metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.3Dic10, C42.55D10, C5⋊6(Q8.Q8), (C4×Q8).4D5, (C5×Q8).3Q8, (C2×C20).65D4, (Q8×C20).4C2, C20.30(C2×Q8), C4⋊C4.250D10, C20⋊3C8.16C2, C20.57(C4○D4), C10.91(C4○D8), C4.64(C4○D20), (C4×C20).93C22, (C2×Q8).157D10, C4.14(C2×Dic10), Q8⋊Dic5.9C2, C20.6Q8.6C2, (C2×C20).344C23, C10.D8.11C2, C10.66(C22⋊Q8), C20.Q8.11C2, C2.8(C20.C23), C10.86(C8.C22), C4⋊Dic5.141C22, (Q8×C10).192C22, C2.12(D4.8D10), C2.17(C20.48D4), (C2×C10).475(C2×D4), (C2×C4).220(C5⋊D4), (C5×C4⋊C4).281C22, (C2×C5⋊2C8).99C22, (C2×C4).444(C22×D5), C22.154(C2×C5⋊D4), SmallGroup(320,649)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.3Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=a2c10, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 262 in 90 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8.Q8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Q8×C10, C20⋊3C8, C10.D8, C20.Q8, Q8⋊Dic5, C20.6Q8, Q8×C20, Q8.3Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8.C22, Dic10, C5⋊D4, C22×D5, Q8.Q8, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C20.C23, D4.8D10, Q8.3Dic10
►Regular action on 320 points
Generators in S320
(1 308 132 112)(2 309 133 113)(3 310 134 114)(4 311 135 115)(5 312 136 116)(6 313 137 117)(7 314 138 118)(8 315 139 119)(9 316 140 120)(10 317 121 101)(11 318 122 102)(12 319 123 103)(13 320 124 104)(14 301 125 105)(15 302 126 106)(16 303 127 107)(17 304 128 108)(18 305 129 109)(19 306 130 110)(20 307 131 111)(21 233 60 244)(22 234 41 245)(23 235 42 246)(24 236 43 247)(25 237 44 248)(26 238 45 249)(27 239 46 250)(28 240 47 251)(29 221 48 252)(30 222 49 253)(31 223 50 254)(32 224 51 255)(33 225 52 256)(34 226 53 257)(35 227 54 258)(36 228 55 259)(37 229 56 260)(38 230 57 241)(39 231 58 242)(40 232 59 243)(61 220 283 192)(62 201 284 193)(63 202 285 194)(64 203 286 195)(65 204 287 196)(66 205 288 197)(67 206 289 198)(68 207 290 199)(69 208 291 200)(70 209 292 181)(71 210 293 182)(72 211 294 183)(73 212 295 184)(74 213 296 185)(75 214 297 186)(76 215 298 187)(77 216 299 188)(78 217 300 189)(79 218 281 190)(80 219 282 191)(81 271 159 174)(82 272 160 175)(83 273 141 176)(84 274 142 177)(85 275 143 178)(86 276 144 179)(87 277 145 180)(88 278 146 161)(89 279 147 162)(90 280 148 163)(91 261 149 164)(92 262 150 165)(93 263 151 166)(94 264 152 167)(95 265 153 168)(96 266 154 169)(97 267 155 170)(98 268 156 171)(99 269 157 172)(100 270 158 173)
(1 40 132 59)(2 21 133 60)(3 22 134 41)(4 23 135 42)(5 24 136 43)(6 25 137 44)(7 26 138 45)(8 27 139 46)(9 28 140 47)(10 29 121 48)(11 30 122 49)(12 31 123 50)(13 32 124 51)(14 33 125 52)(15 34 126 53)(16 35 127 54)(17 36 128 55)(18 37 129 56)(19 38 130 57)(20 39 131 58)(61 161 283 278)(62 162 284 279)(63 163 285 280)(64 164 286 261)(65 165 287 262)(66 166 288 263)(67 167 289 264)(68 168 290 265)(69 169 291 266)(70 170 292 267)(71 171 293 268)(72 172 294 269)(73 173 295 270)(74 174 296 271)(75 175 297 272)(76 176 298 273)(77 177 299 274)(78 178 300 275)(79 179 281 276)(80 180 282 277)(81 213 159 185)(82 214 160 186)(83 215 141 187)(84 216 142 188)(85 217 143 189)(86 218 144 190)(87 219 145 191)(88 220 146 192)(89 201 147 193)(90 202 148 194)(91 203 149 195)(92 204 150 196)(93 205 151 197)(94 206 152 198)(95 207 153 199)(96 208 154 200)(97 209 155 181)(98 210 156 182)(99 211 157 183)(100 212 158 184)(101 221 317 252)(102 222 318 253)(103 223 319 254)(104 224 320 255)(105 225 301 256)(106 226 302 257)(107 227 303 258)(108 228 304 259)(109 229 305 260)(110 230 306 241)(111 231 307 242)(112 232 308 243)(113 233 309 244)(114 234 310 245)(115 235 311 246)(116 236 312 247)(117 237 313 248)(118 238 314 249)(119 239 315 250)(120 240 316 251)
(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)
(1 178 122 265)(2 274 123 167)(3 176 124 263)(4 272 125 165)(5 174 126 261)(6 270 127 163)(7 172 128 279)(8 268 129 161)(9 170 130 277)(10 266 131 179)(11 168 132 275)(12 264 133 177)(13 166 134 273)(14 262 135 175)(15 164 136 271)(16 280 137 173)(17 162 138 269)(18 278 139 171)(19 180 140 267)(20 276 121 169)(21 216 50 198)(22 187 51 205)(23 214 52 196)(24 185 53 203)(25 212 54 194)(26 183 55 201)(27 210 56 192)(28 181 57 219)(29 208 58 190)(30 199 59 217)(31 206 60 188)(32 197 41 215)(33 204 42 186)(34 195 43 213)(35 202 44 184)(36 193 45 211)(37 220 46 182)(38 191 47 209)(39 218 48 200)(40 189 49 207)(61 250 293 229)(62 238 294 259)(63 248 295 227)(64 236 296 257)(65 246 297 225)(66 234 298 255)(67 244 299 223)(68 232 300 253)(69 242 281 221)(70 230 282 251)(71 260 283 239)(72 228 284 249)(73 258 285 237)(74 226 286 247)(75 256 287 235)(76 224 288 245)(77 254 289 233)(78 222 290 243)(79 252 291 231)(80 240 292 241)(81 302 149 116)(82 105 150 311)(83 320 151 114)(84 103 152 309)(85 318 153 112)(86 101 154 307)(87 316 155 110)(88 119 156 305)(89 314 157 108)(90 117 158 303)(91 312 159 106)(92 115 160 301)(93 310 141 104)(94 113 142 319)(95 308 143 102)(96 111 144 317)(97 306 145 120)(98 109 146 315)(99 304 147 118)(100 107 148 313)
G:=sub<Sym(320)| (1,308,132,112)(2,309,133,113)(3,310,134,114)(4,311,135,115)(5,312,136,116)(6,313,137,117)(7,314,138,118)(8,315,139,119)(9,316,140,120)(10,317,121,101)(11,318,122,102)(12,319,123,103)(13,320,124,104)(14,301,125,105)(15,302,126,106)(16,303,127,107)(17,304,128,108)(18,305,129,109)(19,306,130,110)(20,307,131,111)(21,233,60,244)(22,234,41,245)(23,235,42,246)(24,236,43,247)(25,237,44,248)(26,238,45,249)(27,239,46,250)(28,240,47,251)(29,221,48,252)(30,222,49,253)(31,223,50,254)(32,224,51,255)(33,225,52,256)(34,226,53,257)(35,227,54,258)(36,228,55,259)(37,229,56,260)(38,230,57,241)(39,231,58,242)(40,232,59,243)(61,220,283,192)(62,201,284,193)(63,202,285,194)(64,203,286,195)(65,204,287,196)(66,205,288,197)(67,206,289,198)(68,207,290,199)(69,208,291,200)(70,209,292,181)(71,210,293,182)(72,211,294,183)(73,212,295,184)(74,213,296,185)(75,214,297,186)(76,215,298,187)(77,216,299,188)(78,217,300,189)(79,218,281,190)(80,219,282,191)(81,271,159,174)(82,272,160,175)(83,273,141,176)(84,274,142,177)(85,275,143,178)(86,276,144,179)(87,277,145,180)(88,278,146,161)(89,279,147,162)(90,280,148,163)(91,261,149,164)(92,262,150,165)(93,263,151,166)(94,264,152,167)(95,265,153,168)(96,266,154,169)(97,267,155,170)(98,268,156,171)(99,269,157,172)(100,270,158,173), (1,40,132,59)(2,21,133,60)(3,22,134,41)(4,23,135,42)(5,24,136,43)(6,25,137,44)(7,26,138,45)(8,27,139,46)(9,28,140,47)(10,29,121,48)(11,30,122,49)(12,31,123,50)(13,32,124,51)(14,33,125,52)(15,34,126,53)(16,35,127,54)(17,36,128,55)(18,37,129,56)(19,38,130,57)(20,39,131,58)(61,161,283,278)(62,162,284,279)(63,163,285,280)(64,164,286,261)(65,165,287,262)(66,166,288,263)(67,167,289,264)(68,168,290,265)(69,169,291,266)(70,170,292,267)(71,171,293,268)(72,172,294,269)(73,173,295,270)(74,174,296,271)(75,175,297,272)(76,176,298,273)(77,177,299,274)(78,178,300,275)(79,179,281,276)(80,180,282,277)(81,213,159,185)(82,214,160,186)(83,215,141,187)(84,216,142,188)(85,217,143,189)(86,218,144,190)(87,219,145,191)(88,220,146,192)(89,201,147,193)(90,202,148,194)(91,203,149,195)(92,204,150,196)(93,205,151,197)(94,206,152,198)(95,207,153,199)(96,208,154,200)(97,209,155,181)(98,210,156,182)(99,211,157,183)(100,212,158,184)(101,221,317,252)(102,222,318,253)(103,223,319,254)(104,224,320,255)(105,225,301,256)(106,226,302,257)(107,227,303,258)(108,228,304,259)(109,229,305,260)(110,230,306,241)(111,231,307,242)(112,232,308,243)(113,233,309,244)(114,234,310,245)(115,235,311,246)(116,236,312,247)(117,237,313,248)(118,238,314,249)(119,239,315,250)(120,240,316,251), (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), (1,178,122,265)(2,274,123,167)(3,176,124,263)(4,272,125,165)(5,174,126,261)(6,270,127,163)(7,172,128,279)(8,268,129,161)(9,170,130,277)(10,266,131,179)(11,168,132,275)(12,264,133,177)(13,166,134,273)(14,262,135,175)(15,164,136,271)(16,280,137,173)(17,162,138,269)(18,278,139,171)(19,180,140,267)(20,276,121,169)(21,216,50,198)(22,187,51,205)(23,214,52,196)(24,185,53,203)(25,212,54,194)(26,183,55,201)(27,210,56,192)(28,181,57,219)(29,208,58,190)(30,199,59,217)(31,206,60,188)(32,197,41,215)(33,204,42,186)(34,195,43,213)(35,202,44,184)(36,193,45,211)(37,220,46,182)(38,191,47,209)(39,218,48,200)(40,189,49,207)(61,250,293,229)(62,238,294,259)(63,248,295,227)(64,236,296,257)(65,246,297,225)(66,234,298,255)(67,244,299,223)(68,232,300,253)(69,242,281,221)(70,230,282,251)(71,260,283,239)(72,228,284,249)(73,258,285,237)(74,226,286,247)(75,256,287,235)(76,224,288,245)(77,254,289,233)(78,222,290,243)(79,252,291,231)(80,240,292,241)(81,302,149,116)(82,105,150,311)(83,320,151,114)(84,103,152,309)(85,318,153,112)(86,101,154,307)(87,316,155,110)(88,119,156,305)(89,314,157,108)(90,117,158,303)(91,312,159,106)(92,115,160,301)(93,310,141,104)(94,113,142,319)(95,308,143,102)(96,111,144,317)(97,306,145,120)(98,109,146,315)(99,304,147,118)(100,107,148,313)>;
G:=Group( (1,308,132,112)(2,309,133,113)(3,310,134,114)(4,311,135,115)(5,312,136,116)(6,313,137,117)(7,314,138,118)(8,315,139,119)(9,316,140,120)(10,317,121,101)(11,318,122,102)(12,319,123,103)(13,320,124,104)(14,301,125,105)(15,302,126,106)(16,303,127,107)(17,304,128,108)(18,305,129,109)(19,306,130,110)(20,307,131,111)(21,233,60,244)(22,234,41,245)(23,235,42,246)(24,236,43,247)(25,237,44,248)(26,238,45,249)(27,239,46,250)(28,240,47,251)(29,221,48,252)(30,222,49,253)(31,223,50,254)(32,224,51,255)(33,225,52,256)(34,226,53,257)(35,227,54,258)(36,228,55,259)(37,229,56,260)(38,230,57,241)(39,231,58,242)(40,232,59,243)(61,220,283,192)(62,201,284,193)(63,202,285,194)(64,203,286,195)(65,204,287,196)(66,205,288,197)(67,206,289,198)(68,207,290,199)(69,208,291,200)(70,209,292,181)(71,210,293,182)(72,211,294,183)(73,212,295,184)(74,213,296,185)(75,214,297,186)(76,215,298,187)(77,216,299,188)(78,217,300,189)(79,218,281,190)(80,219,282,191)(81,271,159,174)(82,272,160,175)(83,273,141,176)(84,274,142,177)(85,275,143,178)(86,276,144,179)(87,277,145,180)(88,278,146,161)(89,279,147,162)(90,280,148,163)(91,261,149,164)(92,262,150,165)(93,263,151,166)(94,264,152,167)(95,265,153,168)(96,266,154,169)(97,267,155,170)(98,268,156,171)(99,269,157,172)(100,270,158,173), (1,40,132,59)(2,21,133,60)(3,22,134,41)(4,23,135,42)(5,24,136,43)(6,25,137,44)(7,26,138,45)(8,27,139,46)(9,28,140,47)(10,29,121,48)(11,30,122,49)(12,31,123,50)(13,32,124,51)(14,33,125,52)(15,34,126,53)(16,35,127,54)(17,36,128,55)(18,37,129,56)(19,38,130,57)(20,39,131,58)(61,161,283,278)(62,162,284,279)(63,163,285,280)(64,164,286,261)(65,165,287,262)(66,166,288,263)(67,167,289,264)(68,168,290,265)(69,169,291,266)(70,170,292,267)(71,171,293,268)(72,172,294,269)(73,173,295,270)(74,174,296,271)(75,175,297,272)(76,176,298,273)(77,177,299,274)(78,178,300,275)(79,179,281,276)(80,180,282,277)(81,213,159,185)(82,214,160,186)(83,215,141,187)(84,216,142,188)(85,217,143,189)(86,218,144,190)(87,219,145,191)(88,220,146,192)(89,201,147,193)(90,202,148,194)(91,203,149,195)(92,204,150,196)(93,205,151,197)(94,206,152,198)(95,207,153,199)(96,208,154,200)(97,209,155,181)(98,210,156,182)(99,211,157,183)(100,212,158,184)(101,221,317,252)(102,222,318,253)(103,223,319,254)(104,224,320,255)(105,225,301,256)(106,226,302,257)(107,227,303,258)(108,228,304,259)(109,229,305,260)(110,230,306,241)(111,231,307,242)(112,232,308,243)(113,233,309,244)(114,234,310,245)(115,235,311,246)(116,236,312,247)(117,237,313,248)(118,238,314,249)(119,239,315,250)(120,240,316,251), (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), (1,178,122,265)(2,274,123,167)(3,176,124,263)(4,272,125,165)(5,174,126,261)(6,270,127,163)(7,172,128,279)(8,268,129,161)(9,170,130,277)(10,266,131,179)(11,168,132,275)(12,264,133,177)(13,166,134,273)(14,262,135,175)(15,164,136,271)(16,280,137,173)(17,162,138,269)(18,278,139,171)(19,180,140,267)(20,276,121,169)(21,216,50,198)(22,187,51,205)(23,214,52,196)(24,185,53,203)(25,212,54,194)(26,183,55,201)(27,210,56,192)(28,181,57,219)(29,208,58,190)(30,199,59,217)(31,206,60,188)(32,197,41,215)(33,204,42,186)(34,195,43,213)(35,202,44,184)(36,193,45,211)(37,220,46,182)(38,191,47,209)(39,218,48,200)(40,189,49,207)(61,250,293,229)(62,238,294,259)(63,248,295,227)(64,236,296,257)(65,246,297,225)(66,234,298,255)(67,244,299,223)(68,232,300,253)(69,242,281,221)(70,230,282,251)(71,260,283,239)(72,228,284,249)(73,258,285,237)(74,226,286,247)(75,256,287,235)(76,224,288,245)(77,254,289,233)(78,222,290,243)(79,252,291,231)(80,240,292,241)(81,302,149,116)(82,105,150,311)(83,320,151,114)(84,103,152,309)(85,318,153,112)(86,101,154,307)(87,316,155,110)(88,119,156,305)(89,314,157,108)(90,117,158,303)(91,312,159,106)(92,115,160,301)(93,310,141,104)(94,113,142,319)(95,308,143,102)(96,111,144,317)(97,306,145,120)(98,109,146,315)(99,304,147,118)(100,107,148,313) );
G=PermutationGroup([[(1,308,132,112),(2,309,133,113),(3,310,134,114),(4,311,135,115),(5,312,136,116),(6,313,137,117),(7,314,138,118),(8,315,139,119),(9,316,140,120),(10,317,121,101),(11,318,122,102),(12,319,123,103),(13,320,124,104),(14,301,125,105),(15,302,126,106),(16,303,127,107),(17,304,128,108),(18,305,129,109),(19,306,130,110),(20,307,131,111),(21,233,60,244),(22,234,41,245),(23,235,42,246),(24,236,43,247),(25,237,44,248),(26,238,45,249),(27,239,46,250),(28,240,47,251),(29,221,48,252),(30,222,49,253),(31,223,50,254),(32,224,51,255),(33,225,52,256),(34,226,53,257),(35,227,54,258),(36,228,55,259),(37,229,56,260),(38,230,57,241),(39,231,58,242),(40,232,59,243),(61,220,283,192),(62,201,284,193),(63,202,285,194),(64,203,286,195),(65,204,287,196),(66,205,288,197),(67,206,289,198),(68,207,290,199),(69,208,291,200),(70,209,292,181),(71,210,293,182),(72,211,294,183),(73,212,295,184),(74,213,296,185),(75,214,297,186),(76,215,298,187),(77,216,299,188),(78,217,300,189),(79,218,281,190),(80,219,282,191),(81,271,159,174),(82,272,160,175),(83,273,141,176),(84,274,142,177),(85,275,143,178),(86,276,144,179),(87,277,145,180),(88,278,146,161),(89,279,147,162),(90,280,148,163),(91,261,149,164),(92,262,150,165),(93,263,151,166),(94,264,152,167),(95,265,153,168),(96,266,154,169),(97,267,155,170),(98,268,156,171),(99,269,157,172),(100,270,158,173)], [(1,40,132,59),(2,21,133,60),(3,22,134,41),(4,23,135,42),(5,24,136,43),(6,25,137,44),(7,26,138,45),(8,27,139,46),(9,28,140,47),(10,29,121,48),(11,30,122,49),(12,31,123,50),(13,32,124,51),(14,33,125,52),(15,34,126,53),(16,35,127,54),(17,36,128,55),(18,37,129,56),(19,38,130,57),(20,39,131,58),(61,161,283,278),(62,162,284,279),(63,163,285,280),(64,164,286,261),(65,165,287,262),(66,166,288,263),(67,167,289,264),(68,168,290,265),(69,169,291,266),(70,170,292,267),(71,171,293,268),(72,172,294,269),(73,173,295,270),(74,174,296,271),(75,175,297,272),(76,176,298,273),(77,177,299,274),(78,178,300,275),(79,179,281,276),(80,180,282,277),(81,213,159,185),(82,214,160,186),(83,215,141,187),(84,216,142,188),(85,217,143,189),(86,218,144,190),(87,219,145,191),(88,220,146,192),(89,201,147,193),(90,202,148,194),(91,203,149,195),(92,204,150,196),(93,205,151,197),(94,206,152,198),(95,207,153,199),(96,208,154,200),(97,209,155,181),(98,210,156,182),(99,211,157,183),(100,212,158,184),(101,221,317,252),(102,222,318,253),(103,223,319,254),(104,224,320,255),(105,225,301,256),(106,226,302,257),(107,227,303,258),(108,228,304,259),(109,229,305,260),(110,230,306,241),(111,231,307,242),(112,232,308,243),(113,233,309,244),(114,234,310,245),(115,235,311,246),(116,236,312,247),(117,237,313,248),(118,238,314,249),(119,239,315,250),(120,240,316,251)], [(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)], [(1,178,122,265),(2,274,123,167),(3,176,124,263),(4,272,125,165),(5,174,126,261),(6,270,127,163),(7,172,128,279),(8,268,129,161),(9,170,130,277),(10,266,131,179),(11,168,132,275),(12,264,133,177),(13,166,134,273),(14,262,135,175),(15,164,136,271),(16,280,137,173),(17,162,138,269),(18,278,139,171),(19,180,140,267),(20,276,121,169),(21,216,50,198),(22,187,51,205),(23,214,52,196),(24,185,53,203),(25,212,54,194),(26,183,55,201),(27,210,56,192),(28,181,57,219),(29,208,58,190),(30,199,59,217),(31,206,60,188),(32,197,41,215),(33,204,42,186),(34,195,43,213),(35,202,44,184),(36,193,45,211),(37,220,46,182),(38,191,47,209),(39,218,48,200),(40,189,49,207),(61,250,293,229),(62,238,294,259),(63,248,295,227),(64,236,296,257),(65,246,297,225),(66,234,298,255),(67,244,299,223),(68,232,300,253),(69,242,281,221),(70,230,282,251),(71,260,283,239),(72,228,284,249),(73,258,285,237),(74,226,286,247),(75,256,287,235),(76,224,288,245),(77,254,289,233),(78,222,290,243),(79,252,291,231),(80,240,292,241),(81,302,149,116),(82,105,150,311),(83,320,151,114),(84,103,152,309),(85,318,153,112),(86,101,154,307),(87,316,155,110),(88,119,156,305),(89,314,157,108),(90,117,158,303),(91,312,159,106),(92,115,160,301),(93,310,141,104),(94,113,142,319),(95,308,143,102),(96,111,144,317),(97,306,145,120),(98,109,146,315),(99,304,147,118),(100,107,148,313)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | Dic10 | C4○D20 | C8.C22 | C20.C23 | D4.8D10 |
| kernel | Q8.3Dic10 | C20⋊3C8 | C10.D8 | C20.Q8 | Q8⋊Dic5 | C20.6Q8 | Q8×C20 | C2×C20 | C5×Q8 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C2×C4 | Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of Q8.3Dic10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 37 |
| 0 | 0 | 21 | 40 |
| 18 | 35 | 0 | 0 |
| 6 | 23 | 0 | 0 |
| 0 | 0 | 0 | 19 |
| 0 | 0 | 28 | 0 |
| 28 | 39 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 21 | 20 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 0 | 0 | 40 | 4 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,21,0,0,37,40],[18,6,0,0,35,23,0,0,0,0,0,28,0,0,19,0],[28,2,0,0,39,16,0,0,0,0,32,0,0,0,0,32],[21,23,0,0,20,20,0,0,0,0,40,0,0,0,4,1] >;
Q8.3Dic10 in GAP, Magma, Sage, TeX
Q_8._3{\rm Dic}_{10} % in TeX
G:=Group("Q8.3Dic10"); // GroupNames label
G:=SmallGroup(320,649);
// by ID
G=gap.SmallGroup(320,649);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,184,1123,297,136,12550]);
// 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=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations