metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊5Q8, C20.10Q16, C42.84D10, C4⋊Q8.10D5, C4.13(Q8×D5), C4⋊C4.86D10, C5⋊5(C4.Q16), C20.40(C2×Q8), (C2×C20).160D4, C10.43(C2×Q16), C20⋊3C8.22C2, C4.8(C5⋊Q16), C20.86(C4○D4), (C2×C20).409C23, (C4×C20).138C22, C4.36(Q8⋊2D5), (C4×Dic10).18C2, C10.D8.17C2, C10.Q16.14C2, C10.77(C22⋊Q8), C10.100(C8⋊C22), C2.14(D10⋊3Q8), C4⋊Dic5.350C22, C2.21(D4.D10), (C2×Dic10).284C22, (C5×C4⋊Q8).10C2, C2.14(C2×C5⋊Q16), (C2×C10).540(C2×D4), (C2×C4).191(C5⋊D4), (C5×C4⋊C4).133C22, (C2×C4).506(C22×D5), C22.212(C2×C5⋊D4), (C2×C5⋊2C8).141C22, SmallGroup(320,718)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊5Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 294 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4.Q16, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C20⋊3C8, C10.D8, C10.Q16, C4×Dic10, C5×C4⋊Q8, Dic10⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, Q16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×Q16, C8⋊C22, C5⋊D4, C22×D5, C4.Q16, C5⋊Q16, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D4.D10, C2×C5⋊Q16, D10⋊3Q8, Dic10⋊5Q8
►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)
(1 233 11 223)(2 232 12 222)(3 231 13 221)(4 230 14 240)(5 229 15 239)(6 228 16 238)(7 227 17 237)(8 226 18 236)(9 225 19 235)(10 224 20 234)(21 312 31 302)(22 311 32 301)(23 310 33 320)(24 309 34 319)(25 308 35 318)(26 307 36 317)(27 306 37 316)(28 305 38 315)(29 304 39 314)(30 303 40 313)(41 80 51 70)(42 79 52 69)(43 78 53 68)(44 77 54 67)(45 76 55 66)(46 75 56 65)(47 74 57 64)(48 73 58 63)(49 72 59 62)(50 71 60 61)(81 220 91 210)(82 219 92 209)(83 218 93 208)(84 217 94 207)(85 216 95 206)(86 215 96 205)(87 214 97 204)(88 213 98 203)(89 212 99 202)(90 211 100 201)(101 148 111 158)(102 147 112 157)(103 146 113 156)(104 145 114 155)(105 144 115 154)(106 143 116 153)(107 142 117 152)(108 141 118 151)(109 160 119 150)(110 159 120 149)(121 262 131 272)(122 261 132 271)(123 280 133 270)(124 279 134 269)(125 278 135 268)(126 277 136 267)(127 276 137 266)(128 275 138 265)(129 274 139 264)(130 273 140 263)(161 184 171 194)(162 183 172 193)(163 182 173 192)(164 181 174 191)(165 200 175 190)(166 199 176 189)(167 198 177 188)(168 197 178 187)(169 196 179 186)(170 195 180 185)(241 282 251 292)(242 281 252 291)(243 300 253 290)(244 299 254 289)(245 298 255 288)(246 297 256 287)(247 296 257 286)(248 295 258 285)(249 294 259 284)(250 293 260 283)
(1 189 308 203)(2 200 309 214)(3 191 310 205)(4 182 311 216)(5 193 312 207)(6 184 313 218)(7 195 314 209)(8 186 315 220)(9 197 316 211)(10 188 317 202)(11 199 318 213)(12 190 319 204)(13 181 320 215)(14 192 301 206)(15 183 302 217)(16 194 303 208)(17 185 304 219)(18 196 305 210)(19 187 306 201)(20 198 307 212)(21 89 239 167)(22 100 240 178)(23 91 221 169)(24 82 222 180)(25 93 223 171)(26 84 224 162)(27 95 225 173)(28 86 226 164)(29 97 227 175)(30 88 228 166)(31 99 229 177)(32 90 230 168)(33 81 231 179)(34 92 232 170)(35 83 233 161)(36 94 234 172)(37 85 235 163)(38 96 236 174)(39 87 237 165)(40 98 238 176)(41 129 144 287)(42 140 145 298)(43 131 146 289)(44 122 147 300)(45 133 148 291)(46 124 149 282)(47 135 150 293)(48 126 151 284)(49 137 152 295)(50 128 153 286)(51 139 154 297)(52 130 155 288)(53 121 156 299)(54 132 157 290)(55 123 158 281)(56 134 159 292)(57 125 160 283)(58 136 141 294)(59 127 142 285)(60 138 143 296)(61 280 116 252)(62 271 117 243)(63 262 118 254)(64 273 119 245)(65 264 120 256)(66 275 101 247)(67 266 102 258)(68 277 103 249)(69 268 104 260)(70 279 105 251)(71 270 106 242)(72 261 107 253)(73 272 108 244)(74 263 109 255)(75 274 110 246)(76 265 111 257)(77 276 112 248)(78 267 113 259)(79 278 114 250)(80 269 115 241)
(1 292 308 134)(2 293 309 135)(3 294 310 136)(4 295 311 137)(5 296 312 138)(6 297 313 139)(7 298 314 140)(8 299 315 121)(9 300 316 122)(10 281 317 123)(11 282 318 124)(12 283 319 125)(13 284 320 126)(14 285 301 127)(15 286 302 128)(16 287 303 129)(17 288 304 130)(18 289 305 131)(19 290 306 132)(20 291 307 133)(21 265 239 257)(22 266 240 258)(23 267 221 259)(24 268 222 260)(25 269 223 241)(26 270 224 242)(27 271 225 243)(28 272 226 244)(29 273 227 245)(30 274 228 246)(31 275 229 247)(32 276 230 248)(33 277 231 249)(34 278 232 250)(35 279 233 251)(36 280 234 252)(37 261 235 253)(38 262 236 254)(39 263 237 255)(40 264 238 256)(41 194 144 208)(42 195 145 209)(43 196 146 210)(44 197 147 211)(45 198 148 212)(46 199 149 213)(47 200 150 214)(48 181 151 215)(49 182 152 216)(50 183 153 217)(51 184 154 218)(52 185 155 219)(53 186 156 220)(54 187 157 201)(55 188 158 202)(56 189 159 203)(57 190 160 204)(58 191 141 205)(59 192 142 206)(60 193 143 207)(61 172 116 94)(62 173 117 95)(63 174 118 96)(64 175 119 97)(65 176 120 98)(66 177 101 99)(67 178 102 100)(68 179 103 81)(69 180 104 82)(70 161 105 83)(71 162 106 84)(72 163 107 85)(73 164 108 86)(74 165 109 87)(75 166 110 88)(76 167 111 89)(77 168 112 90)(78 169 113 91)(79 170 114 92)(80 171 115 93)
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), (1,233,11,223)(2,232,12,222)(3,231,13,221)(4,230,14,240)(5,229,15,239)(6,228,16,238)(7,227,17,237)(8,226,18,236)(9,225,19,235)(10,224,20,234)(21,312,31,302)(22,311,32,301)(23,310,33,320)(24,309,34,319)(25,308,35,318)(26,307,36,317)(27,306,37,316)(28,305,38,315)(29,304,39,314)(30,303,40,313)(41,80,51,70)(42,79,52,69)(43,78,53,68)(44,77,54,67)(45,76,55,66)(46,75,56,65)(47,74,57,64)(48,73,58,63)(49,72,59,62)(50,71,60,61)(81,220,91,210)(82,219,92,209)(83,218,93,208)(84,217,94,207)(85,216,95,206)(86,215,96,205)(87,214,97,204)(88,213,98,203)(89,212,99,202)(90,211,100,201)(101,148,111,158)(102,147,112,157)(103,146,113,156)(104,145,114,155)(105,144,115,154)(106,143,116,153)(107,142,117,152)(108,141,118,151)(109,160,119,150)(110,159,120,149)(121,262,131,272)(122,261,132,271)(123,280,133,270)(124,279,134,269)(125,278,135,268)(126,277,136,267)(127,276,137,266)(128,275,138,265)(129,274,139,264)(130,273,140,263)(161,184,171,194)(162,183,172,193)(163,182,173,192)(164,181,174,191)(165,200,175,190)(166,199,176,189)(167,198,177,188)(168,197,178,187)(169,196,179,186)(170,195,180,185)(241,282,251,292)(242,281,252,291)(243,300,253,290)(244,299,254,289)(245,298,255,288)(246,297,256,287)(247,296,257,286)(248,295,258,285)(249,294,259,284)(250,293,260,283), (1,189,308,203)(2,200,309,214)(3,191,310,205)(4,182,311,216)(5,193,312,207)(6,184,313,218)(7,195,314,209)(8,186,315,220)(9,197,316,211)(10,188,317,202)(11,199,318,213)(12,190,319,204)(13,181,320,215)(14,192,301,206)(15,183,302,217)(16,194,303,208)(17,185,304,219)(18,196,305,210)(19,187,306,201)(20,198,307,212)(21,89,239,167)(22,100,240,178)(23,91,221,169)(24,82,222,180)(25,93,223,171)(26,84,224,162)(27,95,225,173)(28,86,226,164)(29,97,227,175)(30,88,228,166)(31,99,229,177)(32,90,230,168)(33,81,231,179)(34,92,232,170)(35,83,233,161)(36,94,234,172)(37,85,235,163)(38,96,236,174)(39,87,237,165)(40,98,238,176)(41,129,144,287)(42,140,145,298)(43,131,146,289)(44,122,147,300)(45,133,148,291)(46,124,149,282)(47,135,150,293)(48,126,151,284)(49,137,152,295)(50,128,153,286)(51,139,154,297)(52,130,155,288)(53,121,156,299)(54,132,157,290)(55,123,158,281)(56,134,159,292)(57,125,160,283)(58,136,141,294)(59,127,142,285)(60,138,143,296)(61,280,116,252)(62,271,117,243)(63,262,118,254)(64,273,119,245)(65,264,120,256)(66,275,101,247)(67,266,102,258)(68,277,103,249)(69,268,104,260)(70,279,105,251)(71,270,106,242)(72,261,107,253)(73,272,108,244)(74,263,109,255)(75,274,110,246)(76,265,111,257)(77,276,112,248)(78,267,113,259)(79,278,114,250)(80,269,115,241), (1,292,308,134)(2,293,309,135)(3,294,310,136)(4,295,311,137)(5,296,312,138)(6,297,313,139)(7,298,314,140)(8,299,315,121)(9,300,316,122)(10,281,317,123)(11,282,318,124)(12,283,319,125)(13,284,320,126)(14,285,301,127)(15,286,302,128)(16,287,303,129)(17,288,304,130)(18,289,305,131)(19,290,306,132)(20,291,307,133)(21,265,239,257)(22,266,240,258)(23,267,221,259)(24,268,222,260)(25,269,223,241)(26,270,224,242)(27,271,225,243)(28,272,226,244)(29,273,227,245)(30,274,228,246)(31,275,229,247)(32,276,230,248)(33,277,231,249)(34,278,232,250)(35,279,233,251)(36,280,234,252)(37,261,235,253)(38,262,236,254)(39,263,237,255)(40,264,238,256)(41,194,144,208)(42,195,145,209)(43,196,146,210)(44,197,147,211)(45,198,148,212)(46,199,149,213)(47,200,150,214)(48,181,151,215)(49,182,152,216)(50,183,153,217)(51,184,154,218)(52,185,155,219)(53,186,156,220)(54,187,157,201)(55,188,158,202)(56,189,159,203)(57,190,160,204)(58,191,141,205)(59,192,142,206)(60,193,143,207)(61,172,116,94)(62,173,117,95)(63,174,118,96)(64,175,119,97)(65,176,120,98)(66,177,101,99)(67,178,102,100)(68,179,103,81)(69,180,104,82)(70,161,105,83)(71,162,106,84)(72,163,107,85)(73,164,108,86)(74,165,109,87)(75,166,110,88)(76,167,111,89)(77,168,112,90)(78,169,113,91)(79,170,114,92)(80,171,115,93)>;
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), (1,233,11,223)(2,232,12,222)(3,231,13,221)(4,230,14,240)(5,229,15,239)(6,228,16,238)(7,227,17,237)(8,226,18,236)(9,225,19,235)(10,224,20,234)(21,312,31,302)(22,311,32,301)(23,310,33,320)(24,309,34,319)(25,308,35,318)(26,307,36,317)(27,306,37,316)(28,305,38,315)(29,304,39,314)(30,303,40,313)(41,80,51,70)(42,79,52,69)(43,78,53,68)(44,77,54,67)(45,76,55,66)(46,75,56,65)(47,74,57,64)(48,73,58,63)(49,72,59,62)(50,71,60,61)(81,220,91,210)(82,219,92,209)(83,218,93,208)(84,217,94,207)(85,216,95,206)(86,215,96,205)(87,214,97,204)(88,213,98,203)(89,212,99,202)(90,211,100,201)(101,148,111,158)(102,147,112,157)(103,146,113,156)(104,145,114,155)(105,144,115,154)(106,143,116,153)(107,142,117,152)(108,141,118,151)(109,160,119,150)(110,159,120,149)(121,262,131,272)(122,261,132,271)(123,280,133,270)(124,279,134,269)(125,278,135,268)(126,277,136,267)(127,276,137,266)(128,275,138,265)(129,274,139,264)(130,273,140,263)(161,184,171,194)(162,183,172,193)(163,182,173,192)(164,181,174,191)(165,200,175,190)(166,199,176,189)(167,198,177,188)(168,197,178,187)(169,196,179,186)(170,195,180,185)(241,282,251,292)(242,281,252,291)(243,300,253,290)(244,299,254,289)(245,298,255,288)(246,297,256,287)(247,296,257,286)(248,295,258,285)(249,294,259,284)(250,293,260,283), (1,189,308,203)(2,200,309,214)(3,191,310,205)(4,182,311,216)(5,193,312,207)(6,184,313,218)(7,195,314,209)(8,186,315,220)(9,197,316,211)(10,188,317,202)(11,199,318,213)(12,190,319,204)(13,181,320,215)(14,192,301,206)(15,183,302,217)(16,194,303,208)(17,185,304,219)(18,196,305,210)(19,187,306,201)(20,198,307,212)(21,89,239,167)(22,100,240,178)(23,91,221,169)(24,82,222,180)(25,93,223,171)(26,84,224,162)(27,95,225,173)(28,86,226,164)(29,97,227,175)(30,88,228,166)(31,99,229,177)(32,90,230,168)(33,81,231,179)(34,92,232,170)(35,83,233,161)(36,94,234,172)(37,85,235,163)(38,96,236,174)(39,87,237,165)(40,98,238,176)(41,129,144,287)(42,140,145,298)(43,131,146,289)(44,122,147,300)(45,133,148,291)(46,124,149,282)(47,135,150,293)(48,126,151,284)(49,137,152,295)(50,128,153,286)(51,139,154,297)(52,130,155,288)(53,121,156,299)(54,132,157,290)(55,123,158,281)(56,134,159,292)(57,125,160,283)(58,136,141,294)(59,127,142,285)(60,138,143,296)(61,280,116,252)(62,271,117,243)(63,262,118,254)(64,273,119,245)(65,264,120,256)(66,275,101,247)(67,266,102,258)(68,277,103,249)(69,268,104,260)(70,279,105,251)(71,270,106,242)(72,261,107,253)(73,272,108,244)(74,263,109,255)(75,274,110,246)(76,265,111,257)(77,276,112,248)(78,267,113,259)(79,278,114,250)(80,269,115,241), (1,292,308,134)(2,293,309,135)(3,294,310,136)(4,295,311,137)(5,296,312,138)(6,297,313,139)(7,298,314,140)(8,299,315,121)(9,300,316,122)(10,281,317,123)(11,282,318,124)(12,283,319,125)(13,284,320,126)(14,285,301,127)(15,286,302,128)(16,287,303,129)(17,288,304,130)(18,289,305,131)(19,290,306,132)(20,291,307,133)(21,265,239,257)(22,266,240,258)(23,267,221,259)(24,268,222,260)(25,269,223,241)(26,270,224,242)(27,271,225,243)(28,272,226,244)(29,273,227,245)(30,274,228,246)(31,275,229,247)(32,276,230,248)(33,277,231,249)(34,278,232,250)(35,279,233,251)(36,280,234,252)(37,261,235,253)(38,262,236,254)(39,263,237,255)(40,264,238,256)(41,194,144,208)(42,195,145,209)(43,196,146,210)(44,197,147,211)(45,198,148,212)(46,199,149,213)(47,200,150,214)(48,181,151,215)(49,182,152,216)(50,183,153,217)(51,184,154,218)(52,185,155,219)(53,186,156,220)(54,187,157,201)(55,188,158,202)(56,189,159,203)(57,190,160,204)(58,191,141,205)(59,192,142,206)(60,193,143,207)(61,172,116,94)(62,173,117,95)(63,174,118,96)(64,175,119,97)(65,176,120,98)(66,177,101,99)(67,178,102,100)(68,179,103,81)(69,180,104,82)(70,161,105,83)(71,162,106,84)(72,163,107,85)(73,164,108,86)(74,165,109,87)(75,166,110,88)(76,167,111,89)(77,168,112,90)(78,169,113,91)(79,170,114,92)(80,171,115,93) );
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)], [(1,233,11,223),(2,232,12,222),(3,231,13,221),(4,230,14,240),(5,229,15,239),(6,228,16,238),(7,227,17,237),(8,226,18,236),(9,225,19,235),(10,224,20,234),(21,312,31,302),(22,311,32,301),(23,310,33,320),(24,309,34,319),(25,308,35,318),(26,307,36,317),(27,306,37,316),(28,305,38,315),(29,304,39,314),(30,303,40,313),(41,80,51,70),(42,79,52,69),(43,78,53,68),(44,77,54,67),(45,76,55,66),(46,75,56,65),(47,74,57,64),(48,73,58,63),(49,72,59,62),(50,71,60,61),(81,220,91,210),(82,219,92,209),(83,218,93,208),(84,217,94,207),(85,216,95,206),(86,215,96,205),(87,214,97,204),(88,213,98,203),(89,212,99,202),(90,211,100,201),(101,148,111,158),(102,147,112,157),(103,146,113,156),(104,145,114,155),(105,144,115,154),(106,143,116,153),(107,142,117,152),(108,141,118,151),(109,160,119,150),(110,159,120,149),(121,262,131,272),(122,261,132,271),(123,280,133,270),(124,279,134,269),(125,278,135,268),(126,277,136,267),(127,276,137,266),(128,275,138,265),(129,274,139,264),(130,273,140,263),(161,184,171,194),(162,183,172,193),(163,182,173,192),(164,181,174,191),(165,200,175,190),(166,199,176,189),(167,198,177,188),(168,197,178,187),(169,196,179,186),(170,195,180,185),(241,282,251,292),(242,281,252,291),(243,300,253,290),(244,299,254,289),(245,298,255,288),(246,297,256,287),(247,296,257,286),(248,295,258,285),(249,294,259,284),(250,293,260,283)], [(1,189,308,203),(2,200,309,214),(3,191,310,205),(4,182,311,216),(5,193,312,207),(6,184,313,218),(7,195,314,209),(8,186,315,220),(9,197,316,211),(10,188,317,202),(11,199,318,213),(12,190,319,204),(13,181,320,215),(14,192,301,206),(15,183,302,217),(16,194,303,208),(17,185,304,219),(18,196,305,210),(19,187,306,201),(20,198,307,212),(21,89,239,167),(22,100,240,178),(23,91,221,169),(24,82,222,180),(25,93,223,171),(26,84,224,162),(27,95,225,173),(28,86,226,164),(29,97,227,175),(30,88,228,166),(31,99,229,177),(32,90,230,168),(33,81,231,179),(34,92,232,170),(35,83,233,161),(36,94,234,172),(37,85,235,163),(38,96,236,174),(39,87,237,165),(40,98,238,176),(41,129,144,287),(42,140,145,298),(43,131,146,289),(44,122,147,300),(45,133,148,291),(46,124,149,282),(47,135,150,293),(48,126,151,284),(49,137,152,295),(50,128,153,286),(51,139,154,297),(52,130,155,288),(53,121,156,299),(54,132,157,290),(55,123,158,281),(56,134,159,292),(57,125,160,283),(58,136,141,294),(59,127,142,285),(60,138,143,296),(61,280,116,252),(62,271,117,243),(63,262,118,254),(64,273,119,245),(65,264,120,256),(66,275,101,247),(67,266,102,258),(68,277,103,249),(69,268,104,260),(70,279,105,251),(71,270,106,242),(72,261,107,253),(73,272,108,244),(74,263,109,255),(75,274,110,246),(76,265,111,257),(77,276,112,248),(78,267,113,259),(79,278,114,250),(80,269,115,241)], [(1,292,308,134),(2,293,309,135),(3,294,310,136),(4,295,311,137),(5,296,312,138),(6,297,313,139),(7,298,314,140),(8,299,315,121),(9,300,316,122),(10,281,317,123),(11,282,318,124),(12,283,319,125),(13,284,320,126),(14,285,301,127),(15,286,302,128),(16,287,303,129),(17,288,304,130),(18,289,305,131),(19,290,306,132),(20,291,307,133),(21,265,239,257),(22,266,240,258),(23,267,221,259),(24,268,222,260),(25,269,223,241),(26,270,224,242),(27,271,225,243),(28,272,226,244),(29,273,227,245),(30,274,228,246),(31,275,229,247),(32,276,230,248),(33,277,231,249),(34,278,232,250),(35,279,233,251),(36,280,234,252),(37,261,235,253),(38,262,236,254),(39,263,237,255),(40,264,238,256),(41,194,144,208),(42,195,145,209),(43,196,146,210),(44,197,147,211),(45,198,148,212),(46,199,149,213),(47,200,150,214),(48,181,151,215),(49,182,152,216),(50,183,153,217),(51,184,154,218),(52,185,155,219),(53,186,156,220),(54,187,157,201),(55,188,158,202),(56,189,159,203),(57,190,160,204),(58,191,141,205),(59,192,142,206),(60,193,143,207),(61,172,116,94),(62,173,117,95),(63,174,118,96),(64,175,119,97),(65,176,120,98),(66,177,101,99),(67,178,102,100),(68,179,103,81),(69,180,104,82),(70,161,105,83),(71,162,106,84),(72,163,107,85),(73,164,108,86),(74,165,109,87),(75,166,110,88),(76,167,111,89),(77,168,112,90),(78,169,113,91),(79,170,114,92),(80,171,115,93)]])
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 | ··· | 20L | 20M | ··· | 20T |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 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 | Q8 | D4 | D5 | Q16 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | C5⋊Q16 | Q8×D5 | Q8⋊2D5 | D4.D10 |
| kernel | Dic10⋊5Q8 | C20⋊3C8 | C10.D8 | C10.Q16 | C4×Dic10 | C5×C4⋊Q8 | Dic10 | C2×C20 | C4⋊Q8 | C20 | C20 | C42 | C4⋊C4 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of Dic10⋊5Q8 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 38 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 2 | 0 | 0 |
| 0 | 0 | 11 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 |
| 0 | 0 | 0 | 0 | 17 | 16 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 3 |
| 0 | 0 | 0 | 0 | 34 | 26 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,6,1,0,0,0,0,0,0,17,38,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,11,0,0,0,0,2,31,0,0,0,0,0,0,25,17,0,0,0,0,9,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,15,34,0,0,0,0,3,26] >;
Dic10⋊5Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_5Q_8 % in TeX
G:=Group("Dic10:5Q8"); // GroupNames label
G:=SmallGroup(320,718);
// by ID
G=gap.SmallGroup(320,718);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,268,1123,297,136,12550]);
// 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^11,a*d=d*a,c*b*c^-1=a^5*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations