metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊1Q8, Dic5.20SD16, C4.1(Q8×D5), C20⋊Q8.5C2, C20.9(C2×Q8), C4⋊C4.31D10, C5⋊3(Q8⋊Q8), C4.Q8.4D5, (C2×C8).135D10, C2.20(D5×SD16), C4.70(C4○D20), C10.35(C2×SD16), C10.Q16.5C2, C22.209(D4×D5), C20.Q8.4C2, C20.166(C4○D4), (C2×C20).270C23, (C2×C40).282C22, (C2×Dic5).217D4, Dic5⋊3Q8.5C2, C10.35(C22⋊Q8), C20.8Q8.13C2, C2.12(D10⋊Q8), C20.44D4.13C2, C2.20(SD16⋊D5), C10.38(C8.C22), C4⋊Dic5.102C22, (C4×Dic5).34C22, (C2×Dic10).84C22, (C5×C4.Q8).9C2, (C2×C10).275(C2×D4), (C5×C4⋊C4).63C22, (C2×C5⋊2C8).52C22, (C2×C4).373(C22×D5), SmallGroup(320,481)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a9, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c-1 >
Subgroups: 358 in 96 conjugacy classes, 41 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, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C4×Q8, C4⋊Q8, 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, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C20.Q8, C10.Q16, C20.8Q8, C20.44D4, C5×C4.Q8, Dic5⋊3Q8, C20⋊Q8, Dic10⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8.C22, C22×D5, Q8⋊Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D5×SD16, SD16⋊D5, Dic10⋊Q8
►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)
(1 137 11 127)(2 136 12 126)(3 135 13 125)(4 134 14 124)(5 133 15 123)(6 132 16 122)(7 131 17 121)(8 130 18 140)(9 129 19 139)(10 128 20 138)(21 289 31 299)(22 288 32 298)(23 287 33 297)(24 286 34 296)(25 285 35 295)(26 284 36 294)(27 283 37 293)(28 282 38 292)(29 281 39 291)(30 300 40 290)(41 186 51 196)(42 185 52 195)(43 184 53 194)(44 183 54 193)(45 182 55 192)(46 181 56 191)(47 200 57 190)(48 199 58 189)(49 198 59 188)(50 197 60 187)(61 117 71 107)(62 116 72 106)(63 115 73 105)(64 114 74 104)(65 113 75 103)(66 112 76 102)(67 111 77 101)(68 110 78 120)(69 109 79 119)(70 108 80 118)(81 275 91 265)(82 274 92 264)(83 273 93 263)(84 272 94 262)(85 271 95 261)(86 270 96 280)(87 269 97 279)(88 268 98 278)(89 267 99 277)(90 266 100 276)(141 180 151 170)(142 179 152 169)(143 178 153 168)(144 177 154 167)(145 176 155 166)(146 175 156 165)(147 174 157 164)(148 173 158 163)(149 172 159 162)(150 171 160 161)(201 313 211 303)(202 312 212 302)(203 311 213 301)(204 310 214 320)(205 309 215 319)(206 308 216 318)(207 307 217 317)(208 306 218 316)(209 305 219 315)(210 304 220 314)(221 241 231 251)(222 260 232 250)(223 259 233 249)(224 258 234 248)(225 257 235 247)(226 256 236 246)(227 255 237 245)(228 254 238 244)(229 253 239 243)(230 252 240 242)
(1 105 44 301)(2 114 45 310)(3 103 46 319)(4 112 47 308)(5 101 48 317)(6 110 49 306)(7 119 50 315)(8 108 51 304)(9 117 52 313)(10 106 53 302)(11 115 54 311)(12 104 55 320)(13 113 56 309)(14 102 57 318)(15 111 58 307)(16 120 59 316)(17 109 60 305)(18 118 41 314)(19 107 42 303)(20 116 43 312)(21 268 173 258)(22 277 174 247)(23 266 175 256)(24 275 176 245)(25 264 177 254)(26 273 178 243)(27 262 179 252)(28 271 180 241)(29 280 161 250)(30 269 162 259)(31 278 163 248)(32 267 164 257)(33 276 165 246)(34 265 166 255)(35 274 167 244)(36 263 168 253)(37 272 169 242)(38 261 170 251)(39 270 171 260)(40 279 172 249)(61 185 201 139)(62 194 202 128)(63 183 203 137)(64 192 204 126)(65 181 205 135)(66 190 206 124)(67 199 207 133)(68 188 208 122)(69 197 209 131)(70 186 210 140)(71 195 211 129)(72 184 212 138)(73 193 213 127)(74 182 214 136)(75 191 215 125)(76 200 216 134)(77 189 217 123)(78 198 218 132)(79 187 219 121)(80 196 220 130)(81 145 237 296)(82 154 238 285)(83 143 239 294)(84 152 240 283)(85 141 221 292)(86 150 222 281)(87 159 223 290)(88 148 224 299)(89 157 225 288)(90 146 226 297)(91 155 227 286)(92 144 228 295)(93 153 229 284)(94 142 230 293)(95 151 231 282)(96 160 232 291)(97 149 233 300)(98 158 234 289)(99 147 235 298)(100 156 236 287)
(1 152 44 283)(2 143 45 294)(3 154 46 285)(4 145 47 296)(5 156 48 287)(6 147 49 298)(7 158 50 289)(8 149 51 300)(9 160 52 291)(10 151 53 282)(11 142 54 293)(12 153 55 284)(13 144 56 295)(14 155 57 286)(15 146 58 297)(16 157 59 288)(17 148 60 299)(18 159 41 290)(19 150 42 281)(20 141 43 292)(21 126 173 192)(22 137 174 183)(23 128 175 194)(24 139 176 185)(25 130 177 196)(26 121 178 187)(27 132 179 198)(28 123 180 189)(29 134 161 200)(30 125 162 191)(31 136 163 182)(32 127 164 193)(33 138 165 184)(34 129 166 195)(35 140 167 186)(36 131 168 197)(37 122 169 188)(38 133 170 199)(39 124 171 190)(40 135 172 181)(61 275 201 245)(62 266 202 256)(63 277 203 247)(64 268 204 258)(65 279 205 249)(66 270 206 260)(67 261 207 251)(68 272 208 242)(69 263 209 253)(70 274 210 244)(71 265 211 255)(72 276 212 246)(73 267 213 257)(74 278 214 248)(75 269 215 259)(76 280 216 250)(77 271 217 241)(78 262 218 252)(79 273 219 243)(80 264 220 254)(81 308 237 112)(82 319 238 103)(83 310 239 114)(84 301 240 105)(85 312 221 116)(86 303 222 107)(87 314 223 118)(88 305 224 109)(89 316 225 120)(90 307 226 111)(91 318 227 102)(92 309 228 113)(93 320 229 104)(94 311 230 115)(95 302 231 106)(96 313 232 117)(97 304 233 108)(98 315 234 119)(99 306 235 110)(100 317 236 101)
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), (1,137,11,127)(2,136,12,126)(3,135,13,125)(4,134,14,124)(5,133,15,123)(6,132,16,122)(7,131,17,121)(8,130,18,140)(9,129,19,139)(10,128,20,138)(21,289,31,299)(22,288,32,298)(23,287,33,297)(24,286,34,296)(25,285,35,295)(26,284,36,294)(27,283,37,293)(28,282,38,292)(29,281,39,291)(30,300,40,290)(41,186,51,196)(42,185,52,195)(43,184,53,194)(44,183,54,193)(45,182,55,192)(46,181,56,191)(47,200,57,190)(48,199,58,189)(49,198,59,188)(50,197,60,187)(61,117,71,107)(62,116,72,106)(63,115,73,105)(64,114,74,104)(65,113,75,103)(66,112,76,102)(67,111,77,101)(68,110,78,120)(69,109,79,119)(70,108,80,118)(81,275,91,265)(82,274,92,264)(83,273,93,263)(84,272,94,262)(85,271,95,261)(86,270,96,280)(87,269,97,279)(88,268,98,278)(89,267,99,277)(90,266,100,276)(141,180,151,170)(142,179,152,169)(143,178,153,168)(144,177,154,167)(145,176,155,166)(146,175,156,165)(147,174,157,164)(148,173,158,163)(149,172,159,162)(150,171,160,161)(201,313,211,303)(202,312,212,302)(203,311,213,301)(204,310,214,320)(205,309,215,319)(206,308,216,318)(207,307,217,317)(208,306,218,316)(209,305,219,315)(210,304,220,314)(221,241,231,251)(222,260,232,250)(223,259,233,249)(224,258,234,248)(225,257,235,247)(226,256,236,246)(227,255,237,245)(228,254,238,244)(229,253,239,243)(230,252,240,242), (1,105,44,301)(2,114,45,310)(3,103,46,319)(4,112,47,308)(5,101,48,317)(6,110,49,306)(7,119,50,315)(8,108,51,304)(9,117,52,313)(10,106,53,302)(11,115,54,311)(12,104,55,320)(13,113,56,309)(14,102,57,318)(15,111,58,307)(16,120,59,316)(17,109,60,305)(18,118,41,314)(19,107,42,303)(20,116,43,312)(21,268,173,258)(22,277,174,247)(23,266,175,256)(24,275,176,245)(25,264,177,254)(26,273,178,243)(27,262,179,252)(28,271,180,241)(29,280,161,250)(30,269,162,259)(31,278,163,248)(32,267,164,257)(33,276,165,246)(34,265,166,255)(35,274,167,244)(36,263,168,253)(37,272,169,242)(38,261,170,251)(39,270,171,260)(40,279,172,249)(61,185,201,139)(62,194,202,128)(63,183,203,137)(64,192,204,126)(65,181,205,135)(66,190,206,124)(67,199,207,133)(68,188,208,122)(69,197,209,131)(70,186,210,140)(71,195,211,129)(72,184,212,138)(73,193,213,127)(74,182,214,136)(75,191,215,125)(76,200,216,134)(77,189,217,123)(78,198,218,132)(79,187,219,121)(80,196,220,130)(81,145,237,296)(82,154,238,285)(83,143,239,294)(84,152,240,283)(85,141,221,292)(86,150,222,281)(87,159,223,290)(88,148,224,299)(89,157,225,288)(90,146,226,297)(91,155,227,286)(92,144,228,295)(93,153,229,284)(94,142,230,293)(95,151,231,282)(96,160,232,291)(97,149,233,300)(98,158,234,289)(99,147,235,298)(100,156,236,287), (1,152,44,283)(2,143,45,294)(3,154,46,285)(4,145,47,296)(5,156,48,287)(6,147,49,298)(7,158,50,289)(8,149,51,300)(9,160,52,291)(10,151,53,282)(11,142,54,293)(12,153,55,284)(13,144,56,295)(14,155,57,286)(15,146,58,297)(16,157,59,288)(17,148,60,299)(18,159,41,290)(19,150,42,281)(20,141,43,292)(21,126,173,192)(22,137,174,183)(23,128,175,194)(24,139,176,185)(25,130,177,196)(26,121,178,187)(27,132,179,198)(28,123,180,189)(29,134,161,200)(30,125,162,191)(31,136,163,182)(32,127,164,193)(33,138,165,184)(34,129,166,195)(35,140,167,186)(36,131,168,197)(37,122,169,188)(38,133,170,199)(39,124,171,190)(40,135,172,181)(61,275,201,245)(62,266,202,256)(63,277,203,247)(64,268,204,258)(65,279,205,249)(66,270,206,260)(67,261,207,251)(68,272,208,242)(69,263,209,253)(70,274,210,244)(71,265,211,255)(72,276,212,246)(73,267,213,257)(74,278,214,248)(75,269,215,259)(76,280,216,250)(77,271,217,241)(78,262,218,252)(79,273,219,243)(80,264,220,254)(81,308,237,112)(82,319,238,103)(83,310,239,114)(84,301,240,105)(85,312,221,116)(86,303,222,107)(87,314,223,118)(88,305,224,109)(89,316,225,120)(90,307,226,111)(91,318,227,102)(92,309,228,113)(93,320,229,104)(94,311,230,115)(95,302,231,106)(96,313,232,117)(97,304,233,108)(98,315,234,119)(99,306,235,110)(100,317,236,101)>;
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), (1,137,11,127)(2,136,12,126)(3,135,13,125)(4,134,14,124)(5,133,15,123)(6,132,16,122)(7,131,17,121)(8,130,18,140)(9,129,19,139)(10,128,20,138)(21,289,31,299)(22,288,32,298)(23,287,33,297)(24,286,34,296)(25,285,35,295)(26,284,36,294)(27,283,37,293)(28,282,38,292)(29,281,39,291)(30,300,40,290)(41,186,51,196)(42,185,52,195)(43,184,53,194)(44,183,54,193)(45,182,55,192)(46,181,56,191)(47,200,57,190)(48,199,58,189)(49,198,59,188)(50,197,60,187)(61,117,71,107)(62,116,72,106)(63,115,73,105)(64,114,74,104)(65,113,75,103)(66,112,76,102)(67,111,77,101)(68,110,78,120)(69,109,79,119)(70,108,80,118)(81,275,91,265)(82,274,92,264)(83,273,93,263)(84,272,94,262)(85,271,95,261)(86,270,96,280)(87,269,97,279)(88,268,98,278)(89,267,99,277)(90,266,100,276)(141,180,151,170)(142,179,152,169)(143,178,153,168)(144,177,154,167)(145,176,155,166)(146,175,156,165)(147,174,157,164)(148,173,158,163)(149,172,159,162)(150,171,160,161)(201,313,211,303)(202,312,212,302)(203,311,213,301)(204,310,214,320)(205,309,215,319)(206,308,216,318)(207,307,217,317)(208,306,218,316)(209,305,219,315)(210,304,220,314)(221,241,231,251)(222,260,232,250)(223,259,233,249)(224,258,234,248)(225,257,235,247)(226,256,236,246)(227,255,237,245)(228,254,238,244)(229,253,239,243)(230,252,240,242), (1,105,44,301)(2,114,45,310)(3,103,46,319)(4,112,47,308)(5,101,48,317)(6,110,49,306)(7,119,50,315)(8,108,51,304)(9,117,52,313)(10,106,53,302)(11,115,54,311)(12,104,55,320)(13,113,56,309)(14,102,57,318)(15,111,58,307)(16,120,59,316)(17,109,60,305)(18,118,41,314)(19,107,42,303)(20,116,43,312)(21,268,173,258)(22,277,174,247)(23,266,175,256)(24,275,176,245)(25,264,177,254)(26,273,178,243)(27,262,179,252)(28,271,180,241)(29,280,161,250)(30,269,162,259)(31,278,163,248)(32,267,164,257)(33,276,165,246)(34,265,166,255)(35,274,167,244)(36,263,168,253)(37,272,169,242)(38,261,170,251)(39,270,171,260)(40,279,172,249)(61,185,201,139)(62,194,202,128)(63,183,203,137)(64,192,204,126)(65,181,205,135)(66,190,206,124)(67,199,207,133)(68,188,208,122)(69,197,209,131)(70,186,210,140)(71,195,211,129)(72,184,212,138)(73,193,213,127)(74,182,214,136)(75,191,215,125)(76,200,216,134)(77,189,217,123)(78,198,218,132)(79,187,219,121)(80,196,220,130)(81,145,237,296)(82,154,238,285)(83,143,239,294)(84,152,240,283)(85,141,221,292)(86,150,222,281)(87,159,223,290)(88,148,224,299)(89,157,225,288)(90,146,226,297)(91,155,227,286)(92,144,228,295)(93,153,229,284)(94,142,230,293)(95,151,231,282)(96,160,232,291)(97,149,233,300)(98,158,234,289)(99,147,235,298)(100,156,236,287), (1,152,44,283)(2,143,45,294)(3,154,46,285)(4,145,47,296)(5,156,48,287)(6,147,49,298)(7,158,50,289)(8,149,51,300)(9,160,52,291)(10,151,53,282)(11,142,54,293)(12,153,55,284)(13,144,56,295)(14,155,57,286)(15,146,58,297)(16,157,59,288)(17,148,60,299)(18,159,41,290)(19,150,42,281)(20,141,43,292)(21,126,173,192)(22,137,174,183)(23,128,175,194)(24,139,176,185)(25,130,177,196)(26,121,178,187)(27,132,179,198)(28,123,180,189)(29,134,161,200)(30,125,162,191)(31,136,163,182)(32,127,164,193)(33,138,165,184)(34,129,166,195)(35,140,167,186)(36,131,168,197)(37,122,169,188)(38,133,170,199)(39,124,171,190)(40,135,172,181)(61,275,201,245)(62,266,202,256)(63,277,203,247)(64,268,204,258)(65,279,205,249)(66,270,206,260)(67,261,207,251)(68,272,208,242)(69,263,209,253)(70,274,210,244)(71,265,211,255)(72,276,212,246)(73,267,213,257)(74,278,214,248)(75,269,215,259)(76,280,216,250)(77,271,217,241)(78,262,218,252)(79,273,219,243)(80,264,220,254)(81,308,237,112)(82,319,238,103)(83,310,239,114)(84,301,240,105)(85,312,221,116)(86,303,222,107)(87,314,223,118)(88,305,224,109)(89,316,225,120)(90,307,226,111)(91,318,227,102)(92,309,228,113)(93,320,229,104)(94,311,230,115)(95,302,231,106)(96,313,232,117)(97,304,233,108)(98,315,234,119)(99,306,235,110)(100,317,236,101) );
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)], [(1,137,11,127),(2,136,12,126),(3,135,13,125),(4,134,14,124),(5,133,15,123),(6,132,16,122),(7,131,17,121),(8,130,18,140),(9,129,19,139),(10,128,20,138),(21,289,31,299),(22,288,32,298),(23,287,33,297),(24,286,34,296),(25,285,35,295),(26,284,36,294),(27,283,37,293),(28,282,38,292),(29,281,39,291),(30,300,40,290),(41,186,51,196),(42,185,52,195),(43,184,53,194),(44,183,54,193),(45,182,55,192),(46,181,56,191),(47,200,57,190),(48,199,58,189),(49,198,59,188),(50,197,60,187),(61,117,71,107),(62,116,72,106),(63,115,73,105),(64,114,74,104),(65,113,75,103),(66,112,76,102),(67,111,77,101),(68,110,78,120),(69,109,79,119),(70,108,80,118),(81,275,91,265),(82,274,92,264),(83,273,93,263),(84,272,94,262),(85,271,95,261),(86,270,96,280),(87,269,97,279),(88,268,98,278),(89,267,99,277),(90,266,100,276),(141,180,151,170),(142,179,152,169),(143,178,153,168),(144,177,154,167),(145,176,155,166),(146,175,156,165),(147,174,157,164),(148,173,158,163),(149,172,159,162),(150,171,160,161),(201,313,211,303),(202,312,212,302),(203,311,213,301),(204,310,214,320),(205,309,215,319),(206,308,216,318),(207,307,217,317),(208,306,218,316),(209,305,219,315),(210,304,220,314),(221,241,231,251),(222,260,232,250),(223,259,233,249),(224,258,234,248),(225,257,235,247),(226,256,236,246),(227,255,237,245),(228,254,238,244),(229,253,239,243),(230,252,240,242)], [(1,105,44,301),(2,114,45,310),(3,103,46,319),(4,112,47,308),(5,101,48,317),(6,110,49,306),(7,119,50,315),(8,108,51,304),(9,117,52,313),(10,106,53,302),(11,115,54,311),(12,104,55,320),(13,113,56,309),(14,102,57,318),(15,111,58,307),(16,120,59,316),(17,109,60,305),(18,118,41,314),(19,107,42,303),(20,116,43,312),(21,268,173,258),(22,277,174,247),(23,266,175,256),(24,275,176,245),(25,264,177,254),(26,273,178,243),(27,262,179,252),(28,271,180,241),(29,280,161,250),(30,269,162,259),(31,278,163,248),(32,267,164,257),(33,276,165,246),(34,265,166,255),(35,274,167,244),(36,263,168,253),(37,272,169,242),(38,261,170,251),(39,270,171,260),(40,279,172,249),(61,185,201,139),(62,194,202,128),(63,183,203,137),(64,192,204,126),(65,181,205,135),(66,190,206,124),(67,199,207,133),(68,188,208,122),(69,197,209,131),(70,186,210,140),(71,195,211,129),(72,184,212,138),(73,193,213,127),(74,182,214,136),(75,191,215,125),(76,200,216,134),(77,189,217,123),(78,198,218,132),(79,187,219,121),(80,196,220,130),(81,145,237,296),(82,154,238,285),(83,143,239,294),(84,152,240,283),(85,141,221,292),(86,150,222,281),(87,159,223,290),(88,148,224,299),(89,157,225,288),(90,146,226,297),(91,155,227,286),(92,144,228,295),(93,153,229,284),(94,142,230,293),(95,151,231,282),(96,160,232,291),(97,149,233,300),(98,158,234,289),(99,147,235,298),(100,156,236,287)], [(1,152,44,283),(2,143,45,294),(3,154,46,285),(4,145,47,296),(5,156,48,287),(6,147,49,298),(7,158,50,289),(8,149,51,300),(9,160,52,291),(10,151,53,282),(11,142,54,293),(12,153,55,284),(13,144,56,295),(14,155,57,286),(15,146,58,297),(16,157,59,288),(17,148,60,299),(18,159,41,290),(19,150,42,281),(20,141,43,292),(21,126,173,192),(22,137,174,183),(23,128,175,194),(24,139,176,185),(25,130,177,196),(26,121,178,187),(27,132,179,198),(28,123,180,189),(29,134,161,200),(30,125,162,191),(31,136,163,182),(32,127,164,193),(33,138,165,184),(34,129,166,195),(35,140,167,186),(36,131,168,197),(37,122,169,188),(38,133,170,199),(39,124,171,190),(40,135,172,181),(61,275,201,245),(62,266,202,256),(63,277,203,247),(64,268,204,258),(65,279,205,249),(66,270,206,260),(67,261,207,251),(68,272,208,242),(69,263,209,253),(70,274,210,244),(71,265,211,255),(72,276,212,246),(73,267,213,257),(74,278,214,248),(75,269,215,259),(76,280,216,250),(77,271,217,241),(78,262,218,252),(79,273,219,243),(80,264,220,254),(81,308,237,112),(82,319,238,103),(83,310,239,114),(84,301,240,105),(85,312,221,116),(86,303,222,107),(87,314,223,118),(88,305,224,109),(89,316,225,120),(90,307,226,111),(91,318,227,102),(92,309,228,113),(93,320,229,104),(94,311,230,115),(95,302,231,106),(96,313,232,117),(97,304,233,108),(98,315,234,119),(99,306,235,110),(100,317,236,101)]])
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 | SD16 | C4○D4 | D10 | D10 | C4○D20 | C8.C22 | Q8×D5 | D4×D5 | D5×SD16 | SD16⋊D5 |
| kernel | Dic10⋊Q8 | C20.Q8 | C10.Q16 | C20.8Q8 | C20.44D4 | C5×C4.Q8 | Dic5⋊3Q8 | C20⋊Q8 | Dic10 | C2×Dic5 | C4.Q8 | Dic5 | C20 | C4⋊C4 | C2×C8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10⋊Q8 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 5 | 8 | 0 | 0 |
| 38 | 36 | 0 | 0 |
| 0 | 0 | 23 | 35 |
| 0 | 0 | 20 | 18 |
| 4 | 31 | 0 | 0 |
| 14 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 13 | 0 | 0 |
| 28 | 39 | 0 | 0 |
| 0 | 0 | 8 | 18 |
| 0 | 0 | 17 | 33 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,1,1,0,0,39,40],[5,38,0,0,8,36,0,0,0,0,23,20,0,0,35,18],[4,14,0,0,31,37,0,0,0,0,1,0,0,0,0,1],[2,28,0,0,13,39,0,0,0,0,8,17,0,0,18,33] >;
Dic10⋊Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes Q_8 % in TeX
G:=Group("Dic10:Q8"); // GroupNames label
G:=SmallGroup(320,481);
// by ID
G=gap.SmallGroup(320,481);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations