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