metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊2Q8, Dic5.10Q16, C4.5(Q8×D5), C20⋊Q8.9C2, C5⋊3(C4.Q16), C2.D8.5D5, (C2×C8).25D10, C2.13(D5×Q16), C20.18(C2×Q8), C4⋊C4.168D10, C10.22(C2×Q16), C4.78(C4○D20), C10.Q16.8C2, C10.D8.8C2, C22.224(D4×D5), C20.170(C4○D4), C2.20(D8⋊D5), C10.38(C8⋊C22), (C2×C40).239C22, (C2×C20).291C23, Dic5⋊3Q8.8C2, (C2×Dic5).222D4, C10.39(C22⋊Q8), C20.8Q8.10C2, C2.16(D10⋊Q8), C20.44D4.10C2, C4⋊Dic5.117C22, (C4×Dic5).40C22, (C2×Dic10).91C22, (C5×C2.D8).11C2, (C2×C10).296(C2×D4), (C5×C4⋊C4).84C22, (C2×C5⋊2C8).65C22, (C2×C4).394(C22×D5), SmallGroup(320,502)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊2Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, dad-1=a9, cbc-1=a5b, dbd-1=a10b, 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, C2.D8, C2.D8, C4×Q8, C4⋊Q8, C5⋊2C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4.Q16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C10.D8, C10.Q16, C20.8Q8, C20.44D4, C5×C2.D8, Dic5⋊3Q8, C20⋊Q8, Dic10⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, Q16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×Q16, C8⋊C22, C22×D5, C4.Q16, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D8⋊D5, D5×Q16, Dic10⋊2Q8
►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)
(1 298 11 288)(2 297 12 287)(3 296 13 286)(4 295 14 285)(5 294 15 284)(6 293 16 283)(7 292 17 282)(8 291 18 281)(9 290 19 300)(10 289 20 299)(21 253 31 243)(22 252 32 242)(23 251 33 241)(24 250 34 260)(25 249 35 259)(26 248 36 258)(27 247 37 257)(28 246 38 256)(29 245 39 255)(30 244 40 254)(41 158 51 148)(42 157 52 147)(43 156 53 146)(44 155 54 145)(45 154 55 144)(46 153 56 143)(47 152 57 142)(48 151 58 141)(49 150 59 160)(50 149 60 159)(61 94 71 84)(62 93 72 83)(63 92 73 82)(64 91 74 81)(65 90 75 100)(66 89 76 99)(67 88 77 98)(68 87 78 97)(69 86 79 96)(70 85 80 95)(101 303 111 313)(102 302 112 312)(103 301 113 311)(104 320 114 310)(105 319 115 309)(106 318 116 308)(107 317 117 307)(108 316 118 306)(109 315 119 305)(110 314 120 304)(121 185 131 195)(122 184 132 194)(123 183 133 193)(124 182 134 192)(125 181 135 191)(126 200 136 190)(127 199 137 189)(128 198 138 188)(129 197 139 187)(130 196 140 186)(161 263 171 273)(162 262 172 272)(163 261 173 271)(164 280 174 270)(165 279 175 269)(166 278 176 268)(167 277 177 267)(168 276 178 266)(169 275 179 265)(170 274 180 264)(201 224 211 234)(202 223 212 233)(203 222 213 232)(204 221 214 231)(205 240 215 230)(206 239 216 229)(207 238 217 228)(208 237 218 227)(209 236 219 226)(210 235 220 225)
(1 59 265 82)(2 50 266 93)(3 41 267 84)(4 52 268 95)(5 43 269 86)(6 54 270 97)(7 45 271 88)(8 56 272 99)(9 47 273 90)(10 58 274 81)(11 49 275 92)(12 60 276 83)(13 51 277 94)(14 42 278 85)(15 53 279 96)(16 44 280 87)(17 55 261 98)(18 46 262 89)(19 57 263 100)(20 48 264 91)(21 303 229 131)(22 314 230 122)(23 305 231 133)(24 316 232 124)(25 307 233 135)(26 318 234 126)(27 309 235 137)(28 320 236 128)(29 311 237 139)(30 302 238 130)(31 313 239 121)(32 304 240 132)(33 315 221 123)(34 306 222 134)(35 317 223 125)(36 308 224 136)(37 319 225 127)(38 310 226 138)(39 301 227 129)(40 312 228 140)(61 291 158 162)(62 282 159 173)(63 293 160 164)(64 284 141 175)(65 295 142 166)(66 286 143 177)(67 297 144 168)(68 288 145 179)(69 299 146 170)(70 290 147 161)(71 281 148 172)(72 292 149 163)(73 283 150 174)(74 294 151 165)(75 285 152 176)(76 296 153 167)(77 287 154 178)(78 298 155 169)(79 289 156 180)(80 300 157 171)(101 211 185 258)(102 202 186 249)(103 213 187 260)(104 204 188 251)(105 215 189 242)(106 206 190 253)(107 217 191 244)(108 208 192 255)(109 219 193 246)(110 210 194 257)(111 201 195 248)(112 212 196 259)(113 203 197 250)(114 214 198 241)(115 205 199 252)(116 216 200 243)(117 207 181 254)(118 218 182 245)(119 209 183 256)(120 220 184 247)
(1 229 265 21)(2 238 266 30)(3 227 267 39)(4 236 268 28)(5 225 269 37)(6 234 270 26)(7 223 271 35)(8 232 272 24)(9 221 273 33)(10 230 274 22)(11 239 275 31)(12 228 276 40)(13 237 277 29)(14 226 278 38)(15 235 279 27)(16 224 280 36)(17 233 261 25)(18 222 262 34)(19 231 263 23)(20 240 264 32)(41 301 84 129)(42 310 85 138)(43 319 86 127)(44 308 87 136)(45 317 88 125)(46 306 89 134)(47 315 90 123)(48 304 91 132)(49 313 92 121)(50 302 93 130)(51 311 94 139)(52 320 95 128)(53 309 96 137)(54 318 97 126)(55 307 98 135)(56 316 99 124)(57 305 100 133)(58 314 81 122)(59 303 82 131)(60 312 83 140)(61 187 158 103)(62 196 159 112)(63 185 160 101)(64 194 141 110)(65 183 142 119)(66 192 143 108)(67 181 144 117)(68 190 145 106)(69 199 146 115)(70 188 147 104)(71 197 148 113)(72 186 149 102)(73 195 150 111)(74 184 151 120)(75 193 152 109)(76 182 153 118)(77 191 154 107)(78 200 155 116)(79 189 156 105)(80 198 157 114)(161 251 290 204)(162 260 291 213)(163 249 292 202)(164 258 293 211)(165 247 294 220)(166 256 295 209)(167 245 296 218)(168 254 297 207)(169 243 298 216)(170 252 299 205)(171 241 300 214)(172 250 281 203)(173 259 282 212)(174 248 283 201)(175 257 284 210)(176 246 285 219)(177 255 286 208)(178 244 287 217)(179 253 288 206)(180 242 289 215)
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), (1,298,11,288)(2,297,12,287)(3,296,13,286)(4,295,14,285)(5,294,15,284)(6,293,16,283)(7,292,17,282)(8,291,18,281)(9,290,19,300)(10,289,20,299)(21,253,31,243)(22,252,32,242)(23,251,33,241)(24,250,34,260)(25,249,35,259)(26,248,36,258)(27,247,37,257)(28,246,38,256)(29,245,39,255)(30,244,40,254)(41,158,51,148)(42,157,52,147)(43,156,53,146)(44,155,54,145)(45,154,55,144)(46,153,56,143)(47,152,57,142)(48,151,58,141)(49,150,59,160)(50,149,60,159)(61,94,71,84)(62,93,72,83)(63,92,73,82)(64,91,74,81)(65,90,75,100)(66,89,76,99)(67,88,77,98)(68,87,78,97)(69,86,79,96)(70,85,80,95)(101,303,111,313)(102,302,112,312)(103,301,113,311)(104,320,114,310)(105,319,115,309)(106,318,116,308)(107,317,117,307)(108,316,118,306)(109,315,119,305)(110,314,120,304)(121,185,131,195)(122,184,132,194)(123,183,133,193)(124,182,134,192)(125,181,135,191)(126,200,136,190)(127,199,137,189)(128,198,138,188)(129,197,139,187)(130,196,140,186)(161,263,171,273)(162,262,172,272)(163,261,173,271)(164,280,174,270)(165,279,175,269)(166,278,176,268)(167,277,177,267)(168,276,178,266)(169,275,179,265)(170,274,180,264)(201,224,211,234)(202,223,212,233)(203,222,213,232)(204,221,214,231)(205,240,215,230)(206,239,216,229)(207,238,217,228)(208,237,218,227)(209,236,219,226)(210,235,220,225), (1,59,265,82)(2,50,266,93)(3,41,267,84)(4,52,268,95)(5,43,269,86)(6,54,270,97)(7,45,271,88)(8,56,272,99)(9,47,273,90)(10,58,274,81)(11,49,275,92)(12,60,276,83)(13,51,277,94)(14,42,278,85)(15,53,279,96)(16,44,280,87)(17,55,261,98)(18,46,262,89)(19,57,263,100)(20,48,264,91)(21,303,229,131)(22,314,230,122)(23,305,231,133)(24,316,232,124)(25,307,233,135)(26,318,234,126)(27,309,235,137)(28,320,236,128)(29,311,237,139)(30,302,238,130)(31,313,239,121)(32,304,240,132)(33,315,221,123)(34,306,222,134)(35,317,223,125)(36,308,224,136)(37,319,225,127)(38,310,226,138)(39,301,227,129)(40,312,228,140)(61,291,158,162)(62,282,159,173)(63,293,160,164)(64,284,141,175)(65,295,142,166)(66,286,143,177)(67,297,144,168)(68,288,145,179)(69,299,146,170)(70,290,147,161)(71,281,148,172)(72,292,149,163)(73,283,150,174)(74,294,151,165)(75,285,152,176)(76,296,153,167)(77,287,154,178)(78,298,155,169)(79,289,156,180)(80,300,157,171)(101,211,185,258)(102,202,186,249)(103,213,187,260)(104,204,188,251)(105,215,189,242)(106,206,190,253)(107,217,191,244)(108,208,192,255)(109,219,193,246)(110,210,194,257)(111,201,195,248)(112,212,196,259)(113,203,197,250)(114,214,198,241)(115,205,199,252)(116,216,200,243)(117,207,181,254)(118,218,182,245)(119,209,183,256)(120,220,184,247), (1,229,265,21)(2,238,266,30)(3,227,267,39)(4,236,268,28)(5,225,269,37)(6,234,270,26)(7,223,271,35)(8,232,272,24)(9,221,273,33)(10,230,274,22)(11,239,275,31)(12,228,276,40)(13,237,277,29)(14,226,278,38)(15,235,279,27)(16,224,280,36)(17,233,261,25)(18,222,262,34)(19,231,263,23)(20,240,264,32)(41,301,84,129)(42,310,85,138)(43,319,86,127)(44,308,87,136)(45,317,88,125)(46,306,89,134)(47,315,90,123)(48,304,91,132)(49,313,92,121)(50,302,93,130)(51,311,94,139)(52,320,95,128)(53,309,96,137)(54,318,97,126)(55,307,98,135)(56,316,99,124)(57,305,100,133)(58,314,81,122)(59,303,82,131)(60,312,83,140)(61,187,158,103)(62,196,159,112)(63,185,160,101)(64,194,141,110)(65,183,142,119)(66,192,143,108)(67,181,144,117)(68,190,145,106)(69,199,146,115)(70,188,147,104)(71,197,148,113)(72,186,149,102)(73,195,150,111)(74,184,151,120)(75,193,152,109)(76,182,153,118)(77,191,154,107)(78,200,155,116)(79,189,156,105)(80,198,157,114)(161,251,290,204)(162,260,291,213)(163,249,292,202)(164,258,293,211)(165,247,294,220)(166,256,295,209)(167,245,296,218)(168,254,297,207)(169,243,298,216)(170,252,299,205)(171,241,300,214)(172,250,281,203)(173,259,282,212)(174,248,283,201)(175,257,284,210)(176,246,285,219)(177,255,286,208)(178,244,287,217)(179,253,288,206)(180,242,289,215)>;
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), (1,298,11,288)(2,297,12,287)(3,296,13,286)(4,295,14,285)(5,294,15,284)(6,293,16,283)(7,292,17,282)(8,291,18,281)(9,290,19,300)(10,289,20,299)(21,253,31,243)(22,252,32,242)(23,251,33,241)(24,250,34,260)(25,249,35,259)(26,248,36,258)(27,247,37,257)(28,246,38,256)(29,245,39,255)(30,244,40,254)(41,158,51,148)(42,157,52,147)(43,156,53,146)(44,155,54,145)(45,154,55,144)(46,153,56,143)(47,152,57,142)(48,151,58,141)(49,150,59,160)(50,149,60,159)(61,94,71,84)(62,93,72,83)(63,92,73,82)(64,91,74,81)(65,90,75,100)(66,89,76,99)(67,88,77,98)(68,87,78,97)(69,86,79,96)(70,85,80,95)(101,303,111,313)(102,302,112,312)(103,301,113,311)(104,320,114,310)(105,319,115,309)(106,318,116,308)(107,317,117,307)(108,316,118,306)(109,315,119,305)(110,314,120,304)(121,185,131,195)(122,184,132,194)(123,183,133,193)(124,182,134,192)(125,181,135,191)(126,200,136,190)(127,199,137,189)(128,198,138,188)(129,197,139,187)(130,196,140,186)(161,263,171,273)(162,262,172,272)(163,261,173,271)(164,280,174,270)(165,279,175,269)(166,278,176,268)(167,277,177,267)(168,276,178,266)(169,275,179,265)(170,274,180,264)(201,224,211,234)(202,223,212,233)(203,222,213,232)(204,221,214,231)(205,240,215,230)(206,239,216,229)(207,238,217,228)(208,237,218,227)(209,236,219,226)(210,235,220,225), (1,59,265,82)(2,50,266,93)(3,41,267,84)(4,52,268,95)(5,43,269,86)(6,54,270,97)(7,45,271,88)(8,56,272,99)(9,47,273,90)(10,58,274,81)(11,49,275,92)(12,60,276,83)(13,51,277,94)(14,42,278,85)(15,53,279,96)(16,44,280,87)(17,55,261,98)(18,46,262,89)(19,57,263,100)(20,48,264,91)(21,303,229,131)(22,314,230,122)(23,305,231,133)(24,316,232,124)(25,307,233,135)(26,318,234,126)(27,309,235,137)(28,320,236,128)(29,311,237,139)(30,302,238,130)(31,313,239,121)(32,304,240,132)(33,315,221,123)(34,306,222,134)(35,317,223,125)(36,308,224,136)(37,319,225,127)(38,310,226,138)(39,301,227,129)(40,312,228,140)(61,291,158,162)(62,282,159,173)(63,293,160,164)(64,284,141,175)(65,295,142,166)(66,286,143,177)(67,297,144,168)(68,288,145,179)(69,299,146,170)(70,290,147,161)(71,281,148,172)(72,292,149,163)(73,283,150,174)(74,294,151,165)(75,285,152,176)(76,296,153,167)(77,287,154,178)(78,298,155,169)(79,289,156,180)(80,300,157,171)(101,211,185,258)(102,202,186,249)(103,213,187,260)(104,204,188,251)(105,215,189,242)(106,206,190,253)(107,217,191,244)(108,208,192,255)(109,219,193,246)(110,210,194,257)(111,201,195,248)(112,212,196,259)(113,203,197,250)(114,214,198,241)(115,205,199,252)(116,216,200,243)(117,207,181,254)(118,218,182,245)(119,209,183,256)(120,220,184,247), (1,229,265,21)(2,238,266,30)(3,227,267,39)(4,236,268,28)(5,225,269,37)(6,234,270,26)(7,223,271,35)(8,232,272,24)(9,221,273,33)(10,230,274,22)(11,239,275,31)(12,228,276,40)(13,237,277,29)(14,226,278,38)(15,235,279,27)(16,224,280,36)(17,233,261,25)(18,222,262,34)(19,231,263,23)(20,240,264,32)(41,301,84,129)(42,310,85,138)(43,319,86,127)(44,308,87,136)(45,317,88,125)(46,306,89,134)(47,315,90,123)(48,304,91,132)(49,313,92,121)(50,302,93,130)(51,311,94,139)(52,320,95,128)(53,309,96,137)(54,318,97,126)(55,307,98,135)(56,316,99,124)(57,305,100,133)(58,314,81,122)(59,303,82,131)(60,312,83,140)(61,187,158,103)(62,196,159,112)(63,185,160,101)(64,194,141,110)(65,183,142,119)(66,192,143,108)(67,181,144,117)(68,190,145,106)(69,199,146,115)(70,188,147,104)(71,197,148,113)(72,186,149,102)(73,195,150,111)(74,184,151,120)(75,193,152,109)(76,182,153,118)(77,191,154,107)(78,200,155,116)(79,189,156,105)(80,198,157,114)(161,251,290,204)(162,260,291,213)(163,249,292,202)(164,258,293,211)(165,247,294,220)(166,256,295,209)(167,245,296,218)(168,254,297,207)(169,243,298,216)(170,252,299,205)(171,241,300,214)(172,250,281,203)(173,259,282,212)(174,248,283,201)(175,257,284,210)(176,246,285,219)(177,255,286,208)(178,244,287,217)(179,253,288,206)(180,242,289,215) );
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)], [(1,298,11,288),(2,297,12,287),(3,296,13,286),(4,295,14,285),(5,294,15,284),(6,293,16,283),(7,292,17,282),(8,291,18,281),(9,290,19,300),(10,289,20,299),(21,253,31,243),(22,252,32,242),(23,251,33,241),(24,250,34,260),(25,249,35,259),(26,248,36,258),(27,247,37,257),(28,246,38,256),(29,245,39,255),(30,244,40,254),(41,158,51,148),(42,157,52,147),(43,156,53,146),(44,155,54,145),(45,154,55,144),(46,153,56,143),(47,152,57,142),(48,151,58,141),(49,150,59,160),(50,149,60,159),(61,94,71,84),(62,93,72,83),(63,92,73,82),(64,91,74,81),(65,90,75,100),(66,89,76,99),(67,88,77,98),(68,87,78,97),(69,86,79,96),(70,85,80,95),(101,303,111,313),(102,302,112,312),(103,301,113,311),(104,320,114,310),(105,319,115,309),(106,318,116,308),(107,317,117,307),(108,316,118,306),(109,315,119,305),(110,314,120,304),(121,185,131,195),(122,184,132,194),(123,183,133,193),(124,182,134,192),(125,181,135,191),(126,200,136,190),(127,199,137,189),(128,198,138,188),(129,197,139,187),(130,196,140,186),(161,263,171,273),(162,262,172,272),(163,261,173,271),(164,280,174,270),(165,279,175,269),(166,278,176,268),(167,277,177,267),(168,276,178,266),(169,275,179,265),(170,274,180,264),(201,224,211,234),(202,223,212,233),(203,222,213,232),(204,221,214,231),(205,240,215,230),(206,239,216,229),(207,238,217,228),(208,237,218,227),(209,236,219,226),(210,235,220,225)], [(1,59,265,82),(2,50,266,93),(3,41,267,84),(4,52,268,95),(5,43,269,86),(6,54,270,97),(7,45,271,88),(8,56,272,99),(9,47,273,90),(10,58,274,81),(11,49,275,92),(12,60,276,83),(13,51,277,94),(14,42,278,85),(15,53,279,96),(16,44,280,87),(17,55,261,98),(18,46,262,89),(19,57,263,100),(20,48,264,91),(21,303,229,131),(22,314,230,122),(23,305,231,133),(24,316,232,124),(25,307,233,135),(26,318,234,126),(27,309,235,137),(28,320,236,128),(29,311,237,139),(30,302,238,130),(31,313,239,121),(32,304,240,132),(33,315,221,123),(34,306,222,134),(35,317,223,125),(36,308,224,136),(37,319,225,127),(38,310,226,138),(39,301,227,129),(40,312,228,140),(61,291,158,162),(62,282,159,173),(63,293,160,164),(64,284,141,175),(65,295,142,166),(66,286,143,177),(67,297,144,168),(68,288,145,179),(69,299,146,170),(70,290,147,161),(71,281,148,172),(72,292,149,163),(73,283,150,174),(74,294,151,165),(75,285,152,176),(76,296,153,167),(77,287,154,178),(78,298,155,169),(79,289,156,180),(80,300,157,171),(101,211,185,258),(102,202,186,249),(103,213,187,260),(104,204,188,251),(105,215,189,242),(106,206,190,253),(107,217,191,244),(108,208,192,255),(109,219,193,246),(110,210,194,257),(111,201,195,248),(112,212,196,259),(113,203,197,250),(114,214,198,241),(115,205,199,252),(116,216,200,243),(117,207,181,254),(118,218,182,245),(119,209,183,256),(120,220,184,247)], [(1,229,265,21),(2,238,266,30),(3,227,267,39),(4,236,268,28),(5,225,269,37),(6,234,270,26),(7,223,271,35),(8,232,272,24),(9,221,273,33),(10,230,274,22),(11,239,275,31),(12,228,276,40),(13,237,277,29),(14,226,278,38),(15,235,279,27),(16,224,280,36),(17,233,261,25),(18,222,262,34),(19,231,263,23),(20,240,264,32),(41,301,84,129),(42,310,85,138),(43,319,86,127),(44,308,87,136),(45,317,88,125),(46,306,89,134),(47,315,90,123),(48,304,91,132),(49,313,92,121),(50,302,93,130),(51,311,94,139),(52,320,95,128),(53,309,96,137),(54,318,97,126),(55,307,98,135),(56,316,99,124),(57,305,100,133),(58,314,81,122),(59,303,82,131),(60,312,83,140),(61,187,158,103),(62,196,159,112),(63,185,160,101),(64,194,141,110),(65,183,142,119),(66,192,143,108),(67,181,144,117),(68,190,145,106),(69,199,146,115),(70,188,147,104),(71,197,148,113),(72,186,149,102),(73,195,150,111),(74,184,151,120),(75,193,152,109),(76,182,153,118),(77,191,154,107),(78,200,155,116),(79,189,156,105),(80,198,157,114),(161,251,290,204),(162,260,291,213),(163,249,292,202),(164,258,293,211),(165,247,294,220),(166,256,295,209),(167,245,296,218),(168,254,297,207),(169,243,298,216),(170,252,299,205),(171,241,300,214),(172,250,281,203),(173,259,282,212),(174,248,283,201),(175,257,284,210),(176,246,285,219),(177,255,286,208),(178,244,287,217),(179,253,288,206),(180,242,289,215)]])
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 | Q16 | C4○D4 | D10 | D10 | C4○D20 | C8⋊C22 | Q8×D5 | D4×D5 | D8⋊D5 | D5×Q16 |
| kernel | Dic10⋊2Q8 | C10.D8 | C10.Q16 | C20.8Q8 | C20.44D4 | C5×C2.D8 | Dic5⋊3Q8 | C20⋊Q8 | Dic10 | C2×Dic5 | C2.D8 | 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⋊2Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 20 |
| 0 | 0 | 0 | 0 | 20 | 38 |
| 20 | 5 | 0 | 0 | 0 | 0 |
| 10 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 30 |
| 0 | 0 | 0 | 0 | 30 | 40 |
| 31 | 5 | 0 | 0 | 0 | 0 |
| 29 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,3,20,0,0,0,0,20,38],[20,10,0,0,0,0,5,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,30,0,0,0,0,30,40],[31,29,0,0,0,0,5,10,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;
Dic10⋊2Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_2Q_8 % in TeX
G:=Group("Dic10:2Q8"); // GroupNames label
G:=SmallGroup(320,502);
// by ID
G=gap.SmallGroup(320,502);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,422,135,268,570,297,136,12550]);
// 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^11,d*a*d^-1=a^9,c*b*c^-1=a^5*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations