metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.1Dic10, C5⋊1(Q8.Q8), (C5×Q8).1Q8, C20.7(C2×Q8), C4⋊C4.19D10, (C2×C8).15D10, C40⋊5C4.7C2, Q8⋊C4.4D5, (Q8×Dic5).5C2, C4.7(C2×Dic10), C10.67(C4○D8), (C2×C40).15C22, (C2×Q8).101D10, Q8⋊Dic5.4C2, C22.190(D4×D5), C4.Dic10.2C2, C20.Q8.3C2, C20.8Q8.4C2, C20.159(C4○D4), C2.6(Q8.D10), C4.84(D4⋊2D5), (C2×C20).236C23, (C2×Dic5).206D4, C10.13(C22⋊Q8), C4⋊Dic5.85C22, (Q8×C10).19C22, C2.15(SD16⋊D5), C10.33(C8.C22), (C4×Dic5).26C22, C2.18(Dic5.14D4), (C2×C10).249(C2×D4), (C5×C4⋊C4).37C22, (C5×Q8⋊C4).4C2, (C2×C5⋊2C8).31C22, (C2×C4).343(C22×D5), SmallGroup(320,423)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=a2c-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, C20.Q8, C20.8Q8, C40⋊5C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, Q8×Dic5, Q8.Dic10
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⋊D5, Q8.D10, Q8.Dic10
►Regular action on 320 points
Generators in S320
(1 316 177 127)(2 128 178 317)(3 318 179 129)(4 130 180 319)(5 320 161 131)(6 132 162 301)(7 302 163 133)(8 134 164 303)(9 304 165 135)(10 136 166 305)(11 306 167 137)(12 138 168 307)(13 308 169 139)(14 140 170 309)(15 310 171 121)(16 122 172 311)(17 312 173 123)(18 124 174 313)(19 314 175 125)(20 126 176 315)(21 71 147 300)(22 281 148 72)(23 73 149 282)(24 283 150 74)(25 75 151 284)(26 285 152 76)(27 77 153 286)(28 287 154 78)(29 79 155 288)(30 289 156 80)(31 61 157 290)(32 291 158 62)(33 63 159 292)(34 293 160 64)(35 65 141 294)(36 295 142 66)(37 67 143 296)(38 297 144 68)(39 69 145 298)(40 299 146 70)(41 271 218 82)(42 83 219 272)(43 273 220 84)(44 85 201 274)(45 275 202 86)(46 87 203 276)(47 277 204 88)(48 89 205 278)(49 279 206 90)(50 91 207 280)(51 261 208 92)(52 93 209 262)(53 263 210 94)(54 95 211 264)(55 265 212 96)(56 97 213 266)(57 267 214 98)(58 99 215 268)(59 269 216 100)(60 81 217 270)(101 244 221 190)(102 191 222 245)(103 246 223 192)(104 193 224 247)(105 248 225 194)(106 195 226 249)(107 250 227 196)(108 197 228 251)(109 252 229 198)(110 199 230 253)(111 254 231 200)(112 181 232 255)(113 256 233 182)(114 183 234 257)(115 258 235 184)(116 185 236 259)(117 260 237 186)(118 187 238 241)(119 242 239 188)(120 189 240 243)
(1 292 177 63)(2 160 178 34)(3 294 179 65)(4 142 180 36)(5 296 161 67)(6 144 162 38)(7 298 163 69)(8 146 164 40)(9 300 165 71)(10 148 166 22)(11 282 167 73)(12 150 168 24)(13 284 169 75)(14 152 170 26)(15 286 171 77)(16 154 172 28)(17 288 173 79)(18 156 174 30)(19 290 175 61)(20 158 176 32)(21 304 147 135)(23 306 149 137)(25 308 151 139)(27 310 153 121)(29 312 155 123)(31 314 157 125)(33 316 159 127)(35 318 141 129)(37 320 143 131)(39 302 145 133)(41 232 218 112)(42 182 219 256)(43 234 220 114)(44 184 201 258)(45 236 202 116)(46 186 203 260)(47 238 204 118)(48 188 205 242)(49 240 206 120)(50 190 207 244)(51 222 208 102)(52 192 209 246)(53 224 210 104)(54 194 211 248)(55 226 212 106)(56 196 213 250)(57 228 214 108)(58 198 215 252)(59 230 216 110)(60 200 217 254)(62 126 291 315)(64 128 293 317)(66 130 295 319)(68 132 297 301)(70 134 299 303)(72 136 281 305)(74 138 283 307)(76 140 285 309)(78 122 287 311)(80 124 289 313)(81 231 270 111)(82 255 271 181)(83 233 272 113)(84 257 273 183)(85 235 274 115)(86 259 275 185)(87 237 276 117)(88 241 277 187)(89 239 278 119)(90 243 279 189)(91 221 280 101)(92 245 261 191)(93 223 262 103)(94 247 263 193)(95 225 264 105)(96 249 265 195)(97 227 266 107)(98 251 267 197)(99 229 268 109)(100 253 269 199)
(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)
(1 208 11 218)(2 50 12 60)(3 206 13 216)(4 48 14 58)(5 204 15 214)(6 46 16 56)(7 202 17 212)(8 44 18 54)(9 220 19 210)(10 42 20 52)(21 183 31 193)(22 256 32 246)(23 181 33 191)(24 254 34 244)(25 199 35 189)(26 252 36 242)(27 197 37 187)(28 250 38 260)(29 195 39 185)(30 248 40 258)(41 177 51 167)(43 175 53 165)(45 173 55 163)(47 171 57 161)(49 169 59 179)(61 224 71 234)(62 103 72 113)(63 222 73 232)(64 101 74 111)(65 240 75 230)(66 119 76 109)(67 238 77 228)(68 117 78 107)(69 236 79 226)(70 115 80 105)(81 128 91 138)(82 316 92 306)(83 126 93 136)(84 314 94 304)(85 124 95 134)(86 312 96 302)(87 122 97 132)(88 310 98 320)(89 140 99 130)(90 308 100 318)(102 282 112 292)(104 300 114 290)(106 298 116 288)(108 296 118 286)(110 294 120 284)(121 267 131 277)(123 265 133 275)(125 263 135 273)(127 261 137 271)(129 279 139 269)(141 243 151 253)(142 188 152 198)(143 241 153 251)(144 186 154 196)(145 259 155 249)(146 184 156 194)(147 257 157 247)(148 182 158 192)(149 255 159 245)(150 200 160 190)(162 203 172 213)(164 201 174 211)(166 219 176 209)(168 217 178 207)(170 215 180 205)(221 283 231 293)(223 281 233 291)(225 299 235 289)(227 297 237 287)(229 295 239 285)(262 305 272 315)(264 303 274 313)(266 301 276 311)(268 319 278 309)(270 317 280 307)
G:=sub<Sym(320)| (1,316,177,127)(2,128,178,317)(3,318,179,129)(4,130,180,319)(5,320,161,131)(6,132,162,301)(7,302,163,133)(8,134,164,303)(9,304,165,135)(10,136,166,305)(11,306,167,137)(12,138,168,307)(13,308,169,139)(14,140,170,309)(15,310,171,121)(16,122,172,311)(17,312,173,123)(18,124,174,313)(19,314,175,125)(20,126,176,315)(21,71,147,300)(22,281,148,72)(23,73,149,282)(24,283,150,74)(25,75,151,284)(26,285,152,76)(27,77,153,286)(28,287,154,78)(29,79,155,288)(30,289,156,80)(31,61,157,290)(32,291,158,62)(33,63,159,292)(34,293,160,64)(35,65,141,294)(36,295,142,66)(37,67,143,296)(38,297,144,68)(39,69,145,298)(40,299,146,70)(41,271,218,82)(42,83,219,272)(43,273,220,84)(44,85,201,274)(45,275,202,86)(46,87,203,276)(47,277,204,88)(48,89,205,278)(49,279,206,90)(50,91,207,280)(51,261,208,92)(52,93,209,262)(53,263,210,94)(54,95,211,264)(55,265,212,96)(56,97,213,266)(57,267,214,98)(58,99,215,268)(59,269,216,100)(60,81,217,270)(101,244,221,190)(102,191,222,245)(103,246,223,192)(104,193,224,247)(105,248,225,194)(106,195,226,249)(107,250,227,196)(108,197,228,251)(109,252,229,198)(110,199,230,253)(111,254,231,200)(112,181,232,255)(113,256,233,182)(114,183,234,257)(115,258,235,184)(116,185,236,259)(117,260,237,186)(118,187,238,241)(119,242,239,188)(120,189,240,243), (1,292,177,63)(2,160,178,34)(3,294,179,65)(4,142,180,36)(5,296,161,67)(6,144,162,38)(7,298,163,69)(8,146,164,40)(9,300,165,71)(10,148,166,22)(11,282,167,73)(12,150,168,24)(13,284,169,75)(14,152,170,26)(15,286,171,77)(16,154,172,28)(17,288,173,79)(18,156,174,30)(19,290,175,61)(20,158,176,32)(21,304,147,135)(23,306,149,137)(25,308,151,139)(27,310,153,121)(29,312,155,123)(31,314,157,125)(33,316,159,127)(35,318,141,129)(37,320,143,131)(39,302,145,133)(41,232,218,112)(42,182,219,256)(43,234,220,114)(44,184,201,258)(45,236,202,116)(46,186,203,260)(47,238,204,118)(48,188,205,242)(49,240,206,120)(50,190,207,244)(51,222,208,102)(52,192,209,246)(53,224,210,104)(54,194,211,248)(55,226,212,106)(56,196,213,250)(57,228,214,108)(58,198,215,252)(59,230,216,110)(60,200,217,254)(62,126,291,315)(64,128,293,317)(66,130,295,319)(68,132,297,301)(70,134,299,303)(72,136,281,305)(74,138,283,307)(76,140,285,309)(78,122,287,311)(80,124,289,313)(81,231,270,111)(82,255,271,181)(83,233,272,113)(84,257,273,183)(85,235,274,115)(86,259,275,185)(87,237,276,117)(88,241,277,187)(89,239,278,119)(90,243,279,189)(91,221,280,101)(92,245,261,191)(93,223,262,103)(94,247,263,193)(95,225,264,105)(96,249,265,195)(97,227,266,107)(98,251,267,197)(99,229,268,109)(100,253,269,199), (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), (1,208,11,218)(2,50,12,60)(3,206,13,216)(4,48,14,58)(5,204,15,214)(6,46,16,56)(7,202,17,212)(8,44,18,54)(9,220,19,210)(10,42,20,52)(21,183,31,193)(22,256,32,246)(23,181,33,191)(24,254,34,244)(25,199,35,189)(26,252,36,242)(27,197,37,187)(28,250,38,260)(29,195,39,185)(30,248,40,258)(41,177,51,167)(43,175,53,165)(45,173,55,163)(47,171,57,161)(49,169,59,179)(61,224,71,234)(62,103,72,113)(63,222,73,232)(64,101,74,111)(65,240,75,230)(66,119,76,109)(67,238,77,228)(68,117,78,107)(69,236,79,226)(70,115,80,105)(81,128,91,138)(82,316,92,306)(83,126,93,136)(84,314,94,304)(85,124,95,134)(86,312,96,302)(87,122,97,132)(88,310,98,320)(89,140,99,130)(90,308,100,318)(102,282,112,292)(104,300,114,290)(106,298,116,288)(108,296,118,286)(110,294,120,284)(121,267,131,277)(123,265,133,275)(125,263,135,273)(127,261,137,271)(129,279,139,269)(141,243,151,253)(142,188,152,198)(143,241,153,251)(144,186,154,196)(145,259,155,249)(146,184,156,194)(147,257,157,247)(148,182,158,192)(149,255,159,245)(150,200,160,190)(162,203,172,213)(164,201,174,211)(166,219,176,209)(168,217,178,207)(170,215,180,205)(221,283,231,293)(223,281,233,291)(225,299,235,289)(227,297,237,287)(229,295,239,285)(262,305,272,315)(264,303,274,313)(266,301,276,311)(268,319,278,309)(270,317,280,307)>;
G:=Group( (1,316,177,127)(2,128,178,317)(3,318,179,129)(4,130,180,319)(5,320,161,131)(6,132,162,301)(7,302,163,133)(8,134,164,303)(9,304,165,135)(10,136,166,305)(11,306,167,137)(12,138,168,307)(13,308,169,139)(14,140,170,309)(15,310,171,121)(16,122,172,311)(17,312,173,123)(18,124,174,313)(19,314,175,125)(20,126,176,315)(21,71,147,300)(22,281,148,72)(23,73,149,282)(24,283,150,74)(25,75,151,284)(26,285,152,76)(27,77,153,286)(28,287,154,78)(29,79,155,288)(30,289,156,80)(31,61,157,290)(32,291,158,62)(33,63,159,292)(34,293,160,64)(35,65,141,294)(36,295,142,66)(37,67,143,296)(38,297,144,68)(39,69,145,298)(40,299,146,70)(41,271,218,82)(42,83,219,272)(43,273,220,84)(44,85,201,274)(45,275,202,86)(46,87,203,276)(47,277,204,88)(48,89,205,278)(49,279,206,90)(50,91,207,280)(51,261,208,92)(52,93,209,262)(53,263,210,94)(54,95,211,264)(55,265,212,96)(56,97,213,266)(57,267,214,98)(58,99,215,268)(59,269,216,100)(60,81,217,270)(101,244,221,190)(102,191,222,245)(103,246,223,192)(104,193,224,247)(105,248,225,194)(106,195,226,249)(107,250,227,196)(108,197,228,251)(109,252,229,198)(110,199,230,253)(111,254,231,200)(112,181,232,255)(113,256,233,182)(114,183,234,257)(115,258,235,184)(116,185,236,259)(117,260,237,186)(118,187,238,241)(119,242,239,188)(120,189,240,243), (1,292,177,63)(2,160,178,34)(3,294,179,65)(4,142,180,36)(5,296,161,67)(6,144,162,38)(7,298,163,69)(8,146,164,40)(9,300,165,71)(10,148,166,22)(11,282,167,73)(12,150,168,24)(13,284,169,75)(14,152,170,26)(15,286,171,77)(16,154,172,28)(17,288,173,79)(18,156,174,30)(19,290,175,61)(20,158,176,32)(21,304,147,135)(23,306,149,137)(25,308,151,139)(27,310,153,121)(29,312,155,123)(31,314,157,125)(33,316,159,127)(35,318,141,129)(37,320,143,131)(39,302,145,133)(41,232,218,112)(42,182,219,256)(43,234,220,114)(44,184,201,258)(45,236,202,116)(46,186,203,260)(47,238,204,118)(48,188,205,242)(49,240,206,120)(50,190,207,244)(51,222,208,102)(52,192,209,246)(53,224,210,104)(54,194,211,248)(55,226,212,106)(56,196,213,250)(57,228,214,108)(58,198,215,252)(59,230,216,110)(60,200,217,254)(62,126,291,315)(64,128,293,317)(66,130,295,319)(68,132,297,301)(70,134,299,303)(72,136,281,305)(74,138,283,307)(76,140,285,309)(78,122,287,311)(80,124,289,313)(81,231,270,111)(82,255,271,181)(83,233,272,113)(84,257,273,183)(85,235,274,115)(86,259,275,185)(87,237,276,117)(88,241,277,187)(89,239,278,119)(90,243,279,189)(91,221,280,101)(92,245,261,191)(93,223,262,103)(94,247,263,193)(95,225,264,105)(96,249,265,195)(97,227,266,107)(98,251,267,197)(99,229,268,109)(100,253,269,199), (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), (1,208,11,218)(2,50,12,60)(3,206,13,216)(4,48,14,58)(5,204,15,214)(6,46,16,56)(7,202,17,212)(8,44,18,54)(9,220,19,210)(10,42,20,52)(21,183,31,193)(22,256,32,246)(23,181,33,191)(24,254,34,244)(25,199,35,189)(26,252,36,242)(27,197,37,187)(28,250,38,260)(29,195,39,185)(30,248,40,258)(41,177,51,167)(43,175,53,165)(45,173,55,163)(47,171,57,161)(49,169,59,179)(61,224,71,234)(62,103,72,113)(63,222,73,232)(64,101,74,111)(65,240,75,230)(66,119,76,109)(67,238,77,228)(68,117,78,107)(69,236,79,226)(70,115,80,105)(81,128,91,138)(82,316,92,306)(83,126,93,136)(84,314,94,304)(85,124,95,134)(86,312,96,302)(87,122,97,132)(88,310,98,320)(89,140,99,130)(90,308,100,318)(102,282,112,292)(104,300,114,290)(106,298,116,288)(108,296,118,286)(110,294,120,284)(121,267,131,277)(123,265,133,275)(125,263,135,273)(127,261,137,271)(129,279,139,269)(141,243,151,253)(142,188,152,198)(143,241,153,251)(144,186,154,196)(145,259,155,249)(146,184,156,194)(147,257,157,247)(148,182,158,192)(149,255,159,245)(150,200,160,190)(162,203,172,213)(164,201,174,211)(166,219,176,209)(168,217,178,207)(170,215,180,205)(221,283,231,293)(223,281,233,291)(225,299,235,289)(227,297,237,287)(229,295,239,285)(262,305,272,315)(264,303,274,313)(266,301,276,311)(268,319,278,309)(270,317,280,307) );
G=PermutationGroup([[(1,316,177,127),(2,128,178,317),(3,318,179,129),(4,130,180,319),(5,320,161,131),(6,132,162,301),(7,302,163,133),(8,134,164,303),(9,304,165,135),(10,136,166,305),(11,306,167,137),(12,138,168,307),(13,308,169,139),(14,140,170,309),(15,310,171,121),(16,122,172,311),(17,312,173,123),(18,124,174,313),(19,314,175,125),(20,126,176,315),(21,71,147,300),(22,281,148,72),(23,73,149,282),(24,283,150,74),(25,75,151,284),(26,285,152,76),(27,77,153,286),(28,287,154,78),(29,79,155,288),(30,289,156,80),(31,61,157,290),(32,291,158,62),(33,63,159,292),(34,293,160,64),(35,65,141,294),(36,295,142,66),(37,67,143,296),(38,297,144,68),(39,69,145,298),(40,299,146,70),(41,271,218,82),(42,83,219,272),(43,273,220,84),(44,85,201,274),(45,275,202,86),(46,87,203,276),(47,277,204,88),(48,89,205,278),(49,279,206,90),(50,91,207,280),(51,261,208,92),(52,93,209,262),(53,263,210,94),(54,95,211,264),(55,265,212,96),(56,97,213,266),(57,267,214,98),(58,99,215,268),(59,269,216,100),(60,81,217,270),(101,244,221,190),(102,191,222,245),(103,246,223,192),(104,193,224,247),(105,248,225,194),(106,195,226,249),(107,250,227,196),(108,197,228,251),(109,252,229,198),(110,199,230,253),(111,254,231,200),(112,181,232,255),(113,256,233,182),(114,183,234,257),(115,258,235,184),(116,185,236,259),(117,260,237,186),(118,187,238,241),(119,242,239,188),(120,189,240,243)], [(1,292,177,63),(2,160,178,34),(3,294,179,65),(4,142,180,36),(5,296,161,67),(6,144,162,38),(7,298,163,69),(8,146,164,40),(9,300,165,71),(10,148,166,22),(11,282,167,73),(12,150,168,24),(13,284,169,75),(14,152,170,26),(15,286,171,77),(16,154,172,28),(17,288,173,79),(18,156,174,30),(19,290,175,61),(20,158,176,32),(21,304,147,135),(23,306,149,137),(25,308,151,139),(27,310,153,121),(29,312,155,123),(31,314,157,125),(33,316,159,127),(35,318,141,129),(37,320,143,131),(39,302,145,133),(41,232,218,112),(42,182,219,256),(43,234,220,114),(44,184,201,258),(45,236,202,116),(46,186,203,260),(47,238,204,118),(48,188,205,242),(49,240,206,120),(50,190,207,244),(51,222,208,102),(52,192,209,246),(53,224,210,104),(54,194,211,248),(55,226,212,106),(56,196,213,250),(57,228,214,108),(58,198,215,252),(59,230,216,110),(60,200,217,254),(62,126,291,315),(64,128,293,317),(66,130,295,319),(68,132,297,301),(70,134,299,303),(72,136,281,305),(74,138,283,307),(76,140,285,309),(78,122,287,311),(80,124,289,313),(81,231,270,111),(82,255,271,181),(83,233,272,113),(84,257,273,183),(85,235,274,115),(86,259,275,185),(87,237,276,117),(88,241,277,187),(89,239,278,119),(90,243,279,189),(91,221,280,101),(92,245,261,191),(93,223,262,103),(94,247,263,193),(95,225,264,105),(96,249,265,195),(97,227,266,107),(98,251,267,197),(99,229,268,109),(100,253,269,199)], [(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)], [(1,208,11,218),(2,50,12,60),(3,206,13,216),(4,48,14,58),(5,204,15,214),(6,46,16,56),(7,202,17,212),(8,44,18,54),(9,220,19,210),(10,42,20,52),(21,183,31,193),(22,256,32,246),(23,181,33,191),(24,254,34,244),(25,199,35,189),(26,252,36,242),(27,197,37,187),(28,250,38,260),(29,195,39,185),(30,248,40,258),(41,177,51,167),(43,175,53,165),(45,173,55,163),(47,171,57,161),(49,169,59,179),(61,224,71,234),(62,103,72,113),(63,222,73,232),(64,101,74,111),(65,240,75,230),(66,119,76,109),(67,238,77,228),(68,117,78,107),(69,236,79,226),(70,115,80,105),(81,128,91,138),(82,316,92,306),(83,126,93,136),(84,314,94,304),(85,124,95,134),(86,312,96,302),(87,122,97,132),(88,310,98,320),(89,140,99,130),(90,308,100,318),(102,282,112,292),(104,300,114,290),(106,298,116,288),(108,296,118,286),(110,294,120,284),(121,267,131,277),(123,265,133,275),(125,263,135,273),(127,261,137,271),(129,279,139,269),(141,243,151,253),(142,188,152,198),(143,241,153,251),(144,186,154,196),(145,259,155,249),(146,184,156,194),(147,257,157,247),(148,182,158,192),(149,255,159,245),(150,200,160,190),(162,203,172,213),(164,201,174,211),(166,219,176,209),(168,217,178,207),(170,215,180,205),(221,283,231,293),(223,281,233,291),(225,299,235,289),(227,297,237,287),(229,295,239,285),(262,305,272,315),(264,303,274,313),(266,301,276,311),(268,319,278,309),(270,317,280,307)]])
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⋊D5 | Q8.D10 |
| kernel | Q8.Dic10 | C20.Q8 | C20.8Q8 | C40⋊5C4 | 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.Dic10 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 38 | 21 | 0 | 0 |
| 21 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 11 | 0 | 0 |
| 11 | 1 | 0 | 0 |
| 0 | 0 | 16 | 39 |
| 0 | 0 | 2 | 28 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 15 | 10 |
| 0 | 0 | 2 | 26 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[38,21,0,0,21,3,0,0,0,0,1,0,0,0,0,1],[40,11,0,0,11,1,0,0,0,0,16,2,0,0,39,28],[9,0,0,0,0,9,0,0,0,0,15,2,0,0,10,26] >;
Q8.Dic10 in GAP, Magma, Sage, TeX
Q_8.{\rm Dic}_{10} % in TeX
G:=Group("Q8.Dic10"); // GroupNames label
G:=SmallGroup(320,423);
// by ID
G=gap.SmallGroup(320,423);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,926,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^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=a^2*c^-1>;
// generators/relations