metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.2Q8, C4.6(Q8×D5), C5⋊5(Q8.Q8), C4⋊C4.45D10, (C2×C8).26D10, C2.D8.6D5, C20.19(C2×Q8), C4.79(C4○D20), C10.27(C4○D8), (C2×Dic5).55D4, C10.Q16.9C2, C22.226(D4×D5), C4.Dic10.9C2, C20.Q8.8C2, C20.171(C4○D4), C2.12(D8⋊3D5), (C2×C20).293C23, (C2×C40).240C22, Dic5⋊3Q8.9C2, C10.40(C22⋊Q8), C20.8Q8.11C2, C2.17(D10⋊Q8), C2.21(Q16⋊D5), C20.44D4.11C2, C10.68(C8.C22), C4⋊Dic5.119C22, (C4×Dic5).42C22, (C2×Dic10).92C22, (C5×C2.D8).12C2, (C2×C10).298(C2×D4), (C5×C4⋊C4).86C22, (C2×C5⋊2C8).67C22, (C2×C4).396(C22×D5), SmallGroup(320,504)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.2Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=a10c2, bab-1=a-1, cac-1=a11, dad-1=a9, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 310 in 90 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C5⋊2C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, Q8.Q8, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C20.Q8, C10.Q16, C20.8Q8, C20.44D4, C5×C2.D8, Dic5⋊3Q8, C4.Dic10, Dic10.2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8.C22, C22×D5, Q8.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D8⋊3D5, Q16⋊D5, Dic10.2Q8
►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)
(1 102 11 112)(2 101 12 111)(3 120 13 110)(4 119 14 109)(5 118 15 108)(6 117 16 107)(7 116 17 106)(8 115 18 105)(9 114 19 104)(10 113 20 103)(21 78 31 68)(22 77 32 67)(23 76 33 66)(24 75 34 65)(25 74 35 64)(26 73 36 63)(27 72 37 62)(28 71 38 61)(29 70 39 80)(30 69 40 79)(41 159 51 149)(42 158 52 148)(43 157 53 147)(44 156 54 146)(45 155 55 145)(46 154 56 144)(47 153 57 143)(48 152 58 142)(49 151 59 141)(50 150 60 160)(81 200 91 190)(82 199 92 189)(83 198 93 188)(84 197 94 187)(85 196 95 186)(86 195 96 185)(87 194 97 184)(88 193 98 183)(89 192 99 182)(90 191 100 181)(121 253 131 243)(122 252 132 242)(123 251 133 241)(124 250 134 260)(125 249 135 259)(126 248 136 258)(127 247 137 257)(128 246 138 256)(129 245 139 255)(130 244 140 254)(161 268 171 278)(162 267 172 277)(163 266 173 276)(164 265 174 275)(165 264 175 274)(166 263 176 273)(167 262 177 272)(168 261 178 271)(169 280 179 270)(170 279 180 269)(201 301 211 311)(202 320 212 310)(203 319 213 309)(204 318 214 308)(205 317 215 307)(206 316 216 306)(207 315 217 305)(208 314 218 304)(209 313 219 303)(210 312 220 302)(221 282 231 292)(222 281 232 291)(223 300 233 290)(224 299 234 289)(225 298 235 288)(226 297 236 287)(227 296 237 286)(228 295 238 285)(229 294 239 284)(230 293 240 283)
(1 50 123 89)(2 41 124 100)(3 52 125 91)(4 43 126 82)(5 54 127 93)(6 45 128 84)(7 56 129 95)(8 47 130 86)(9 58 131 97)(10 49 132 88)(11 60 133 99)(12 51 134 90)(13 42 135 81)(14 53 136 92)(15 44 137 83)(16 55 138 94)(17 46 139 85)(18 57 140 96)(19 48 121 87)(20 59 122 98)(21 219 168 295)(22 210 169 286)(23 201 170 297)(24 212 171 288)(25 203 172 299)(26 214 173 290)(27 205 174 281)(28 216 175 292)(29 207 176 283)(30 218 177 294)(31 209 178 285)(32 220 179 296)(33 211 180 287)(34 202 161 298)(35 213 162 289)(36 204 163 300)(37 215 164 291)(38 206 165 282)(39 217 166 293)(40 208 167 284)(61 301 264 236)(62 312 265 227)(63 303 266 238)(64 314 267 229)(65 305 268 240)(66 316 269 231)(67 307 270 222)(68 318 271 233)(69 309 272 224)(70 320 273 235)(71 311 274 226)(72 302 275 237)(73 313 276 228)(74 304 277 239)(75 315 278 230)(76 306 279 221)(77 317 280 232)(78 308 261 223)(79 319 262 234)(80 310 263 225)(101 144 250 186)(102 155 251 197)(103 146 252 188)(104 157 253 199)(105 148 254 190)(106 159 255 181)(107 150 256 192)(108 141 257 183)(109 152 258 194)(110 143 259 185)(111 154 260 196)(112 145 241 187)(113 156 242 198)(114 147 243 189)(115 158 244 200)(116 149 245 191)(117 160 246 182)(118 151 247 193)(119 142 248 184)(120 153 249 195)
(1 30 133 167)(2 39 134 176)(3 28 135 165)(4 37 136 174)(5 26 137 163)(6 35 138 172)(7 24 139 161)(8 33 140 170)(9 22 121 179)(10 31 122 168)(11 40 123 177)(12 29 124 166)(13 38 125 175)(14 27 126 164)(15 36 127 173)(16 25 128 162)(17 34 129 171)(18 23 130 180)(19 32 131 169)(20 21 132 178)(41 293 90 207)(42 282 91 216)(43 291 92 205)(44 300 93 214)(45 289 94 203)(46 298 95 212)(47 287 96 201)(48 296 97 210)(49 285 98 219)(50 294 99 208)(51 283 100 217)(52 292 81 206)(53 281 82 215)(54 290 83 204)(55 299 84 213)(56 288 85 202)(57 297 86 211)(58 286 87 220)(59 295 88 209)(60 284 89 218)(61 249 274 110)(62 258 275 119)(63 247 276 108)(64 256 277 117)(65 245 278 106)(66 254 279 115)(67 243 280 104)(68 252 261 113)(69 241 262 102)(70 250 263 111)(71 259 264 120)(72 248 265 109)(73 257 266 118)(74 246 267 107)(75 255 268 116)(76 244 269 105)(77 253 270 114)(78 242 271 103)(79 251 272 112)(80 260 273 101)(141 238 193 313)(142 227 194 302)(143 236 195 311)(144 225 196 320)(145 234 197 309)(146 223 198 318)(147 232 199 307)(148 221 200 316)(149 230 181 305)(150 239 182 314)(151 228 183 303)(152 237 184 312)(153 226 185 301)(154 235 186 310)(155 224 187 319)(156 233 188 308)(157 222 189 317)(158 231 190 306)(159 240 191 315)(160 229 192 304)
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), (1,102,11,112)(2,101,12,111)(3,120,13,110)(4,119,14,109)(5,118,15,108)(6,117,16,107)(7,116,17,106)(8,115,18,105)(9,114,19,104)(10,113,20,103)(21,78,31,68)(22,77,32,67)(23,76,33,66)(24,75,34,65)(25,74,35,64)(26,73,36,63)(27,72,37,62)(28,71,38,61)(29,70,39,80)(30,69,40,79)(41,159,51,149)(42,158,52,148)(43,157,53,147)(44,156,54,146)(45,155,55,145)(46,154,56,144)(47,153,57,143)(48,152,58,142)(49,151,59,141)(50,150,60,160)(81,200,91,190)(82,199,92,189)(83,198,93,188)(84,197,94,187)(85,196,95,186)(86,195,96,185)(87,194,97,184)(88,193,98,183)(89,192,99,182)(90,191,100,181)(121,253,131,243)(122,252,132,242)(123,251,133,241)(124,250,134,260)(125,249,135,259)(126,248,136,258)(127,247,137,257)(128,246,138,256)(129,245,139,255)(130,244,140,254)(161,268,171,278)(162,267,172,277)(163,266,173,276)(164,265,174,275)(165,264,175,274)(166,263,176,273)(167,262,177,272)(168,261,178,271)(169,280,179,270)(170,279,180,269)(201,301,211,311)(202,320,212,310)(203,319,213,309)(204,318,214,308)(205,317,215,307)(206,316,216,306)(207,315,217,305)(208,314,218,304)(209,313,219,303)(210,312,220,302)(221,282,231,292)(222,281,232,291)(223,300,233,290)(224,299,234,289)(225,298,235,288)(226,297,236,287)(227,296,237,286)(228,295,238,285)(229,294,239,284)(230,293,240,283), (1,50,123,89)(2,41,124,100)(3,52,125,91)(4,43,126,82)(5,54,127,93)(6,45,128,84)(7,56,129,95)(8,47,130,86)(9,58,131,97)(10,49,132,88)(11,60,133,99)(12,51,134,90)(13,42,135,81)(14,53,136,92)(15,44,137,83)(16,55,138,94)(17,46,139,85)(18,57,140,96)(19,48,121,87)(20,59,122,98)(21,219,168,295)(22,210,169,286)(23,201,170,297)(24,212,171,288)(25,203,172,299)(26,214,173,290)(27,205,174,281)(28,216,175,292)(29,207,176,283)(30,218,177,294)(31,209,178,285)(32,220,179,296)(33,211,180,287)(34,202,161,298)(35,213,162,289)(36,204,163,300)(37,215,164,291)(38,206,165,282)(39,217,166,293)(40,208,167,284)(61,301,264,236)(62,312,265,227)(63,303,266,238)(64,314,267,229)(65,305,268,240)(66,316,269,231)(67,307,270,222)(68,318,271,233)(69,309,272,224)(70,320,273,235)(71,311,274,226)(72,302,275,237)(73,313,276,228)(74,304,277,239)(75,315,278,230)(76,306,279,221)(77,317,280,232)(78,308,261,223)(79,319,262,234)(80,310,263,225)(101,144,250,186)(102,155,251,197)(103,146,252,188)(104,157,253,199)(105,148,254,190)(106,159,255,181)(107,150,256,192)(108,141,257,183)(109,152,258,194)(110,143,259,185)(111,154,260,196)(112,145,241,187)(113,156,242,198)(114,147,243,189)(115,158,244,200)(116,149,245,191)(117,160,246,182)(118,151,247,193)(119,142,248,184)(120,153,249,195), (1,30,133,167)(2,39,134,176)(3,28,135,165)(4,37,136,174)(5,26,137,163)(6,35,138,172)(7,24,139,161)(8,33,140,170)(9,22,121,179)(10,31,122,168)(11,40,123,177)(12,29,124,166)(13,38,125,175)(14,27,126,164)(15,36,127,173)(16,25,128,162)(17,34,129,171)(18,23,130,180)(19,32,131,169)(20,21,132,178)(41,293,90,207)(42,282,91,216)(43,291,92,205)(44,300,93,214)(45,289,94,203)(46,298,95,212)(47,287,96,201)(48,296,97,210)(49,285,98,219)(50,294,99,208)(51,283,100,217)(52,292,81,206)(53,281,82,215)(54,290,83,204)(55,299,84,213)(56,288,85,202)(57,297,86,211)(58,286,87,220)(59,295,88,209)(60,284,89,218)(61,249,274,110)(62,258,275,119)(63,247,276,108)(64,256,277,117)(65,245,278,106)(66,254,279,115)(67,243,280,104)(68,252,261,113)(69,241,262,102)(70,250,263,111)(71,259,264,120)(72,248,265,109)(73,257,266,118)(74,246,267,107)(75,255,268,116)(76,244,269,105)(77,253,270,114)(78,242,271,103)(79,251,272,112)(80,260,273,101)(141,238,193,313)(142,227,194,302)(143,236,195,311)(144,225,196,320)(145,234,197,309)(146,223,198,318)(147,232,199,307)(148,221,200,316)(149,230,181,305)(150,239,182,314)(151,228,183,303)(152,237,184,312)(153,226,185,301)(154,235,186,310)(155,224,187,319)(156,233,188,308)(157,222,189,317)(158,231,190,306)(159,240,191,315)(160,229,192,304)>;
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), (1,102,11,112)(2,101,12,111)(3,120,13,110)(4,119,14,109)(5,118,15,108)(6,117,16,107)(7,116,17,106)(8,115,18,105)(9,114,19,104)(10,113,20,103)(21,78,31,68)(22,77,32,67)(23,76,33,66)(24,75,34,65)(25,74,35,64)(26,73,36,63)(27,72,37,62)(28,71,38,61)(29,70,39,80)(30,69,40,79)(41,159,51,149)(42,158,52,148)(43,157,53,147)(44,156,54,146)(45,155,55,145)(46,154,56,144)(47,153,57,143)(48,152,58,142)(49,151,59,141)(50,150,60,160)(81,200,91,190)(82,199,92,189)(83,198,93,188)(84,197,94,187)(85,196,95,186)(86,195,96,185)(87,194,97,184)(88,193,98,183)(89,192,99,182)(90,191,100,181)(121,253,131,243)(122,252,132,242)(123,251,133,241)(124,250,134,260)(125,249,135,259)(126,248,136,258)(127,247,137,257)(128,246,138,256)(129,245,139,255)(130,244,140,254)(161,268,171,278)(162,267,172,277)(163,266,173,276)(164,265,174,275)(165,264,175,274)(166,263,176,273)(167,262,177,272)(168,261,178,271)(169,280,179,270)(170,279,180,269)(201,301,211,311)(202,320,212,310)(203,319,213,309)(204,318,214,308)(205,317,215,307)(206,316,216,306)(207,315,217,305)(208,314,218,304)(209,313,219,303)(210,312,220,302)(221,282,231,292)(222,281,232,291)(223,300,233,290)(224,299,234,289)(225,298,235,288)(226,297,236,287)(227,296,237,286)(228,295,238,285)(229,294,239,284)(230,293,240,283), (1,50,123,89)(2,41,124,100)(3,52,125,91)(4,43,126,82)(5,54,127,93)(6,45,128,84)(7,56,129,95)(8,47,130,86)(9,58,131,97)(10,49,132,88)(11,60,133,99)(12,51,134,90)(13,42,135,81)(14,53,136,92)(15,44,137,83)(16,55,138,94)(17,46,139,85)(18,57,140,96)(19,48,121,87)(20,59,122,98)(21,219,168,295)(22,210,169,286)(23,201,170,297)(24,212,171,288)(25,203,172,299)(26,214,173,290)(27,205,174,281)(28,216,175,292)(29,207,176,283)(30,218,177,294)(31,209,178,285)(32,220,179,296)(33,211,180,287)(34,202,161,298)(35,213,162,289)(36,204,163,300)(37,215,164,291)(38,206,165,282)(39,217,166,293)(40,208,167,284)(61,301,264,236)(62,312,265,227)(63,303,266,238)(64,314,267,229)(65,305,268,240)(66,316,269,231)(67,307,270,222)(68,318,271,233)(69,309,272,224)(70,320,273,235)(71,311,274,226)(72,302,275,237)(73,313,276,228)(74,304,277,239)(75,315,278,230)(76,306,279,221)(77,317,280,232)(78,308,261,223)(79,319,262,234)(80,310,263,225)(101,144,250,186)(102,155,251,197)(103,146,252,188)(104,157,253,199)(105,148,254,190)(106,159,255,181)(107,150,256,192)(108,141,257,183)(109,152,258,194)(110,143,259,185)(111,154,260,196)(112,145,241,187)(113,156,242,198)(114,147,243,189)(115,158,244,200)(116,149,245,191)(117,160,246,182)(118,151,247,193)(119,142,248,184)(120,153,249,195), (1,30,133,167)(2,39,134,176)(3,28,135,165)(4,37,136,174)(5,26,137,163)(6,35,138,172)(7,24,139,161)(8,33,140,170)(9,22,121,179)(10,31,122,168)(11,40,123,177)(12,29,124,166)(13,38,125,175)(14,27,126,164)(15,36,127,173)(16,25,128,162)(17,34,129,171)(18,23,130,180)(19,32,131,169)(20,21,132,178)(41,293,90,207)(42,282,91,216)(43,291,92,205)(44,300,93,214)(45,289,94,203)(46,298,95,212)(47,287,96,201)(48,296,97,210)(49,285,98,219)(50,294,99,208)(51,283,100,217)(52,292,81,206)(53,281,82,215)(54,290,83,204)(55,299,84,213)(56,288,85,202)(57,297,86,211)(58,286,87,220)(59,295,88,209)(60,284,89,218)(61,249,274,110)(62,258,275,119)(63,247,276,108)(64,256,277,117)(65,245,278,106)(66,254,279,115)(67,243,280,104)(68,252,261,113)(69,241,262,102)(70,250,263,111)(71,259,264,120)(72,248,265,109)(73,257,266,118)(74,246,267,107)(75,255,268,116)(76,244,269,105)(77,253,270,114)(78,242,271,103)(79,251,272,112)(80,260,273,101)(141,238,193,313)(142,227,194,302)(143,236,195,311)(144,225,196,320)(145,234,197,309)(146,223,198,318)(147,232,199,307)(148,221,200,316)(149,230,181,305)(150,239,182,314)(151,228,183,303)(152,237,184,312)(153,226,185,301)(154,235,186,310)(155,224,187,319)(156,233,188,308)(157,222,189,317)(158,231,190,306)(159,240,191,315)(160,229,192,304) );
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)], [(1,102,11,112),(2,101,12,111),(3,120,13,110),(4,119,14,109),(5,118,15,108),(6,117,16,107),(7,116,17,106),(8,115,18,105),(9,114,19,104),(10,113,20,103),(21,78,31,68),(22,77,32,67),(23,76,33,66),(24,75,34,65),(25,74,35,64),(26,73,36,63),(27,72,37,62),(28,71,38,61),(29,70,39,80),(30,69,40,79),(41,159,51,149),(42,158,52,148),(43,157,53,147),(44,156,54,146),(45,155,55,145),(46,154,56,144),(47,153,57,143),(48,152,58,142),(49,151,59,141),(50,150,60,160),(81,200,91,190),(82,199,92,189),(83,198,93,188),(84,197,94,187),(85,196,95,186),(86,195,96,185),(87,194,97,184),(88,193,98,183),(89,192,99,182),(90,191,100,181),(121,253,131,243),(122,252,132,242),(123,251,133,241),(124,250,134,260),(125,249,135,259),(126,248,136,258),(127,247,137,257),(128,246,138,256),(129,245,139,255),(130,244,140,254),(161,268,171,278),(162,267,172,277),(163,266,173,276),(164,265,174,275),(165,264,175,274),(166,263,176,273),(167,262,177,272),(168,261,178,271),(169,280,179,270),(170,279,180,269),(201,301,211,311),(202,320,212,310),(203,319,213,309),(204,318,214,308),(205,317,215,307),(206,316,216,306),(207,315,217,305),(208,314,218,304),(209,313,219,303),(210,312,220,302),(221,282,231,292),(222,281,232,291),(223,300,233,290),(224,299,234,289),(225,298,235,288),(226,297,236,287),(227,296,237,286),(228,295,238,285),(229,294,239,284),(230,293,240,283)], [(1,50,123,89),(2,41,124,100),(3,52,125,91),(4,43,126,82),(5,54,127,93),(6,45,128,84),(7,56,129,95),(8,47,130,86),(9,58,131,97),(10,49,132,88),(11,60,133,99),(12,51,134,90),(13,42,135,81),(14,53,136,92),(15,44,137,83),(16,55,138,94),(17,46,139,85),(18,57,140,96),(19,48,121,87),(20,59,122,98),(21,219,168,295),(22,210,169,286),(23,201,170,297),(24,212,171,288),(25,203,172,299),(26,214,173,290),(27,205,174,281),(28,216,175,292),(29,207,176,283),(30,218,177,294),(31,209,178,285),(32,220,179,296),(33,211,180,287),(34,202,161,298),(35,213,162,289),(36,204,163,300),(37,215,164,291),(38,206,165,282),(39,217,166,293),(40,208,167,284),(61,301,264,236),(62,312,265,227),(63,303,266,238),(64,314,267,229),(65,305,268,240),(66,316,269,231),(67,307,270,222),(68,318,271,233),(69,309,272,224),(70,320,273,235),(71,311,274,226),(72,302,275,237),(73,313,276,228),(74,304,277,239),(75,315,278,230),(76,306,279,221),(77,317,280,232),(78,308,261,223),(79,319,262,234),(80,310,263,225),(101,144,250,186),(102,155,251,197),(103,146,252,188),(104,157,253,199),(105,148,254,190),(106,159,255,181),(107,150,256,192),(108,141,257,183),(109,152,258,194),(110,143,259,185),(111,154,260,196),(112,145,241,187),(113,156,242,198),(114,147,243,189),(115,158,244,200),(116,149,245,191),(117,160,246,182),(118,151,247,193),(119,142,248,184),(120,153,249,195)], [(1,30,133,167),(2,39,134,176),(3,28,135,165),(4,37,136,174),(5,26,137,163),(6,35,138,172),(7,24,139,161),(8,33,140,170),(9,22,121,179),(10,31,122,168),(11,40,123,177),(12,29,124,166),(13,38,125,175),(14,27,126,164),(15,36,127,173),(16,25,128,162),(17,34,129,171),(18,23,130,180),(19,32,131,169),(20,21,132,178),(41,293,90,207),(42,282,91,216),(43,291,92,205),(44,300,93,214),(45,289,94,203),(46,298,95,212),(47,287,96,201),(48,296,97,210),(49,285,98,219),(50,294,99,208),(51,283,100,217),(52,292,81,206),(53,281,82,215),(54,290,83,204),(55,299,84,213),(56,288,85,202),(57,297,86,211),(58,286,87,220),(59,295,88,209),(60,284,89,218),(61,249,274,110),(62,258,275,119),(63,247,276,108),(64,256,277,117),(65,245,278,106),(66,254,279,115),(67,243,280,104),(68,252,261,113),(69,241,262,102),(70,250,263,111),(71,259,264,120),(72,248,265,109),(73,257,266,118),(74,246,267,107),(75,255,268,116),(76,244,269,105),(77,253,270,114),(78,242,271,103),(79,251,272,112),(80,260,273,101),(141,238,193,313),(142,227,194,302),(143,236,195,311),(144,225,196,320),(145,234,197,309),(146,223,198,318),(147,232,199,307),(148,221,200,316),(149,230,181,305),(150,239,182,314),(151,228,183,303),(152,237,184,312),(153,226,185,301),(154,235,186,310),(155,224,187,319),(156,233,188,308),(157,222,189,317),(158,231,190,306),(159,240,191,315),(160,229,192,304)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C4○D20 | C8.C22 | Q8×D5 | D4×D5 | D8⋊3D5 | Q16⋊D5 |
| kernel | Dic10.2Q8 | C20.Q8 | C10.Q16 | C20.8Q8 | C20.44D4 | C5×C2.D8 | Dic5⋊3Q8 | C4.Dic10 | Dic10 | C2×Dic5 | C2.D8 | C20 | C4⋊C4 | C2×C8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10.2Q8 ►in GL4(𝔽41) generated by
| 1 | 40 | 0 | 0 |
| 8 | 34 | 0 | 0 |
| 0 | 0 | 40 | 23 |
| 0 | 0 | 32 | 1 |
| 35 | 29 | 0 | 0 |
| 20 | 6 | 0 | 0 |
| 0 | 0 | 21 | 6 |
| 0 | 0 | 22 | 20 |
| 11 | 28 | 0 | 0 |
| 22 | 30 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 39 | 36 |
| 28 | 15 | 0 | 0 |
| 16 | 13 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [1,8,0,0,40,34,0,0,0,0,40,32,0,0,23,1],[35,20,0,0,29,6,0,0,0,0,21,22,0,0,6,20],[11,22,0,0,28,30,0,0,0,0,5,39,0,0,12,36],[28,16,0,0,15,13,0,0,0,0,32,0,0,0,0,32] >;
Dic10.2Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._2Q_8 % in TeX
G:=Group("Dic10.2Q8"); // GroupNames label
G:=SmallGroup(320,504);
// by ID
G=gap.SmallGroup(320,504);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=a^10*c^2,b*a*b^-1=a^-1,c*a*c^-1=a^11,d*a*d^-1=a^9,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations