metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊8Q8, C42.169D10, C10.332- (1+4), C4⋊Q8.14D5, C4.17(Q8×D5), C5⋊5(Q8⋊3Q8), C20.52(C2×Q8), C4⋊C4.121D10, (C2×C20).98C23, (C2×Q8).141D10, Dic5.26(C2×Q8), (Q8×Dic5).13C2, C20.135(C4○D4), C4.18(D4⋊2D5), C10.44(C22×Q8), (C2×C10).265C24, (C4×C20).206C22, (C4×Dic10).25C2, C4.Dic10.15C2, Dic5⋊3Q8.12C2, Dic5⋊Q8.10C2, C4⋊Dic5.382C22, Dic5.Q8.4C2, (Q8×C10).132C22, C22.286(C23×D5), (C4×Dic5).165C22, (C2×Dic5).280C23, C10.D4.57C22, C2.34(Q8.10D10), (C2×Dic10).310C22, C2.27(C2×Q8×D5), (C5×C4⋊Q8).14C2, C10.99(C2×C4○D4), C2.63(C2×D4⋊2D5), (C2×C4).90(C22×D5), (C5×C4⋊C4).208C22, SmallGroup(320,1393)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 534 in 200 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C4 [×4], C4 [×15], C22, C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C10 [×3], C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×18], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×5], C2×C10, C4×Q8 [×6], C42.C2 [×6], C4⋊Q8, C4⋊Q8 [×2], Dic10 [×4], Dic10 [×2], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], Q8⋊3Q8, C4×Dic5 [×8], C10.D4 [×12], C4⋊Dic5 [×2], C4⋊Dic5 [×4], C4×C20, C5×C4⋊C4 [×4], C2×Dic10 [×2], Q8×C10 [×2], C4×Dic10 [×2], Dic5⋊3Q8 [×2], Dic5.Q8 [×4], C4.Dic10 [×2], Dic5⋊Q8 [×2], Q8×Dic5 [×2], C5×C4⋊Q8, Dic10⋊8Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C22×D5 [×7], Q8⋊3Q8, D4⋊2D5 [×2], Q8×D5 [×2], C23×D5, C2×D4⋊2D5, C2×Q8×D5, Q8.10D10, Dic10⋊8Q8
Generators and relations
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, bd=db, 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)
(1 257 11 247)(2 256 12 246)(3 255 13 245)(4 254 14 244)(5 253 15 243)(6 252 16 242)(7 251 17 241)(8 250 18 260)(9 249 19 259)(10 248 20 258)(21 196 31 186)(22 195 32 185)(23 194 33 184)(24 193 34 183)(25 192 35 182)(26 191 36 181)(27 190 37 200)(28 189 38 199)(29 188 39 198)(30 187 40 197)(41 180 51 170)(42 179 52 169)(43 178 53 168)(44 177 54 167)(45 176 55 166)(46 175 56 165)(47 174 57 164)(48 173 58 163)(49 172 59 162)(50 171 60 161)(61 312 71 302)(62 311 72 301)(63 310 73 320)(64 309 74 319)(65 308 75 318)(66 307 76 317)(67 306 77 316)(68 305 78 315)(69 304 79 314)(70 303 80 313)(81 115 91 105)(82 114 92 104)(83 113 93 103)(84 112 94 102)(85 111 95 101)(86 110 96 120)(87 109 97 119)(88 108 98 118)(89 107 99 117)(90 106 100 116)(121 213 131 203)(122 212 132 202)(123 211 133 201)(124 210 134 220)(125 209 135 219)(126 208 136 218)(127 207 137 217)(128 206 138 216)(129 205 139 215)(130 204 140 214)(141 278 151 268)(142 277 152 267)(143 276 153 266)(144 275 154 265)(145 274 155 264)(146 273 156 263)(147 272 157 262)(148 271 158 261)(149 270 159 280)(150 269 160 279)(221 295 231 285)(222 294 232 284)(223 293 233 283)(224 292 234 282)(225 291 235 281)(226 290 236 300)(227 289 237 299)(228 288 238 298)(229 287 239 297)(230 286 240 296)
(1 77 42 99)(2 78 43 100)(3 79 44 81)(4 80 45 82)(5 61 46 83)(6 62 47 84)(7 63 48 85)(8 64 49 86)(9 65 50 87)(10 66 51 88)(11 67 52 89)(12 68 53 90)(13 69 54 91)(14 70 55 92)(15 71 56 93)(16 72 57 94)(17 73 58 95)(18 74 59 96)(19 75 60 97)(20 76 41 98)(21 298 265 135)(22 299 266 136)(23 300 267 137)(24 281 268 138)(25 282 269 139)(26 283 270 140)(27 284 271 121)(28 285 272 122)(29 286 273 123)(30 287 274 124)(31 288 275 125)(32 289 276 126)(33 290 277 127)(34 291 278 128)(35 292 279 129)(36 293 280 130)(37 294 261 131)(38 295 262 132)(39 296 263 133)(40 297 264 134)(101 241 320 163)(102 242 301 164)(103 243 302 165)(104 244 303 166)(105 245 304 167)(106 246 305 168)(107 247 306 169)(108 248 307 170)(109 249 308 171)(110 250 309 172)(111 251 310 173)(112 252 311 174)(113 253 312 175)(114 254 313 176)(115 255 314 177)(116 256 315 178)(117 257 316 179)(118 258 317 180)(119 259 318 161)(120 260 319 162)(141 216 193 225)(142 217 194 226)(143 218 195 227)(144 219 196 228)(145 220 197 229)(146 201 198 230)(147 202 199 231)(148 203 200 232)(149 204 181 233)(150 205 182 234)(151 206 183 235)(152 207 184 236)(153 208 185 237)(154 209 186 238)(155 210 187 239)(156 211 188 240)(157 212 189 221)(158 213 190 222)(159 214 191 223)(160 215 192 224)
(1 146 42 198)(2 157 43 189)(3 148 44 200)(4 159 45 191)(5 150 46 182)(6 141 47 193)(7 152 48 184)(8 143 49 195)(9 154 50 186)(10 145 51 197)(11 156 52 188)(12 147 53 199)(13 158 54 190)(14 149 55 181)(15 160 56 192)(16 151 57 183)(17 142 58 194)(18 153 59 185)(19 144 60 196)(20 155 41 187)(21 249 265 171)(22 260 266 162)(23 251 267 173)(24 242 268 164)(25 253 269 175)(26 244 270 166)(27 255 271 177)(28 246 272 168)(29 257 273 179)(30 248 274 170)(31 259 275 161)(32 250 276 172)(33 241 277 163)(34 252 278 174)(35 243 279 165)(36 254 280 176)(37 245 261 167)(38 256 262 178)(39 247 263 169)(40 258 264 180)(61 234 83 205)(62 225 84 216)(63 236 85 207)(64 227 86 218)(65 238 87 209)(66 229 88 220)(67 240 89 211)(68 231 90 202)(69 222 91 213)(70 233 92 204)(71 224 93 215)(72 235 94 206)(73 226 95 217)(74 237 96 208)(75 228 97 219)(76 239 98 210)(77 230 99 201)(78 221 100 212)(79 232 81 203)(80 223 82 214)(101 127 320 290)(102 138 301 281)(103 129 302 292)(104 140 303 283)(105 131 304 294)(106 122 305 285)(107 133 306 296)(108 124 307 287)(109 135 308 298)(110 126 309 289)(111 137 310 300)(112 128 311 291)(113 139 312 282)(114 130 313 293)(115 121 314 284)(116 132 315 295)(117 123 316 286)(118 134 317 297)(119 125 318 288)(120 136 319 299)
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), (1,257,11,247)(2,256,12,246)(3,255,13,245)(4,254,14,244)(5,253,15,243)(6,252,16,242)(7,251,17,241)(8,250,18,260)(9,249,19,259)(10,248,20,258)(21,196,31,186)(22,195,32,185)(23,194,33,184)(24,193,34,183)(25,192,35,182)(26,191,36,181)(27,190,37,200)(28,189,38,199)(29,188,39,198)(30,187,40,197)(41,180,51,170)(42,179,52,169)(43,178,53,168)(44,177,54,167)(45,176,55,166)(46,175,56,165)(47,174,57,164)(48,173,58,163)(49,172,59,162)(50,171,60,161)(61,312,71,302)(62,311,72,301)(63,310,73,320)(64,309,74,319)(65,308,75,318)(66,307,76,317)(67,306,77,316)(68,305,78,315)(69,304,79,314)(70,303,80,313)(81,115,91,105)(82,114,92,104)(83,113,93,103)(84,112,94,102)(85,111,95,101)(86,110,96,120)(87,109,97,119)(88,108,98,118)(89,107,99,117)(90,106,100,116)(121,213,131,203)(122,212,132,202)(123,211,133,201)(124,210,134,220)(125,209,135,219)(126,208,136,218)(127,207,137,217)(128,206,138,216)(129,205,139,215)(130,204,140,214)(141,278,151,268)(142,277,152,267)(143,276,153,266)(144,275,154,265)(145,274,155,264)(146,273,156,263)(147,272,157,262)(148,271,158,261)(149,270,159,280)(150,269,160,279)(221,295,231,285)(222,294,232,284)(223,293,233,283)(224,292,234,282)(225,291,235,281)(226,290,236,300)(227,289,237,299)(228,288,238,298)(229,287,239,297)(230,286,240,296), (1,77,42,99)(2,78,43,100)(3,79,44,81)(4,80,45,82)(5,61,46,83)(6,62,47,84)(7,63,48,85)(8,64,49,86)(9,65,50,87)(10,66,51,88)(11,67,52,89)(12,68,53,90)(13,69,54,91)(14,70,55,92)(15,71,56,93)(16,72,57,94)(17,73,58,95)(18,74,59,96)(19,75,60,97)(20,76,41,98)(21,298,265,135)(22,299,266,136)(23,300,267,137)(24,281,268,138)(25,282,269,139)(26,283,270,140)(27,284,271,121)(28,285,272,122)(29,286,273,123)(30,287,274,124)(31,288,275,125)(32,289,276,126)(33,290,277,127)(34,291,278,128)(35,292,279,129)(36,293,280,130)(37,294,261,131)(38,295,262,132)(39,296,263,133)(40,297,264,134)(101,241,320,163)(102,242,301,164)(103,243,302,165)(104,244,303,166)(105,245,304,167)(106,246,305,168)(107,247,306,169)(108,248,307,170)(109,249,308,171)(110,250,309,172)(111,251,310,173)(112,252,311,174)(113,253,312,175)(114,254,313,176)(115,255,314,177)(116,256,315,178)(117,257,316,179)(118,258,317,180)(119,259,318,161)(120,260,319,162)(141,216,193,225)(142,217,194,226)(143,218,195,227)(144,219,196,228)(145,220,197,229)(146,201,198,230)(147,202,199,231)(148,203,200,232)(149,204,181,233)(150,205,182,234)(151,206,183,235)(152,207,184,236)(153,208,185,237)(154,209,186,238)(155,210,187,239)(156,211,188,240)(157,212,189,221)(158,213,190,222)(159,214,191,223)(160,215,192,224), (1,146,42,198)(2,157,43,189)(3,148,44,200)(4,159,45,191)(5,150,46,182)(6,141,47,193)(7,152,48,184)(8,143,49,195)(9,154,50,186)(10,145,51,197)(11,156,52,188)(12,147,53,199)(13,158,54,190)(14,149,55,181)(15,160,56,192)(16,151,57,183)(17,142,58,194)(18,153,59,185)(19,144,60,196)(20,155,41,187)(21,249,265,171)(22,260,266,162)(23,251,267,173)(24,242,268,164)(25,253,269,175)(26,244,270,166)(27,255,271,177)(28,246,272,168)(29,257,273,179)(30,248,274,170)(31,259,275,161)(32,250,276,172)(33,241,277,163)(34,252,278,174)(35,243,279,165)(36,254,280,176)(37,245,261,167)(38,256,262,178)(39,247,263,169)(40,258,264,180)(61,234,83,205)(62,225,84,216)(63,236,85,207)(64,227,86,218)(65,238,87,209)(66,229,88,220)(67,240,89,211)(68,231,90,202)(69,222,91,213)(70,233,92,204)(71,224,93,215)(72,235,94,206)(73,226,95,217)(74,237,96,208)(75,228,97,219)(76,239,98,210)(77,230,99,201)(78,221,100,212)(79,232,81,203)(80,223,82,214)(101,127,320,290)(102,138,301,281)(103,129,302,292)(104,140,303,283)(105,131,304,294)(106,122,305,285)(107,133,306,296)(108,124,307,287)(109,135,308,298)(110,126,309,289)(111,137,310,300)(112,128,311,291)(113,139,312,282)(114,130,313,293)(115,121,314,284)(116,132,315,295)(117,123,316,286)(118,134,317,297)(119,125,318,288)(120,136,319,299)>;
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), (1,257,11,247)(2,256,12,246)(3,255,13,245)(4,254,14,244)(5,253,15,243)(6,252,16,242)(7,251,17,241)(8,250,18,260)(9,249,19,259)(10,248,20,258)(21,196,31,186)(22,195,32,185)(23,194,33,184)(24,193,34,183)(25,192,35,182)(26,191,36,181)(27,190,37,200)(28,189,38,199)(29,188,39,198)(30,187,40,197)(41,180,51,170)(42,179,52,169)(43,178,53,168)(44,177,54,167)(45,176,55,166)(46,175,56,165)(47,174,57,164)(48,173,58,163)(49,172,59,162)(50,171,60,161)(61,312,71,302)(62,311,72,301)(63,310,73,320)(64,309,74,319)(65,308,75,318)(66,307,76,317)(67,306,77,316)(68,305,78,315)(69,304,79,314)(70,303,80,313)(81,115,91,105)(82,114,92,104)(83,113,93,103)(84,112,94,102)(85,111,95,101)(86,110,96,120)(87,109,97,119)(88,108,98,118)(89,107,99,117)(90,106,100,116)(121,213,131,203)(122,212,132,202)(123,211,133,201)(124,210,134,220)(125,209,135,219)(126,208,136,218)(127,207,137,217)(128,206,138,216)(129,205,139,215)(130,204,140,214)(141,278,151,268)(142,277,152,267)(143,276,153,266)(144,275,154,265)(145,274,155,264)(146,273,156,263)(147,272,157,262)(148,271,158,261)(149,270,159,280)(150,269,160,279)(221,295,231,285)(222,294,232,284)(223,293,233,283)(224,292,234,282)(225,291,235,281)(226,290,236,300)(227,289,237,299)(228,288,238,298)(229,287,239,297)(230,286,240,296), (1,77,42,99)(2,78,43,100)(3,79,44,81)(4,80,45,82)(5,61,46,83)(6,62,47,84)(7,63,48,85)(8,64,49,86)(9,65,50,87)(10,66,51,88)(11,67,52,89)(12,68,53,90)(13,69,54,91)(14,70,55,92)(15,71,56,93)(16,72,57,94)(17,73,58,95)(18,74,59,96)(19,75,60,97)(20,76,41,98)(21,298,265,135)(22,299,266,136)(23,300,267,137)(24,281,268,138)(25,282,269,139)(26,283,270,140)(27,284,271,121)(28,285,272,122)(29,286,273,123)(30,287,274,124)(31,288,275,125)(32,289,276,126)(33,290,277,127)(34,291,278,128)(35,292,279,129)(36,293,280,130)(37,294,261,131)(38,295,262,132)(39,296,263,133)(40,297,264,134)(101,241,320,163)(102,242,301,164)(103,243,302,165)(104,244,303,166)(105,245,304,167)(106,246,305,168)(107,247,306,169)(108,248,307,170)(109,249,308,171)(110,250,309,172)(111,251,310,173)(112,252,311,174)(113,253,312,175)(114,254,313,176)(115,255,314,177)(116,256,315,178)(117,257,316,179)(118,258,317,180)(119,259,318,161)(120,260,319,162)(141,216,193,225)(142,217,194,226)(143,218,195,227)(144,219,196,228)(145,220,197,229)(146,201,198,230)(147,202,199,231)(148,203,200,232)(149,204,181,233)(150,205,182,234)(151,206,183,235)(152,207,184,236)(153,208,185,237)(154,209,186,238)(155,210,187,239)(156,211,188,240)(157,212,189,221)(158,213,190,222)(159,214,191,223)(160,215,192,224), (1,146,42,198)(2,157,43,189)(3,148,44,200)(4,159,45,191)(5,150,46,182)(6,141,47,193)(7,152,48,184)(8,143,49,195)(9,154,50,186)(10,145,51,197)(11,156,52,188)(12,147,53,199)(13,158,54,190)(14,149,55,181)(15,160,56,192)(16,151,57,183)(17,142,58,194)(18,153,59,185)(19,144,60,196)(20,155,41,187)(21,249,265,171)(22,260,266,162)(23,251,267,173)(24,242,268,164)(25,253,269,175)(26,244,270,166)(27,255,271,177)(28,246,272,168)(29,257,273,179)(30,248,274,170)(31,259,275,161)(32,250,276,172)(33,241,277,163)(34,252,278,174)(35,243,279,165)(36,254,280,176)(37,245,261,167)(38,256,262,178)(39,247,263,169)(40,258,264,180)(61,234,83,205)(62,225,84,216)(63,236,85,207)(64,227,86,218)(65,238,87,209)(66,229,88,220)(67,240,89,211)(68,231,90,202)(69,222,91,213)(70,233,92,204)(71,224,93,215)(72,235,94,206)(73,226,95,217)(74,237,96,208)(75,228,97,219)(76,239,98,210)(77,230,99,201)(78,221,100,212)(79,232,81,203)(80,223,82,214)(101,127,320,290)(102,138,301,281)(103,129,302,292)(104,140,303,283)(105,131,304,294)(106,122,305,285)(107,133,306,296)(108,124,307,287)(109,135,308,298)(110,126,309,289)(111,137,310,300)(112,128,311,291)(113,139,312,282)(114,130,313,293)(115,121,314,284)(116,132,315,295)(117,123,316,286)(118,134,317,297)(119,125,318,288)(120,136,319,299) );
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)], [(1,257,11,247),(2,256,12,246),(3,255,13,245),(4,254,14,244),(5,253,15,243),(6,252,16,242),(7,251,17,241),(8,250,18,260),(9,249,19,259),(10,248,20,258),(21,196,31,186),(22,195,32,185),(23,194,33,184),(24,193,34,183),(25,192,35,182),(26,191,36,181),(27,190,37,200),(28,189,38,199),(29,188,39,198),(30,187,40,197),(41,180,51,170),(42,179,52,169),(43,178,53,168),(44,177,54,167),(45,176,55,166),(46,175,56,165),(47,174,57,164),(48,173,58,163),(49,172,59,162),(50,171,60,161),(61,312,71,302),(62,311,72,301),(63,310,73,320),(64,309,74,319),(65,308,75,318),(66,307,76,317),(67,306,77,316),(68,305,78,315),(69,304,79,314),(70,303,80,313),(81,115,91,105),(82,114,92,104),(83,113,93,103),(84,112,94,102),(85,111,95,101),(86,110,96,120),(87,109,97,119),(88,108,98,118),(89,107,99,117),(90,106,100,116),(121,213,131,203),(122,212,132,202),(123,211,133,201),(124,210,134,220),(125,209,135,219),(126,208,136,218),(127,207,137,217),(128,206,138,216),(129,205,139,215),(130,204,140,214),(141,278,151,268),(142,277,152,267),(143,276,153,266),(144,275,154,265),(145,274,155,264),(146,273,156,263),(147,272,157,262),(148,271,158,261),(149,270,159,280),(150,269,160,279),(221,295,231,285),(222,294,232,284),(223,293,233,283),(224,292,234,282),(225,291,235,281),(226,290,236,300),(227,289,237,299),(228,288,238,298),(229,287,239,297),(230,286,240,296)], [(1,77,42,99),(2,78,43,100),(3,79,44,81),(4,80,45,82),(5,61,46,83),(6,62,47,84),(7,63,48,85),(8,64,49,86),(9,65,50,87),(10,66,51,88),(11,67,52,89),(12,68,53,90),(13,69,54,91),(14,70,55,92),(15,71,56,93),(16,72,57,94),(17,73,58,95),(18,74,59,96),(19,75,60,97),(20,76,41,98),(21,298,265,135),(22,299,266,136),(23,300,267,137),(24,281,268,138),(25,282,269,139),(26,283,270,140),(27,284,271,121),(28,285,272,122),(29,286,273,123),(30,287,274,124),(31,288,275,125),(32,289,276,126),(33,290,277,127),(34,291,278,128),(35,292,279,129),(36,293,280,130),(37,294,261,131),(38,295,262,132),(39,296,263,133),(40,297,264,134),(101,241,320,163),(102,242,301,164),(103,243,302,165),(104,244,303,166),(105,245,304,167),(106,246,305,168),(107,247,306,169),(108,248,307,170),(109,249,308,171),(110,250,309,172),(111,251,310,173),(112,252,311,174),(113,253,312,175),(114,254,313,176),(115,255,314,177),(116,256,315,178),(117,257,316,179),(118,258,317,180),(119,259,318,161),(120,260,319,162),(141,216,193,225),(142,217,194,226),(143,218,195,227),(144,219,196,228),(145,220,197,229),(146,201,198,230),(147,202,199,231),(148,203,200,232),(149,204,181,233),(150,205,182,234),(151,206,183,235),(152,207,184,236),(153,208,185,237),(154,209,186,238),(155,210,187,239),(156,211,188,240),(157,212,189,221),(158,213,190,222),(159,214,191,223),(160,215,192,224)], [(1,146,42,198),(2,157,43,189),(3,148,44,200),(4,159,45,191),(5,150,46,182),(6,141,47,193),(7,152,48,184),(8,143,49,195),(9,154,50,186),(10,145,51,197),(11,156,52,188),(12,147,53,199),(13,158,54,190),(14,149,55,181),(15,160,56,192),(16,151,57,183),(17,142,58,194),(18,153,59,185),(19,144,60,196),(20,155,41,187),(21,249,265,171),(22,260,266,162),(23,251,267,173),(24,242,268,164),(25,253,269,175),(26,244,270,166),(27,255,271,177),(28,246,272,168),(29,257,273,179),(30,248,274,170),(31,259,275,161),(32,250,276,172),(33,241,277,163),(34,252,278,174),(35,243,279,165),(36,254,280,176),(37,245,261,167),(38,256,262,178),(39,247,263,169),(40,258,264,180),(61,234,83,205),(62,225,84,216),(63,236,85,207),(64,227,86,218),(65,238,87,209),(66,229,88,220),(67,240,89,211),(68,231,90,202),(69,222,91,213),(70,233,92,204),(71,224,93,215),(72,235,94,206),(73,226,95,217),(74,237,96,208),(75,228,97,219),(76,239,98,210),(77,230,99,201),(78,221,100,212),(79,232,81,203),(80,223,82,214),(101,127,320,290),(102,138,301,281),(103,129,302,292),(104,140,303,283),(105,131,304,294),(106,122,305,285),(107,133,306,296),(108,124,307,287),(109,135,308,298),(110,126,309,289),(111,137,310,300),(112,128,311,291),(113,139,312,282),(114,130,313,293),(115,121,314,284),(116,132,315,295),(117,123,316,286),(118,134,317,297),(119,125,318,288),(120,136,319,299)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 10 | 4 | 0 | 0 | 0 | 0 |
| 26 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 24 | 15 | 0 | 0 | 0 | 0 |
| 8 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 39 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 36 |
| 0 | 0 | 0 | 0 | 15 | 19 |
| 11 | 12 | 0 | 0 | 0 | 0 |
| 31 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 7 |
| 0 | 0 | 0 | 0 | 23 | 24 |
G:=sub<GL(6,GF(41))| [10,26,0,0,0,0,4,31,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,8,0,0,0,0,15,17,0,0,0,0,0,0,21,15,0,0,0,0,39,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,15,0,0,0,0,36,19],[11,31,0,0,0,0,12,30,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,23,0,0,0,0,7,24] >;
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D4⋊2D5 | Q8×D5 | Q8.10D10 |
| kernel | Dic10⋊8Q8 | C4×Dic10 | Dic5⋊3Q8 | Dic5.Q8 | C4.Dic10 | Dic5⋊Q8 | Q8×Dic5 | C5×C4⋊Q8 | Dic10 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_8Q_8 % in TeX
G:=Group("Dic10:8Q8"); // GroupNames label
G:=SmallGroup(320,1393);
// by ID
G=gap.SmallGroup(320,1393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,297,136,12550]);
// 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,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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ℤ
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