,315)(186,304,196,314)(187,303,197,313)(188,302,198,312)(189,301,199,311)(190,320,200,310)(201,225,211,235)(202,224,212,234)(203,223,213,233)(204,222,214,232)(205,221,215,231)(206,240,216,230)(207,239,217,229)(208,238,218,228)(209,237,219,227)(210,236,220,226), (1,63,301,84)(2,74,302,95)(3,65,303,86)(4,76,304,97)(5,67,305,88)(6,78,306,99)(7,69,307,90)(8,80,308,81)(9,71,309,92)(10,62,310,83)(11,73,311,94)(12,64,312,85)(13,75,313,96)(14,66,314,87)(15,77,315,98)(16,68,316,89)(17,79,317,100)(18,70,318,91)(19,61,319,82)(20,72,320,93)(21,220,251,284)(22,211,252,295)(23,202,253,286)(24,213,254,297)(25,204,255,288)(26,215,256,299)(27,206,257,290)(28,217,258,281)(29,208,259,292)(30,219,260,283)(31,210,241,294)(32,201,242,285)(33,212,243,296)(34,203,244,287)(35,214,245,298)(36,205,246,289)(37,216,247,300)(38,207,248,291)(39,218,249,282)(40,209,250,293)(41,114,226,162)(42,105,227,173)(43,116,228,164)(44,107,229,175)(45,118,230,166)(46,109,231,177)(47,120,232,168)(48,111,233,179)(49,102,234,170)(50,113,235,161)(51,104,236,172)(52,115,237,163)(53,106,238,174)(54,117,239,165)(55,108,240,176)(56,119,221,167)(57,110,222,178)(58,101,223,169)(59,112,224,180)(60,103,225,171)(121,149,187,271)(122,160,188,262)(123,151,189,273)(124,142,190,264)(125,153,191,275)(126,144,192,266)(127,155,193,277)(128,146,194,268)(129,157,195,279)(130,148,196,270)(131,159,197,261)(132,150,198,272)(133,141,199,263)(134,152,200,274)(135,143,181,265)(136,154,182,276)(137,145,183,267)(138,156,184,278)(139,147,185,269)(140,158,186,280), (1,287,301,203)(2,288,302,204)(3,289,303,205)(4,290,304,206)(5,291,305,207)(6,292,306,208)(7,293,307,209)(8,294,308,210)(9,295,309,211)(10,296,310,212)(11,297,311,213)(12,298,312,214)(13,299,313,215)(14,300,314,216)(15,281,315,217)(16,282,316,218)(17,283,317,219)(18,284,318,220)(19,285,319,201)(20,286,320,202)(21,70,251,91)(22,71,252,92)(23,72,253,93)(24,73,254,94)(25,74,255,95)(26,75,256,96)(27,76,257,97)(28,77,258,98)(29,78,259,99)(30,79,260,100)(31,80,241,81)(32,61,242,82)(33,62,243,83)(34,63,244,84)(35,64,245,85)(36,65,246,86)(37,66,247,87)(38,67,248,88)(39,68,249,89)(40,69,250,90)(41,192,226,126)(42,193,227,127)(43,194,228,128)(44,195,229,129)(45,196,230,130)(46,197,231,131)(47,198,232,132)(48,199,233,133)(49,200,234,134)(50,181,235,135)(51,182,236,136)(52,183,237,137)(53,184,238,138)(54,185,239,139)(55,186,240,140)(56,187,221,121)(57,188,222,122)(58,189,223,123)(59,190,224,124)(60,191,225,125)(101,151,169,273)(102,152,170,274)(103,153,171,275)(104,154,172,276)(105,155,173,277)(106,156,174,278)(107,157,175,279)(108,158,176,280)(109,159,177,261)(110,160,178,262)(111,141,179,263)(112,142,180,264)(113,143,161,265)(114,144,162,266)(115,145,163,267)(116,146,164,268)(117,147,165,269)(118,148,166,270)(119,149,167,271)(120,150,168,272) );
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)], [(1,133,11,123),(2,132,12,122),(3,131,13,121),(4,130,14,140),(5,129,15,139),(6,128,16,138),(7,127,17,137),(8,126,18,136),(9,125,19,135),(10,124,20,134),(21,114,31,104),(22,113,32,103),(23,112,33,102),(24,111,34,101),(25,110,35,120),(26,109,36,119),(27,108,37,118),(28,107,38,117),(29,106,39,116),(30,105,40,115),(41,294,51,284),(42,293,52,283),(43,292,53,282),(44,291,54,281),(45,290,55,300),(46,289,56,299),(47,288,57,298),(48,287,58,297),(49,286,59,296),(50,285,60,295),(61,143,71,153),(62,142,72,152),(63,141,73,151),(64,160,74,150),(65,159,75,149),(66,158,76,148),(67,157,77,147),(68,156,78,146),(69,155,79,145),(70,154,80,144),(81,266,91,276),(82,265,92,275),(83,264,93,274),(84,263,94,273),(85,262,95,272),(86,261,96,271),(87,280,97,270),(88,279,98,269),(89,278,99,268),(90,277,100,267),(161,242,171,252),(162,241,172,251),(163,260,173,250),(164,259,174,249),(165,258,175,248),(166,257,176,247),(167,256,177,246),(168,255,178,245),(169,254,179,244),(170,253,180,243),(181,309,191,319),(182,308,192,318),(183,307,193,317),(184,306,194,316),(185,305,195,315),(186,304,196,314),(187,303,197,313),(188,302,198,312),(189,301,199,311),(190,320,200,310),(201,225,211,235),(202,224,212,234),(203,223,213,233),(204,222,214,232),(205,221,215,231),(206,240,216,230),(207,239,217,229),(208,238,218,228),(209,237,219,227),(210,236,220,226)], [(1,63,301,84),(2,74,302,95),(3,65,303,86),(4,76,304,97),(5,67,305,88),(6,78,306,99),(7,69,307,90),(8,80,308,81),(9,71,309,92),(10,62,310,83),(11,73,311,94),(12,64,312,85),(13,75,313,96),(14,66,314,87),(15,77,315,98),(16,68,316,89),(17,79,317,100),(18,70,318,91),(19,61,319,82),(20,72,320,93),(21,220,251,284),(22,211,252,295),(23,202,253,286),(24,213,254,297),(25,204,255,288),(26,215,256,299),(27,206,257,290),(28,217,258,281),(29,208,259,292),(30,219,260,283),(31,210,241,294),(32,201,242,285),(33,212,243,296),(34,203,244,287),(35,214,245,298),(36,205,246,289),(37,216,247,300),(38,207,248,291),(39,218,249,282),(40,209,250,293),(41,114,226,162),(42,105,227,173),(43,116,228,164),(44,107,229,175),(45,118,230,166),(46,109,231,177),(47,120,232,168),(48,111,233,179),(49,102,234,170),(50,113,235,161),(51,104,236,172),(52,115,237,163),(53,106,238,174),(54,117,239,165),(55,108,240,176),(56,119,221,167),(57,110,222,178),(58,101,223,169),(59,112,224,180),(60,103,225,171),(121,149,187,271),(122,160,188,262),(123,151,189,273),(124,142,190,264),(125,153,191,275),(126,144,192,266),(127,155,193,277),(128,146,194,268),(129,157,195,279),(130,148,196,270),(131,159,197,261),(132,150,198,272),(133,141,199,263),(134,152,200,274),(135,143,181,265),(136,154,182,276),(137,145,183,267),(138,156,184,278),(139,147,185,269),(140,158,186,280)], [(1,287,301,203),(2,288,302,204),(3,289,303,205),(4,290,304,206),(5,291,305,207),(6,292,306,208),(7,293,307,209),(8,294,308,210),(9,295,309,211),(10,296,310,212),(11,297,311,213),(12,298,312,214),(13,299,313,215),(14,300,314,216),(15,281,315,217),(16,282,316,218),(17,283,317,219),(18,284,318,220),(19,285,319,201),(20,286,320,202),(21,70,251,91),(22,71,252,92),(23,72,253,93),(24,73,254,94),(25,74,255,95),(26,75,256,96),(27,76,257,97),(28,77,258,98),(29,78,259,99),(30,79,260,100),(31,80,241,81),(32,61,242,82),(33,62,243,83),(34,63,244,84),(35,64,245,85),(36,65,246,86),(37,66,247,87),(38,67,248,88),(39,68,249,89),(40,69,250,90),(41,192,226,126),(42,193,227,127),(43,194,228,128),(44,195,229,129),(45,196,230,130),(46,197,231,131),(47,198,232,132),(48,199,233,133),(49,200,234,134),(50,181,235,135),(51,182,236,136),(52,183,237,137),(53,184,238,138),(54,185,239,139),(55,186,240,140),(56,187,221,121),(57,188,222,122),(58,189,223,123),(59,190,224,124),(60,191,225,125),(101,151,169,273),(102,152,170,274),(103,153,171,275),(104,154,172,276),(105,155,173,277),(106,156,174,278),(107,157,175,279),(108,158,176,280),(109,159,177,261),(110,160,178,262),(111,141,179,263),(112,142,180,264),(113,143,161,265),(114,144,162,266),(115,145,163,267),(116,146,164,268),(117,147,165,269),(118,148,166,270),(119,149,167,271),(120,150,168,272)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | D10 | 2+ 1+4 | Q8×D5 | D4⋊6D10 |
| kernel | Dic10⋊9Q8 | C4×Dic10 | Dic5⋊3Q8 | C20⋊Q8 | Dic5⋊Q8 | C5×C4⋊Q8 | Dic10 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 8 | 2 | 2 | 8 | 4 | 1 | 8 | 4 |
Matrix representation of Dic10⋊9Q8 ►in GL6(𝔽41)
| 10 | 34 | 0 | 0 | 0 | 0 |
| 32 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 23 | 0 | 0 | 0 | 0 |
| 32 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 31 | 7 | 0 | 0 | 0 | 0 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 11 | 40 |
G:=sub<GL(6,GF(41))| [10,32,0,0,0,0,34,31,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,9,0,0,0,0,18,40,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,32,0,0,0,0,23,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[31,9,0,0,0,0,7,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,11,40] >;
Dic10⋊9Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_9Q_8 % in TeX
G:=Group("Dic10:9Q8"); // GroupNames label
G:=SmallGroup(320,1394);
// by ID
G=gap.SmallGroup(320,1394);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,570,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations