metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.2Dic10, C5⋊2(Q8.Q8), (C5×Q8).2Q8, C20.8(C2×Q8), C4⋊C4.21D10, C40⋊6C4.7C2, (C2×C8).121D10, Q8⋊C4.6D5, (Q8×Dic5).6C2, C4.8(C2×Dic10), C10.46(C4○D8), (C2×Q8).102D10, Q8⋊Dic5.6C2, C10.D8.3C2, C22.193(D4×D5), C4.Dic10.4C2, C20.8Q8.6C2, C20.160(C4○D4), C4.85(D4⋊2D5), (C2×C20).239C23, (C2×C40).132C22, (C2×Dic5).207D4, C10.14(C22⋊Q8), C4⋊Dic5.87C22, (Q8×C10).22C22, C2.12(Q16⋊D5), C10.57(C8.C22), (C4×Dic5).29C22, C2.15(SD16⋊3D5), C2.19(Dic5.14D4), (C2×C10).252(C2×D4), (C5×C4⋊C4).40C22, (C5×Q8⋊C4).6C2, (C2×C5⋊2C8).34C22, (C2×C4).346(C22×D5), SmallGroup(320,426)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.2Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=a2c10, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >
Subgroups: 294 in 90 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8.Q8, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, Q8×C10, C10.D8, C20.8Q8, C40⋊6C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, Q8×Dic5, Q8.2Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8.C22, Dic10, C22×D5, Q8.Q8, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4, SD16⋊3D5, Q16⋊D5, Q8.2Dic10
►Regular action on 320 points
Generators in S320
(1 172 52 262)(2 263 53 173)(3 174 54 264)(4 265 55 175)(5 176 56 266)(6 267 57 177)(7 178 58 268)(8 269 59 179)(9 180 60 270)(10 271 41 161)(11 162 42 272)(12 273 43 163)(13 164 44 274)(14 275 45 165)(15 166 46 276)(16 277 47 167)(17 168 48 278)(18 279 49 169)(19 170 50 280)(20 261 51 171)(21 284 98 154)(22 155 99 285)(23 286 100 156)(24 157 81 287)(25 288 82 158)(26 159 83 289)(27 290 84 160)(28 141 85 291)(29 292 86 142)(30 143 87 293)(31 294 88 144)(32 145 89 295)(33 296 90 146)(34 147 91 297)(35 298 92 148)(36 149 93 299)(37 300 94 150)(38 151 95 281)(39 282 96 152)(40 153 97 283)(61 244 310 227)(62 228 311 245)(63 246 312 229)(64 230 313 247)(65 248 314 231)(66 232 315 249)(67 250 316 233)(68 234 317 251)(69 252 318 235)(70 236 319 253)(71 254 320 237)(72 238 301 255)(73 256 302 239)(74 240 303 257)(75 258 304 221)(76 222 305 259)(77 260 306 223)(78 224 307 241)(79 242 308 225)(80 226 309 243)(101 187 133 211)(102 212 134 188)(103 189 135 213)(104 214 136 190)(105 191 137 215)(106 216 138 192)(107 193 139 217)(108 218 140 194)(109 195 121 219)(110 220 122 196)(111 197 123 201)(112 202 124 198)(113 199 125 203)(114 204 126 200)(115 181 127 205)(116 206 128 182)(117 183 129 207)(118 208 130 184)(119 185 131 209)(120 210 132 186)
(1 188 52 212)(2 135 53 103)(3 190 54 214)(4 137 55 105)(5 192 56 216)(6 139 57 107)(7 194 58 218)(8 121 59 109)(9 196 60 220)(10 123 41 111)(11 198 42 202)(12 125 43 113)(13 200 44 204)(14 127 45 115)(15 182 46 206)(16 129 47 117)(17 184 48 208)(18 131 49 119)(19 186 50 210)(20 133 51 101)(21 318 98 69)(22 253 99 236)(23 320 100 71)(24 255 81 238)(25 302 82 73)(26 257 83 240)(27 304 84 75)(28 259 85 222)(29 306 86 77)(30 241 87 224)(31 308 88 79)(32 243 89 226)(33 310 90 61)(34 245 91 228)(35 312 92 63)(36 247 93 230)(37 314 94 65)(38 249 95 232)(39 316 96 67)(40 251 97 234)(62 147 311 297)(64 149 313 299)(66 151 315 281)(68 153 317 283)(70 155 319 285)(72 157 301 287)(74 159 303 289)(76 141 305 291)(78 143 307 293)(80 145 309 295)(102 172 134 262)(104 174 136 264)(106 176 138 266)(108 178 140 268)(110 180 122 270)(112 162 124 272)(114 164 126 274)(116 166 128 276)(118 168 130 278)(120 170 132 280)(142 223 292 260)(144 225 294 242)(146 227 296 244)(148 229 298 246)(150 231 300 248)(152 233 282 250)(154 235 284 252)(156 237 286 254)(158 239 288 256)(160 221 290 258)(161 201 271 197)(163 203 273 199)(165 205 275 181)(167 207 277 183)(169 209 279 185)(171 211 261 187)(173 213 263 189)(175 215 265 191)(177 217 267 193)(179 219 269 195)
(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)
(1 159 42 299)(2 158 43 298)(3 157 44 297)(4 156 45 296)(5 155 46 295)(6 154 47 294)(7 153 48 293)(8 152 49 292)(9 151 50 291)(10 150 51 290)(11 149 52 289)(12 148 53 288)(13 147 54 287)(14 146 55 286)(15 145 56 285)(16 144 57 284)(17 143 58 283)(18 142 59 282)(19 141 60 281)(20 160 41 300)(21 167 88 267)(22 166 89 266)(23 165 90 265)(24 164 91 264)(25 163 92 263)(26 162 93 262)(27 161 94 261)(28 180 95 280)(29 179 96 279)(30 178 97 278)(31 177 98 277)(32 176 99 276)(33 175 100 275)(34 174 81 274)(35 173 82 273)(36 172 83 272)(37 171 84 271)(38 170 85 270)(39 169 86 269)(40 168 87 268)(61 191 320 205)(62 190 301 204)(63 189 302 203)(64 188 303 202)(65 187 304 201)(66 186 305 220)(67 185 306 219)(68 184 307 218)(69 183 308 217)(70 182 309 216)(71 181 310 215)(72 200 311 214)(73 199 312 213)(74 198 313 212)(75 197 314 211)(76 196 315 210)(77 195 316 209)(78 194 317 208)(79 193 318 207)(80 192 319 206)(101 258 123 231)(102 257 124 230)(103 256 125 229)(104 255 126 228)(105 254 127 227)(106 253 128 226)(107 252 129 225)(108 251 130 224)(109 250 131 223)(110 249 132 222)(111 248 133 221)(112 247 134 240)(113 246 135 239)(114 245 136 238)(115 244 137 237)(116 243 138 236)(117 242 139 235)(118 241 140 234)(119 260 121 233)(120 259 122 232)
G:=sub<Sym(320)| (1,172,52,262)(2,263,53,173)(3,174,54,264)(4,265,55,175)(5,176,56,266)(6,267,57,177)(7,178,58,268)(8,269,59,179)(9,180,60,270)(10,271,41,161)(11,162,42,272)(12,273,43,163)(13,164,44,274)(14,275,45,165)(15,166,46,276)(16,277,47,167)(17,168,48,278)(18,279,49,169)(19,170,50,280)(20,261,51,171)(21,284,98,154)(22,155,99,285)(23,286,100,156)(24,157,81,287)(25,288,82,158)(26,159,83,289)(27,290,84,160)(28,141,85,291)(29,292,86,142)(30,143,87,293)(31,294,88,144)(32,145,89,295)(33,296,90,146)(34,147,91,297)(35,298,92,148)(36,149,93,299)(37,300,94,150)(38,151,95,281)(39,282,96,152)(40,153,97,283)(61,244,310,227)(62,228,311,245)(63,246,312,229)(64,230,313,247)(65,248,314,231)(66,232,315,249)(67,250,316,233)(68,234,317,251)(69,252,318,235)(70,236,319,253)(71,254,320,237)(72,238,301,255)(73,256,302,239)(74,240,303,257)(75,258,304,221)(76,222,305,259)(77,260,306,223)(78,224,307,241)(79,242,308,225)(80,226,309,243)(101,187,133,211)(102,212,134,188)(103,189,135,213)(104,214,136,190)(105,191,137,215)(106,216,138,192)(107,193,139,217)(108,218,140,194)(109,195,121,219)(110,220,122,196)(111,197,123,201)(112,202,124,198)(113,199,125,203)(114,204,126,200)(115,181,127,205)(116,206,128,182)(117,183,129,207)(118,208,130,184)(119,185,131,209)(120,210,132,186), (1,188,52,212)(2,135,53,103)(3,190,54,214)(4,137,55,105)(5,192,56,216)(6,139,57,107)(7,194,58,218)(8,121,59,109)(9,196,60,220)(10,123,41,111)(11,198,42,202)(12,125,43,113)(13,200,44,204)(14,127,45,115)(15,182,46,206)(16,129,47,117)(17,184,48,208)(18,131,49,119)(19,186,50,210)(20,133,51,101)(21,318,98,69)(22,253,99,236)(23,320,100,71)(24,255,81,238)(25,302,82,73)(26,257,83,240)(27,304,84,75)(28,259,85,222)(29,306,86,77)(30,241,87,224)(31,308,88,79)(32,243,89,226)(33,310,90,61)(34,245,91,228)(35,312,92,63)(36,247,93,230)(37,314,94,65)(38,249,95,232)(39,316,96,67)(40,251,97,234)(62,147,311,297)(64,149,313,299)(66,151,315,281)(68,153,317,283)(70,155,319,285)(72,157,301,287)(74,159,303,289)(76,141,305,291)(78,143,307,293)(80,145,309,295)(102,172,134,262)(104,174,136,264)(106,176,138,266)(108,178,140,268)(110,180,122,270)(112,162,124,272)(114,164,126,274)(116,166,128,276)(118,168,130,278)(120,170,132,280)(142,223,292,260)(144,225,294,242)(146,227,296,244)(148,229,298,246)(150,231,300,248)(152,233,282,250)(154,235,284,252)(156,237,286,254)(158,239,288,256)(160,221,290,258)(161,201,271,197)(163,203,273,199)(165,205,275,181)(167,207,277,183)(169,209,279,185)(171,211,261,187)(173,213,263,189)(175,215,265,191)(177,217,267,193)(179,219,269,195), (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), (1,159,42,299)(2,158,43,298)(3,157,44,297)(4,156,45,296)(5,155,46,295)(6,154,47,294)(7,153,48,293)(8,152,49,292)(9,151,50,291)(10,150,51,290)(11,149,52,289)(12,148,53,288)(13,147,54,287)(14,146,55,286)(15,145,56,285)(16,144,57,284)(17,143,58,283)(18,142,59,282)(19,141,60,281)(20,160,41,300)(21,167,88,267)(22,166,89,266)(23,165,90,265)(24,164,91,264)(25,163,92,263)(26,162,93,262)(27,161,94,261)(28,180,95,280)(29,179,96,279)(30,178,97,278)(31,177,98,277)(32,176,99,276)(33,175,100,275)(34,174,81,274)(35,173,82,273)(36,172,83,272)(37,171,84,271)(38,170,85,270)(39,169,86,269)(40,168,87,268)(61,191,320,205)(62,190,301,204)(63,189,302,203)(64,188,303,202)(65,187,304,201)(66,186,305,220)(67,185,306,219)(68,184,307,218)(69,183,308,217)(70,182,309,216)(71,181,310,215)(72,200,311,214)(73,199,312,213)(74,198,313,212)(75,197,314,211)(76,196,315,210)(77,195,316,209)(78,194,317,208)(79,193,318,207)(80,192,319,206)(101,258,123,231)(102,257,124,230)(103,256,125,229)(104,255,126,228)(105,254,127,227)(106,253,128,226)(107,252,129,225)(108,251,130,224)(109,250,131,223)(110,249,132,222)(111,248,133,221)(112,247,134,240)(113,246,135,239)(114,245,136,238)(115,244,137,237)(116,243,138,236)(117,242,139,235)(118,241,140,234)(119,260,121,233)(120,259,122,232)>;
G:=Group( (1,172,52,262)(2,263,53,173)(3,174,54,264)(4,265,55,175)(5,176,56,266)(6,267,57,177)(7,178,58,268)(8,269,59,179)(9,180,60,270)(10,271,41,161)(11,162,42,272)(12,273,43,163)(13,164,44,274)(14,275,45,165)(15,166,46,276)(16,277,47,167)(17,168,48,278)(18,279,49,169)(19,170,50,280)(20,261,51,171)(21,284,98,154)(22,155,99,285)(23,286,100,156)(24,157,81,287)(25,288,82,158)(26,159,83,289)(27,290,84,160)(28,141,85,291)(29,292,86,142)(30,143,87,293)(31,294,88,144)(32,145,89,295)(33,296,90,146)(34,147,91,297)(35,298,92,148)(36,149,93,299)(37,300,94,150)(38,151,95,281)(39,282,96,152)(40,153,97,283)(61,244,310,227)(62,228,311,245)(63,246,312,229)(64,230,313,247)(65,248,314,231)(66,232,315,249)(67,250,316,233)(68,234,317,251)(69,252,318,235)(70,236,319,253)(71,254,320,237)(72,238,301,255)(73,256,302,239)(74,240,303,257)(75,258,304,221)(76,222,305,259)(77,260,306,223)(78,224,307,241)(79,242,308,225)(80,226,309,243)(101,187,133,211)(102,212,134,188)(103,189,135,213)(104,214,136,190)(105,191,137,215)(106,216,138,192)(107,193,139,217)(108,218,140,194)(109,195,121,219)(110,220,122,196)(111,197,123,201)(112,202,124,198)(113,199,125,203)(114,204,126,200)(115,181,127,205)(116,206,128,182)(117,183,129,207)(118,208,130,184)(119,185,131,209)(120,210,132,186), (1,188,52,212)(2,135,53,103)(3,190,54,214)(4,137,55,105)(5,192,56,216)(6,139,57,107)(7,194,58,218)(8,121,59,109)(9,196,60,220)(10,123,41,111)(11,198,42,202)(12,125,43,113)(13,200,44,204)(14,127,45,115)(15,182,46,206)(16,129,47,117)(17,184,48,208)(18,131,49,119)(19,186,50,210)(20,133,51,101)(21,318,98,69)(22,253,99,236)(23,320,100,71)(24,255,81,238)(25,302,82,73)(26,257,83,240)(27,304,84,75)(28,259,85,222)(29,306,86,77)(30,241,87,224)(31,308,88,79)(32,243,89,226)(33,310,90,61)(34,245,91,228)(35,312,92,63)(36,247,93,230)(37,314,94,65)(38,249,95,232)(39,316,96,67)(40,251,97,234)(62,147,311,297)(64,149,313,299)(66,151,315,281)(68,153,317,283)(70,155,319,285)(72,157,301,287)(74,159,303,289)(76,141,305,291)(78,143,307,293)(80,145,309,295)(102,172,134,262)(104,174,136,264)(106,176,138,266)(108,178,140,268)(110,180,122,270)(112,162,124,272)(114,164,126,274)(116,166,128,276)(118,168,130,278)(120,170,132,280)(142,223,292,260)(144,225,294,242)(146,227,296,244)(148,229,298,246)(150,231,300,248)(152,233,282,250)(154,235,284,252)(156,237,286,254)(158,239,288,256)(160,221,290,258)(161,201,271,197)(163,203,273,199)(165,205,275,181)(167,207,277,183)(169,209,279,185)(171,211,261,187)(173,213,263,189)(175,215,265,191)(177,217,267,193)(179,219,269,195), (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), (1,159,42,299)(2,158,43,298)(3,157,44,297)(4,156,45,296)(5,155,46,295)(6,154,47,294)(7,153,48,293)(8,152,49,292)(9,151,50,291)(10,150,51,290)(11,149,52,289)(12,148,53,288)(13,147,54,287)(14,146,55,286)(15,145,56,285)(16,144,57,284)(17,143,58,283)(18,142,59,282)(19,141,60,281)(20,160,41,300)(21,167,88,267)(22,166,89,266)(23,165,90,265)(24,164,91,264)(25,163,92,263)(26,162,93,262)(27,161,94,261)(28,180,95,280)(29,179,96,279)(30,178,97,278)(31,177,98,277)(32,176,99,276)(33,175,100,275)(34,174,81,274)(35,173,82,273)(36,172,83,272)(37,171,84,271)(38,170,85,270)(39,169,86,269)(40,168,87,268)(61,191,320,205)(62,190,301,204)(63,189,302,203)(64,188,303,202)(65,187,304,201)(66,186,305,220)(67,185,306,219)(68,184,307,218)(69,183,308,217)(70,182,309,216)(71,181,310,215)(72,200,311,214)(73,199,312,213)(74,198,313,212)(75,197,314,211)(76,196,315,210)(77,195,316,209)(78,194,317,208)(79,193,318,207)(80,192,319,206)(101,258,123,231)(102,257,124,230)(103,256,125,229)(104,255,126,228)(105,254,127,227)(106,253,128,226)(107,252,129,225)(108,251,130,224)(109,250,131,223)(110,249,132,222)(111,248,133,221)(112,247,134,240)(113,246,135,239)(114,245,136,238)(115,244,137,237)(116,243,138,236)(117,242,139,235)(118,241,140,234)(119,260,121,233)(120,259,122,232) );
G=PermutationGroup([[(1,172,52,262),(2,263,53,173),(3,174,54,264),(4,265,55,175),(5,176,56,266),(6,267,57,177),(7,178,58,268),(8,269,59,179),(9,180,60,270),(10,271,41,161),(11,162,42,272),(12,273,43,163),(13,164,44,274),(14,275,45,165),(15,166,46,276),(16,277,47,167),(17,168,48,278),(18,279,49,169),(19,170,50,280),(20,261,51,171),(21,284,98,154),(22,155,99,285),(23,286,100,156),(24,157,81,287),(25,288,82,158),(26,159,83,289),(27,290,84,160),(28,141,85,291),(29,292,86,142),(30,143,87,293),(31,294,88,144),(32,145,89,295),(33,296,90,146),(34,147,91,297),(35,298,92,148),(36,149,93,299),(37,300,94,150),(38,151,95,281),(39,282,96,152),(40,153,97,283),(61,244,310,227),(62,228,311,245),(63,246,312,229),(64,230,313,247),(65,248,314,231),(66,232,315,249),(67,250,316,233),(68,234,317,251),(69,252,318,235),(70,236,319,253),(71,254,320,237),(72,238,301,255),(73,256,302,239),(74,240,303,257),(75,258,304,221),(76,222,305,259),(77,260,306,223),(78,224,307,241),(79,242,308,225),(80,226,309,243),(101,187,133,211),(102,212,134,188),(103,189,135,213),(104,214,136,190),(105,191,137,215),(106,216,138,192),(107,193,139,217),(108,218,140,194),(109,195,121,219),(110,220,122,196),(111,197,123,201),(112,202,124,198),(113,199,125,203),(114,204,126,200),(115,181,127,205),(116,206,128,182),(117,183,129,207),(118,208,130,184),(119,185,131,209),(120,210,132,186)], [(1,188,52,212),(2,135,53,103),(3,190,54,214),(4,137,55,105),(5,192,56,216),(6,139,57,107),(7,194,58,218),(8,121,59,109),(9,196,60,220),(10,123,41,111),(11,198,42,202),(12,125,43,113),(13,200,44,204),(14,127,45,115),(15,182,46,206),(16,129,47,117),(17,184,48,208),(18,131,49,119),(19,186,50,210),(20,133,51,101),(21,318,98,69),(22,253,99,236),(23,320,100,71),(24,255,81,238),(25,302,82,73),(26,257,83,240),(27,304,84,75),(28,259,85,222),(29,306,86,77),(30,241,87,224),(31,308,88,79),(32,243,89,226),(33,310,90,61),(34,245,91,228),(35,312,92,63),(36,247,93,230),(37,314,94,65),(38,249,95,232),(39,316,96,67),(40,251,97,234),(62,147,311,297),(64,149,313,299),(66,151,315,281),(68,153,317,283),(70,155,319,285),(72,157,301,287),(74,159,303,289),(76,141,305,291),(78,143,307,293),(80,145,309,295),(102,172,134,262),(104,174,136,264),(106,176,138,266),(108,178,140,268),(110,180,122,270),(112,162,124,272),(114,164,126,274),(116,166,128,276),(118,168,130,278),(120,170,132,280),(142,223,292,260),(144,225,294,242),(146,227,296,244),(148,229,298,246),(150,231,300,248),(152,233,282,250),(154,235,284,252),(156,237,286,254),(158,239,288,256),(160,221,290,258),(161,201,271,197),(163,203,273,199),(165,205,275,181),(167,207,277,183),(169,209,279,185),(171,211,261,187),(173,213,263,189),(175,215,265,191),(177,217,267,193),(179,219,269,195)], [(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)], [(1,159,42,299),(2,158,43,298),(3,157,44,297),(4,156,45,296),(5,155,46,295),(6,154,47,294),(7,153,48,293),(8,152,49,292),(9,151,50,291),(10,150,51,290),(11,149,52,289),(12,148,53,288),(13,147,54,287),(14,146,55,286),(15,145,56,285),(16,144,57,284),(17,143,58,283),(18,142,59,282),(19,141,60,281),(20,160,41,300),(21,167,88,267),(22,166,89,266),(23,165,90,265),(24,164,91,264),(25,163,92,263),(26,162,93,262),(27,161,94,261),(28,180,95,280),(29,179,96,279),(30,178,97,278),(31,177,98,277),(32,176,99,276),(33,175,100,275),(34,174,81,274),(35,173,82,273),(36,172,83,272),(37,171,84,271),(38,170,85,270),(39,169,86,269),(40,168,87,268),(61,191,320,205),(62,190,301,204),(63,189,302,203),(64,188,303,202),(65,187,304,201),(66,186,305,220),(67,185,306,219),(68,184,307,218),(69,183,308,217),(70,182,309,216),(71,181,310,215),(72,200,311,214),(73,199,312,213),(74,198,313,212),(75,197,314,211),(76,196,315,210),(77,195,316,209),(78,194,317,208),(79,193,318,207),(80,192,319,206),(101,258,123,231),(102,257,124,230),(103,256,125,229),(104,255,126,228),(105,254,127,227),(106,253,128,226),(107,252,129,225),(108,251,130,224),(109,250,131,223),(110,249,132,222),(111,248,133,221),(112,247,134,240),(113,246,135,239),(114,245,136,238),(115,244,137,237),(116,243,138,236),(117,242,139,235),(118,241,140,234),(119,260,121,233),(120,259,122,232)]])
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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | Dic10 | C8.C22 | D4⋊2D5 | D4×D5 | SD16⋊3D5 | Q16⋊D5 |
| kernel | Q8.2Dic10 | C10.D8 | C20.8Q8 | C40⋊6C4 | Q8⋊Dic5 | C5×Q8⋊C4 | C4.Dic10 | Q8×Dic5 | C2×Dic5 | C5×Q8 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Q8.2Dic10 ►in GL4(𝔽41) generated by
| 1 | 9 | 0 | 0 |
| 18 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 13 | 0 | 0 |
| 36 | 33 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 19 | 14 | 0 | 0 |
| 27 | 22 | 0 | 0 |
| 0 | 0 | 9 | 11 |
| 0 | 0 | 30 | 14 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 22 | 9 |
G:=sub<GL(4,GF(41))| [1,18,0,0,9,40,0,0,0,0,1,0,0,0,0,1],[8,36,0,0,13,33,0,0,0,0,1,0,0,0,0,1],[19,27,0,0,14,22,0,0,0,0,9,30,0,0,11,14],[32,0,0,0,0,32,0,0,0,0,32,22,0,0,0,9] >;
Q8.2Dic10 in GAP, Magma, Sage, TeX
Q_8._2{\rm Dic}_{10} % in TeX
G:=Group("Q8.2Dic10"); // GroupNames label
G:=SmallGroup(320,426);
// by ID
G=gap.SmallGroup(320,426);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,926,219,226,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=a^2*c^10,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations