metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊8Q8, C42.169D10, C10.332- 1+4, C4⋊Q8.14D5, C4.17(Q8×D5), C5⋊5(Q8⋊3Q8), C20.52(C2×Q8), C4⋊C4.121D10, (C2×C20).98C23, (C2×Q8).141D10, (Q8×Dic5).13C2, Dic5.26(C2×Q8), C20.135(C4○D4), C4.18(D4⋊2D5), C10.44(C22×Q8), (C2×C10).265C24, (C4×C20).206C22, (C4×Dic10).25C2, C4.Dic10.15C2, Dic5⋊3Q8.12C2, Dic5⋊Q8.10C2, C4⋊Dic5.382C22, Dic5.Q8.4C2, (Q8×C10).132C22, C22.286(C23×D5), (C2×Dic5).280C23, (C4×Dic5).165C22, C10.D4.57C22, C2.34(Q8.10D10), (C2×Dic10).310C22, C2.27(C2×Q8×D5), (C5×C4⋊Q8).14C2, C10.99(C2×C4○D4), C2.63(C2×D4⋊2D5), (C2×C4).90(C22×D5), (C5×C4⋊C4).208C22, SmallGroup(320,1393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊8Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 534 in 200 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, C42.C2, C4⋊Q8, C4⋊Q8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Q8⋊3Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×Dic10, Dic5⋊3Q8, Dic5.Q8, C4.Dic10, Dic5⋊Q8, Q8×Dic5, C5×C4⋊Q8, Dic10⋊8Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2- 1+4, C22×D5, Q8⋊3Q8, D4⋊2D5, Q8×D5, C23×D5, C2×D4⋊2D5, C2×Q8×D5, Q8.10D10, Dic10⋊8Q8
►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)
(1 263 11 273)(2 262 12 272)(3 261 13 271)(4 280 14 270)(5 279 15 269)(6 278 16 268)(7 277 17 267)(8 276 18 266)(9 275 19 265)(10 274 20 264)(21 126 31 136)(22 125 32 135)(23 124 33 134)(24 123 34 133)(25 122 35 132)(26 121 36 131)(27 140 37 130)(28 139 38 129)(29 138 39 128)(30 137 40 127)(41 90 51 100)(42 89 52 99)(43 88 53 98)(44 87 54 97)(45 86 55 96)(46 85 56 95)(47 84 57 94)(48 83 58 93)(49 82 59 92)(50 81 60 91)(61 152 71 142)(62 151 72 141)(63 150 73 160)(64 149 74 159)(65 148 75 158)(66 147 76 157)(67 146 77 156)(68 145 78 155)(69 144 79 154)(70 143 80 153)(101 283 111 293)(102 282 112 292)(103 281 113 291)(104 300 114 290)(105 299 115 289)(106 298 116 288)(107 297 117 287)(108 296 118 286)(109 295 119 285)(110 294 120 284)(161 205 171 215)(162 204 172 214)(163 203 173 213)(164 202 174 212)(165 201 175 211)(166 220 176 210)(167 219 177 209)(168 218 178 208)(169 217 179 207)(170 216 180 206)(181 310 191 320)(182 309 192 319)(183 308 193 318)(184 307 194 317)(185 306 195 316)(186 305 196 315)(187 304 197 314)(188 303 198 313)(189 302 199 312)(190 301 200 311)(221 260 231 250)(222 259 232 249)(223 258 233 248)(224 257 234 247)(225 256 235 246)(226 255 236 245)(227 254 237 244)(228 253 238 243)(229 252 239 242)(230 251 240 241)
(1 97 165 128)(2 98 166 129)(3 99 167 130)(4 100 168 131)(5 81 169 132)(6 82 170 133)(7 83 171 134)(8 84 172 135)(9 85 173 136)(10 86 174 137)(11 87 175 138)(12 88 176 139)(13 89 177 140)(14 90 178 121)(15 91 179 122)(16 92 180 123)(17 93 161 124)(18 94 162 125)(19 95 163 126)(20 96 164 127)(21 275 56 213)(22 276 57 214)(23 277 58 215)(24 278 59 216)(25 279 60 217)(26 280 41 218)(27 261 42 219)(28 262 43 220)(29 263 44 201)(30 264 45 202)(31 265 46 203)(32 266 47 204)(33 267 48 205)(34 268 49 206)(35 269 50 207)(36 270 51 208)(37 271 52 209)(38 272 53 210)(39 273 54 211)(40 274 55 212)(61 240 195 107)(62 221 196 108)(63 222 197 109)(64 223 198 110)(65 224 199 111)(66 225 200 112)(67 226 181 113)(68 227 182 114)(69 228 183 115)(70 229 184 116)(71 230 185 117)(72 231 186 118)(73 232 187 119)(74 233 188 120)(75 234 189 101)(76 235 190 102)(77 236 191 103)(78 237 192 104)(79 238 193 105)(80 239 194 106)(141 250 305 286)(142 251 306 287)(143 252 307 288)(144 253 308 289)(145 254 309 290)(146 255 310 291)(147 256 311 292)(148 257 312 293)(149 258 313 294)(150 259 314 295)(151 260 315 296)(152 241 316 297)(153 242 317 298)(154 243 318 299)(155 244 319 300)(156 245 320 281)(157 246 301 282)(158 247 302 283)(159 248 303 284)(160 249 304 285)
(1 297 165 241)(2 288 166 252)(3 299 167 243)(4 290 168 254)(5 281 169 245)(6 292 170 256)(7 283 171 247)(8 294 172 258)(9 285 173 249)(10 296 174 260)(11 287 175 251)(12 298 176 242)(13 289 177 253)(14 300 178 244)(15 291 179 255)(16 282 180 246)(17 293 161 257)(18 284 162 248)(19 295 163 259)(20 286 164 250)(21 63 56 197)(22 74 57 188)(23 65 58 199)(24 76 59 190)(25 67 60 181)(26 78 41 192)(27 69 42 183)(28 80 43 194)(29 71 44 185)(30 62 45 196)(31 73 46 187)(32 64 47 198)(33 75 48 189)(34 66 49 200)(35 77 50 191)(36 68 51 182)(37 79 52 193)(38 70 53 184)(39 61 54 195)(40 72 55 186)(81 320 132 156)(82 311 133 147)(83 302 134 158)(84 313 135 149)(85 304 136 160)(86 315 137 151)(87 306 138 142)(88 317 139 153)(89 308 140 144)(90 319 121 155)(91 310 122 146)(92 301 123 157)(93 312 124 148)(94 303 125 159)(95 314 126 150)(96 305 127 141)(97 316 128 152)(98 307 129 143)(99 318 130 154)(100 309 131 145)(101 205 234 267)(102 216 235 278)(103 207 236 269)(104 218 237 280)(105 209 238 271)(106 220 239 262)(107 211 240 273)(108 202 221 264)(109 213 222 275)(110 204 223 266)(111 215 224 277)(112 206 225 268)(113 217 226 279)(114 208 227 270)(115 219 228 261)(116 210 229 272)(117 201 230 263)(118 212 231 274)(119 203 232 265)(120 214 233 276)
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), (1,263,11,273)(2,262,12,272)(3,261,13,271)(4,280,14,270)(5,279,15,269)(6,278,16,268)(7,277,17,267)(8,276,18,266)(9,275,19,265)(10,274,20,264)(21,126,31,136)(22,125,32,135)(23,124,33,134)(24,123,34,133)(25,122,35,132)(26,121,36,131)(27,140,37,130)(28,139,38,129)(29,138,39,128)(30,137,40,127)(41,90,51,100)(42,89,52,99)(43,88,53,98)(44,87,54,97)(45,86,55,96)(46,85,56,95)(47,84,57,94)(48,83,58,93)(49,82,59,92)(50,81,60,91)(61,152,71,142)(62,151,72,141)(63,150,73,160)(64,149,74,159)(65,148,75,158)(66,147,76,157)(67,146,77,156)(68,145,78,155)(69,144,79,154)(70,143,80,153)(101,283,111,293)(102,282,112,292)(103,281,113,291)(104,300,114,290)(105,299,115,289)(106,298,116,288)(107,297,117,287)(108,296,118,286)(109,295,119,285)(110,294,120,284)(161,205,171,215)(162,204,172,214)(163,203,173,213)(164,202,174,212)(165,201,175,211)(166,220,176,210)(167,219,177,209)(168,218,178,208)(169,217,179,207)(170,216,180,206)(181,310,191,320)(182,309,192,319)(183,308,193,318)(184,307,194,317)(185,306,195,316)(186,305,196,315)(187,304,197,314)(188,303,198,313)(189,302,199,312)(190,301,200,311)(221,260,231,250)(222,259,232,249)(223,258,233,248)(224,257,234,247)(225,256,235,246)(226,255,236,245)(227,254,237,244)(228,253,238,243)(229,252,239,242)(230,251,240,241), (1,97,165,128)(2,98,166,129)(3,99,167,130)(4,100,168,131)(5,81,169,132)(6,82,170,133)(7,83,171,134)(8,84,172,135)(9,85,173,136)(10,86,174,137)(11,87,175,138)(12,88,176,139)(13,89,177,140)(14,90,178,121)(15,91,179,122)(16,92,180,123)(17,93,161,124)(18,94,162,125)(19,95,163,126)(20,96,164,127)(21,275,56,213)(22,276,57,214)(23,277,58,215)(24,278,59,216)(25,279,60,217)(26,280,41,218)(27,261,42,219)(28,262,43,220)(29,263,44,201)(30,264,45,202)(31,265,46,203)(32,266,47,204)(33,267,48,205)(34,268,49,206)(35,269,50,207)(36,270,51,208)(37,271,52,209)(38,272,53,210)(39,273,54,211)(40,274,55,212)(61,240,195,107)(62,221,196,108)(63,222,197,109)(64,223,198,110)(65,224,199,111)(66,225,200,112)(67,226,181,113)(68,227,182,114)(69,228,183,115)(70,229,184,116)(71,230,185,117)(72,231,186,118)(73,232,187,119)(74,233,188,120)(75,234,189,101)(76,235,190,102)(77,236,191,103)(78,237,192,104)(79,238,193,105)(80,239,194,106)(141,250,305,286)(142,251,306,287)(143,252,307,288)(144,253,308,289)(145,254,309,290)(146,255,310,291)(147,256,311,292)(148,257,312,293)(149,258,313,294)(150,259,314,295)(151,260,315,296)(152,241,316,297)(153,242,317,298)(154,243,318,299)(155,244,319,300)(156,245,320,281)(157,246,301,282)(158,247,302,283)(159,248,303,284)(160,249,304,285), (1,297,165,241)(2,288,166,252)(3,299,167,243)(4,290,168,254)(5,281,169,245)(6,292,170,256)(7,283,171,247)(8,294,172,258)(9,285,173,249)(10,296,174,260)(11,287,175,251)(12,298,176,242)(13,289,177,253)(14,300,178,244)(15,291,179,255)(16,282,180,246)(17,293,161,257)(18,284,162,248)(19,295,163,259)(20,286,164,250)(21,63,56,197)(22,74,57,188)(23,65,58,199)(24,76,59,190)(25,67,60,181)(26,78,41,192)(27,69,42,183)(28,80,43,194)(29,71,44,185)(30,62,45,196)(31,73,46,187)(32,64,47,198)(33,75,48,189)(34,66,49,200)(35,77,50,191)(36,68,51,182)(37,79,52,193)(38,70,53,184)(39,61,54,195)(40,72,55,186)(81,320,132,156)(82,311,133,147)(83,302,134,158)(84,313,135,149)(85,304,136,160)(86,315,137,151)(87,306,138,142)(88,317,139,153)(89,308,140,144)(90,319,121,155)(91,310,122,146)(92,301,123,157)(93,312,124,148)(94,303,125,159)(95,314,126,150)(96,305,127,141)(97,316,128,152)(98,307,129,143)(99,318,130,154)(100,309,131,145)(101,205,234,267)(102,216,235,278)(103,207,236,269)(104,218,237,280)(105,209,238,271)(106,220,239,262)(107,211,240,273)(108,202,221,264)(109,213,222,275)(110,204,223,266)(111,215,224,277)(112,206,225,268)(113,217,226,279)(114,208,227,270)(115,219,228,261)(116,210,229,272)(117,201,230,263)(118,212,231,274)(119,203,232,265)(120,214,233,276)>;
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), (1,263,11,273)(2,262,12,272)(3,261,13,271)(4,280,14,270)(5,279,15,269)(6,278,16,268)(7,277,17,267)(8,276,18,266)(9,275,19,265)(10,274,20,264)(21,126,31,136)(22,125,32,135)(23,124,33,134)(24,123,34,133)(25,122,35,132)(26,121,36,131)(27,140,37,130)(28,139,38,129)(29,138,39,128)(30,137,40,127)(41,90,51,100)(42,89,52,99)(43,88,53,98)(44,87,54,97)(45,86,55,96)(46,85,56,95)(47,84,57,94)(48,83,58,93)(49,82,59,92)(50,81,60,91)(61,152,71,142)(62,151,72,141)(63,150,73,160)(64,149,74,159)(65,148,75,158)(66,147,76,157)(67,146,77,156)(68,145,78,155)(69,144,79,154)(70,143,80,153)(101,283,111,293)(102,282,112,292)(103,281,113,291)(104,300,114,290)(105,299,115,289)(106,298,116,288)(107,297,117,287)(108,296,118,286)(109,295,119,285)(110,294,120,284)(161,205,171,215)(162,204,172,214)(163,203,173,213)(164,202,174,212)(165,201,175,211)(166,220,176,210)(167,219,177,209)(168,218,178,208)(169,217,179,207)(170,216,180,206)(181,310,191,320)(182,309,192,319)(183,308,193,318)(184,307,194,317)(185,306,195,316)(186,305,196,315)(187,304,197,314)(188,303,198,313)(189,302,199,312)(190,301,200,311)(221,260,231,250)(222,259,232,249)(223,258,233,248)(224,257,234,247)(225,256,235,246)(226,255,236,245)(227,254,237,244)(228,253,238,243)(229,252,239,242)(230,251,240,241), (1,97,165,128)(2,98,166,129)(3,99,167,130)(4,100,168,131)(5,81,169,132)(6,82,170,133)(7,83,171,134)(8,84,172,135)(9,85,173,136)(10,86,174,137)(11,87,175,138)(12,88,176,139)(13,89,177,140)(14,90,178,121)(15,91,179,122)(16,92,180,123)(17,93,161,124)(18,94,162,125)(19,95,163,126)(20,96,164,127)(21,275,56,213)(22,276,57,214)(23,277,58,215)(24,278,59,216)(25,279,60,217)(26,280,41,218)(27,261,42,219)(28,262,43,220)(29,263,44,201)(30,264,45,202)(31,265,46,203)(32,266,47,204)(33,267,48,205)(34,268,49,206)(35,269,50,207)(36,270,51,208)(37,271,52,209)(38,272,53,210)(39,273,54,211)(40,274,55,212)(61,240,195,107)(62,221,196,108)(63,222,197,109)(64,223,198,110)(65,224,199,111)(66,225,200,112)(67,226,181,113)(68,227,182,114)(69,228,183,115)(70,229,184,116)(71,230,185,117)(72,231,186,118)(73,232,187,119)(74,233,188,120)(75,234,189,101)(76,235,190,102)(77,236,191,103)(78,237,192,104)(79,238,193,105)(80,239,194,106)(141,250,305,286)(142,251,306,287)(143,252,307,288)(144,253,308,289)(145,254,309,290)(146,255,310,291)(147,256,311,292)(148,257,312,293)(149,258,313,294)(150,259,314,295)(151,260,315,296)(152,241,316,297)(153,242,317,298)(154,243,318,299)(155,244,319,300)(156,245,320,281)(157,246,301,282)(158,247,302,283)(159,248,303,284)(160,249,304,285), (1,297,165,241)(2,288,166,252)(3,299,167,243)(4,290,168,254)(5,281,169,245)(6,292,170,256)(7,283,171,247)(8,294,172,258)(9,285,173,249)(10,296,174,260)(11,287,175,251)(12,298,176,242)(13,289,177,253)(14,300,178,244)(15,291,179,255)(16,282,180,246)(17,293,161,257)(18,284,162,248)(19,295,163,259)(20,286,164,250)(21,63,56,197)(22,74,57,188)(23,65,58,199)(24,76,59,190)(25,67,60,181)(26,78,41,192)(27,69,42,183)(28,80,43,194)(29,71,44,185)(30,62,45,196)(31,73,46,187)(32,64,47,198)(33,75,48,189)(34,66,49,200)(35,77,50,191)(36,68,51,182)(37,79,52,193)(38,70,53,184)(39,61,54,195)(40,72,55,186)(81,320,132,156)(82,311,133,147)(83,302,134,158)(84,313,135,149)(85,304,136,160)(86,315,137,151)(87,306,138,142)(88,317,139,153)(89,308,140,144)(90,319,121,155)(91,310,122,146)(92,301,123,157)(93,312,124,148)(94,303,125,159)(95,314,126,150)(96,305,127,141)(97,316,128,152)(98,307,129,143)(99,318,130,154)(100,309,131,145)(101,205,234,267)(102,216,235,278)(103,207,236,269)(104,218,237,280)(105,209,238,271)(106,220,239,262)(107,211,240,273)(108,202,221,264)(109,213,222,275)(110,204,223,266)(111,215,224,277)(112,206,225,268)(113,217,226,279)(114,208,227,270)(115,219,228,261)(116,210,229,272)(117,201,230,263)(118,212,231,274)(119,203,232,265)(120,214,233,276) );
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)], [(1,263,11,273),(2,262,12,272),(3,261,13,271),(4,280,14,270),(5,279,15,269),(6,278,16,268),(7,277,17,267),(8,276,18,266),(9,275,19,265),(10,274,20,264),(21,126,31,136),(22,125,32,135),(23,124,33,134),(24,123,34,133),(25,122,35,132),(26,121,36,131),(27,140,37,130),(28,139,38,129),(29,138,39,128),(30,137,40,127),(41,90,51,100),(42,89,52,99),(43,88,53,98),(44,87,54,97),(45,86,55,96),(46,85,56,95),(47,84,57,94),(48,83,58,93),(49,82,59,92),(50,81,60,91),(61,152,71,142),(62,151,72,141),(63,150,73,160),(64,149,74,159),(65,148,75,158),(66,147,76,157),(67,146,77,156),(68,145,78,155),(69,144,79,154),(70,143,80,153),(101,283,111,293),(102,282,112,292),(103,281,113,291),(104,300,114,290),(105,299,115,289),(106,298,116,288),(107,297,117,287),(108,296,118,286),(109,295,119,285),(110,294,120,284),(161,205,171,215),(162,204,172,214),(163,203,173,213),(164,202,174,212),(165,201,175,211),(166,220,176,210),(167,219,177,209),(168,218,178,208),(169,217,179,207),(170,216,180,206),(181,310,191,320),(182,309,192,319),(183,308,193,318),(184,307,194,317),(185,306,195,316),(186,305,196,315),(187,304,197,314),(188,303,198,313),(189,302,199,312),(190,301,200,311),(221,260,231,250),(222,259,232,249),(223,258,233,248),(224,257,234,247),(225,256,235,246),(226,255,236,245),(227,254,237,244),(228,253,238,243),(229,252,239,242),(230,251,240,241)], [(1,97,165,128),(2,98,166,129),(3,99,167,130),(4,100,168,131),(5,81,169,132),(6,82,170,133),(7,83,171,134),(8,84,172,135),(9,85,173,136),(10,86,174,137),(11,87,175,138),(12,88,176,139),(13,89,177,140),(14,90,178,121),(15,91,179,122),(16,92,180,123),(17,93,161,124),(18,94,162,125),(19,95,163,126),(20,96,164,127),(21,275,56,213),(22,276,57,214),(23,277,58,215),(24,278,59,216),(25,279,60,217),(26,280,41,218),(27,261,42,219),(28,262,43,220),(29,263,44,201),(30,264,45,202),(31,265,46,203),(32,266,47,204),(33,267,48,205),(34,268,49,206),(35,269,50,207),(36,270,51,208),(37,271,52,209),(38,272,53,210),(39,273,54,211),(40,274,55,212),(61,240,195,107),(62,221,196,108),(63,222,197,109),(64,223,198,110),(65,224,199,111),(66,225,200,112),(67,226,181,113),(68,227,182,114),(69,228,183,115),(70,229,184,116),(71,230,185,117),(72,231,186,118),(73,232,187,119),(74,233,188,120),(75,234,189,101),(76,235,190,102),(77,236,191,103),(78,237,192,104),(79,238,193,105),(80,239,194,106),(141,250,305,286),(142,251,306,287),(143,252,307,288),(144,253,308,289),(145,254,309,290),(146,255,310,291),(147,256,311,292),(148,257,312,293),(149,258,313,294),(150,259,314,295),(151,260,315,296),(152,241,316,297),(153,242,317,298),(154,243,318,299),(155,244,319,300),(156,245,320,281),(157,246,301,282),(158,247,302,283),(159,248,303,284),(160,249,304,285)], [(1,297,165,241),(2,288,166,252),(3,299,167,243),(4,290,168,254),(5,281,169,245),(6,292,170,256),(7,283,171,247),(8,294,172,258),(9,285,173,249),(10,296,174,260),(11,287,175,251),(12,298,176,242),(13,289,177,253),(14,300,178,244),(15,291,179,255),(16,282,180,246),(17,293,161,257),(18,284,162,248),(19,295,163,259),(20,286,164,250),(21,63,56,197),(22,74,57,188),(23,65,58,199),(24,76,59,190),(25,67,60,181),(26,78,41,192),(27,69,42,183),(28,80,43,194),(29,71,44,185),(30,62,45,196),(31,73,46,187),(32,64,47,198),(33,75,48,189),(34,66,49,200),(35,77,50,191),(36,68,51,182),(37,79,52,193),(38,70,53,184),(39,61,54,195),(40,72,55,186),(81,320,132,156),(82,311,133,147),(83,302,134,158),(84,313,135,149),(85,304,136,160),(86,315,137,151),(87,306,138,142),(88,317,139,153),(89,308,140,144),(90,319,121,155),(91,310,122,146),(92,301,123,157),(93,312,124,148),(94,303,125,159),(95,314,126,150),(96,305,127,141),(97,316,128,152),(98,307,129,143),(99,318,130,154),(100,309,131,145),(101,205,234,267),(102,216,235,278),(103,207,236,269),(104,218,237,280),(105,209,238,271),(106,220,239,262),(107,211,240,273),(108,202,221,264),(109,213,222,275),(110,204,223,266),(111,215,224,277),(112,206,225,268),(113,217,226,279),(114,208,227,270),(115,219,228,261),(116,210,229,272),(117,201,230,263),(118,212,231,274),(119,203,232,265),(120,214,233,276)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | 2- 1+4 | D4⋊2D5 | Q8×D5 | Q8.10D10 |
| kernel | Dic10⋊8Q8 | C4×Dic10 | Dic5⋊3Q8 | Dic5.Q8 | C4.Dic10 | Dic5⋊Q8 | Q8×Dic5 | C5×C4⋊Q8 | Dic10 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊8Q8 ►in GL6(𝔽41)
| 10 | 4 | 0 | 0 | 0 | 0 |
| 26 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 24 | 15 | 0 | 0 | 0 | 0 |
| 8 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 39 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 36 |
| 0 | 0 | 0 | 0 | 15 | 19 |
| 11 | 12 | 0 | 0 | 0 | 0 |
| 31 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 7 |
| 0 | 0 | 0 | 0 | 23 | 24 |
G:=sub<GL(6,GF(41))| [10,26,0,0,0,0,4,31,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,8,0,0,0,0,15,17,0,0,0,0,0,0,21,15,0,0,0,0,39,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,15,0,0,0,0,36,19],[11,31,0,0,0,0,12,30,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,23,0,0,0,0,7,24] >;
Dic10⋊8Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_8Q_8 % in TeX
G:=Group("Dic10:8Q8"); // GroupNames label
G:=SmallGroup(320,1393);
// by ID
G=gap.SmallGroup(320,1393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,297,136,12550]);
// 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,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations