metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊6Q8, C20.21SD16, C42.85D10, C4⋊Q8.13D5, C4.14(Q8×D5), C4⋊C4.88D10, C5⋊5(Q8⋊Q8), C20.41(C2×Q8), (C2×C20).163D4, C4.8(D4.D5), C20⋊3C8.23C2, C20.88(C4○D4), C10.62(C2×SD16), (C2×C20).412C23, (C4×C20).141C22, C4.37(Q8⋊2D5), (C4×Dic10).19C2, C10.Q16.16C2, C10.78(C22⋊Q8), C20.Q8.18C2, C2.15(D10⋊3Q8), C4⋊Dic5.351C22, C2.21(C20.C23), C10.100(C8.C22), (C2×Dic10).285C22, (C5×C4⋊Q8).13C2, C2.16(C2×D4.D5), (C2×C10).543(C2×D4), (C2×C4).192(C5⋊D4), (C5×C4⋊C4).135C22, (C2×C4).509(C22×D5), C22.215(C2×C5⋊D4), (C2×C5⋊2C8).142C22, SmallGroup(320,721)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊6Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, bd=db, 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, C4.Q8, C4×Q8, C4⋊Q8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Q8⋊Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C20⋊3C8, C20.Q8, C10.Q16, C4×Dic10, C5×C4⋊Q8, Dic10⋊6Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8.C22, C5⋊D4, C22×D5, Q8⋊Q8, D4.D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, C2×D4.D5, C20.C23, D10⋊3Q8, Dic10⋊6Q8
►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)
(1 303 11 313)(2 302 12 312)(3 301 13 311)(4 320 14 310)(5 319 15 309)(6 318 16 308)(7 317 17 307)(8 316 18 306)(9 315 19 305)(10 314 20 304)(21 119 31 109)(22 118 32 108)(23 117 33 107)(24 116 34 106)(25 115 35 105)(26 114 36 104)(27 113 37 103)(28 112 38 102)(29 111 39 101)(30 110 40 120)(41 130 51 140)(42 129 52 139)(43 128 53 138)(44 127 54 137)(45 126 55 136)(46 125 56 135)(47 124 57 134)(48 123 58 133)(49 122 59 132)(50 121 60 131)(61 243 71 253)(62 242 72 252)(63 241 73 251)(64 260 74 250)(65 259 75 249)(66 258 76 248)(67 257 77 247)(68 256 78 246)(69 255 79 245)(70 254 80 244)(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)(141 293 151 283)(142 292 152 282)(143 291 153 281)(144 290 154 300)(145 289 155 299)(146 288 156 298)(147 287 157 297)(148 286 158 296)(149 285 159 295)(150 284 160 294)(161 272 171 262)(162 271 172 261)(163 270 173 280)(164 269 174 279)(165 268 175 278)(166 267 176 277)(167 266 177 276)(168 265 178 275)(169 264 179 274)(170 263 180 273)(181 222 191 232)(182 221 192 231)(183 240 193 230)(184 239 194 229)(185 238 195 228)(186 237 196 227)(187 236 197 226)(188 235 198 225)(189 234 199 224)(190 233 200 223)
(1 297 120 257)(2 288 101 248)(3 299 102 259)(4 290 103 250)(5 281 104 241)(6 292 105 252)(7 283 106 243)(8 294 107 254)(9 285 108 245)(10 296 109 256)(11 287 110 247)(12 298 111 258)(13 289 112 249)(14 300 113 260)(15 291 114 251)(16 282 115 242)(17 293 116 253)(18 284 117 244)(19 295 118 255)(20 286 119 246)(21 73 314 143)(22 64 315 154)(23 75 316 145)(24 66 317 156)(25 77 318 147)(26 68 319 158)(27 79 320 149)(28 70 301 160)(29 61 302 151)(30 72 303 142)(31 63 304 153)(32 74 305 144)(33 65 306 155)(34 76 307 146)(35 67 308 157)(36 78 309 148)(37 69 310 159)(38 80 311 150)(39 71 312 141)(40 62 313 152)(41 178 225 100)(42 169 226 91)(43 180 227 82)(44 171 228 93)(45 162 229 84)(46 173 230 95)(47 164 231 86)(48 175 232 97)(49 166 233 88)(50 177 234 99)(51 168 235 90)(52 179 236 81)(53 170 237 92)(54 161 238 83)(55 172 239 94)(56 163 240 85)(57 174 221 96)(58 165 222 87)(59 176 223 98)(60 167 224 89)(121 271 199 217)(122 262 200 208)(123 273 181 219)(124 264 182 210)(125 275 183 201)(126 266 184 212)(127 277 185 203)(128 268 186 214)(129 279 187 205)(130 270 188 216)(131 261 189 207)(132 272 190 218)(133 263 191 209)(134 274 192 220)(135 265 193 211)(136 276 194 202)(137 267 195 213)(138 278 196 204)(139 269 197 215)(140 280 198 206)
(1 227 120 43)(2 228 101 44)(3 229 102 45)(4 230 103 46)(5 231 104 47)(6 232 105 48)(7 233 106 49)(8 234 107 50)(9 235 108 51)(10 236 109 52)(11 237 110 53)(12 238 111 54)(13 239 112 55)(14 240 113 56)(15 221 114 57)(16 222 115 58)(17 223 116 59)(18 224 117 60)(19 225 118 41)(20 226 119 42)(21 139 314 197)(22 140 315 198)(23 121 316 199)(24 122 317 200)(25 123 318 181)(26 124 319 182)(27 125 320 183)(28 126 301 184)(29 127 302 185)(30 128 303 186)(31 129 304 187)(32 130 305 188)(33 131 306 189)(34 132 307 190)(35 133 308 191)(36 134 309 192)(37 135 310 193)(38 136 311 194)(39 137 312 195)(40 138 313 196)(61 203 151 277)(62 204 152 278)(63 205 153 279)(64 206 154 280)(65 207 155 261)(66 208 156 262)(67 209 157 263)(68 210 158 264)(69 211 159 265)(70 212 160 266)(71 213 141 267)(72 214 142 268)(73 215 143 269)(74 216 144 270)(75 217 145 271)(76 218 146 272)(77 219 147 273)(78 220 148 274)(79 201 149 275)(80 202 150 276)(81 296 179 256)(82 297 180 257)(83 298 161 258)(84 299 162 259)(85 300 163 260)(86 281 164 241)(87 282 165 242)(88 283 166 243)(89 284 167 244)(90 285 168 245)(91 286 169 246)(92 287 170 247)(93 288 171 248)(94 289 172 249)(95 290 173 250)(96 291 174 251)(97 292 175 252)(98 293 176 253)(99 294 177 254)(100 295 178 255)
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), (1,303,11,313)(2,302,12,312)(3,301,13,311)(4,320,14,310)(5,319,15,309)(6,318,16,308)(7,317,17,307)(8,316,18,306)(9,315,19,305)(10,314,20,304)(21,119,31,109)(22,118,32,108)(23,117,33,107)(24,116,34,106)(25,115,35,105)(26,114,36,104)(27,113,37,103)(28,112,38,102)(29,111,39,101)(30,110,40,120)(41,130,51,140)(42,129,52,139)(43,128,53,138)(44,127,54,137)(45,126,55,136)(46,125,56,135)(47,124,57,134)(48,123,58,133)(49,122,59,132)(50,121,60,131)(61,243,71,253)(62,242,72,252)(63,241,73,251)(64,260,74,250)(65,259,75,249)(66,258,76,248)(67,257,77,247)(68,256,78,246)(69,255,79,245)(70,254,80,244)(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)(141,293,151,283)(142,292,152,282)(143,291,153,281)(144,290,154,300)(145,289,155,299)(146,288,156,298)(147,287,157,297)(148,286,158,296)(149,285,159,295)(150,284,160,294)(161,272,171,262)(162,271,172,261)(163,270,173,280)(164,269,174,279)(165,268,175,278)(166,267,176,277)(167,266,177,276)(168,265,178,275)(169,264,179,274)(170,263,180,273)(181,222,191,232)(182,221,192,231)(183,240,193,230)(184,239,194,229)(185,238,195,228)(186,237,196,227)(187,236,197,226)(188,235,198,225)(189,234,199,224)(190,233,200,223), (1,297,120,257)(2,288,101,248)(3,299,102,259)(4,290,103,250)(5,281,104,241)(6,292,105,252)(7,283,106,243)(8,294,107,254)(9,285,108,245)(10,296,109,256)(11,287,110,247)(12,298,111,258)(13,289,112,249)(14,300,113,260)(15,291,114,251)(16,282,115,242)(17,293,116,253)(18,284,117,244)(19,295,118,255)(20,286,119,246)(21,73,314,143)(22,64,315,154)(23,75,316,145)(24,66,317,156)(25,77,318,147)(26,68,319,158)(27,79,320,149)(28,70,301,160)(29,61,302,151)(30,72,303,142)(31,63,304,153)(32,74,305,144)(33,65,306,155)(34,76,307,146)(35,67,308,157)(36,78,309,148)(37,69,310,159)(38,80,311,150)(39,71,312,141)(40,62,313,152)(41,178,225,100)(42,169,226,91)(43,180,227,82)(44,171,228,93)(45,162,229,84)(46,173,230,95)(47,164,231,86)(48,175,232,97)(49,166,233,88)(50,177,234,99)(51,168,235,90)(52,179,236,81)(53,170,237,92)(54,161,238,83)(55,172,239,94)(56,163,240,85)(57,174,221,96)(58,165,222,87)(59,176,223,98)(60,167,224,89)(121,271,199,217)(122,262,200,208)(123,273,181,219)(124,264,182,210)(125,275,183,201)(126,266,184,212)(127,277,185,203)(128,268,186,214)(129,279,187,205)(130,270,188,216)(131,261,189,207)(132,272,190,218)(133,263,191,209)(134,274,192,220)(135,265,193,211)(136,276,194,202)(137,267,195,213)(138,278,196,204)(139,269,197,215)(140,280,198,206), (1,227,120,43)(2,228,101,44)(3,229,102,45)(4,230,103,46)(5,231,104,47)(6,232,105,48)(7,233,106,49)(8,234,107,50)(9,235,108,51)(10,236,109,52)(11,237,110,53)(12,238,111,54)(13,239,112,55)(14,240,113,56)(15,221,114,57)(16,222,115,58)(17,223,116,59)(18,224,117,60)(19,225,118,41)(20,226,119,42)(21,139,314,197)(22,140,315,198)(23,121,316,199)(24,122,317,200)(25,123,318,181)(26,124,319,182)(27,125,320,183)(28,126,301,184)(29,127,302,185)(30,128,303,186)(31,129,304,187)(32,130,305,188)(33,131,306,189)(34,132,307,190)(35,133,308,191)(36,134,309,192)(37,135,310,193)(38,136,311,194)(39,137,312,195)(40,138,313,196)(61,203,151,277)(62,204,152,278)(63,205,153,279)(64,206,154,280)(65,207,155,261)(66,208,156,262)(67,209,157,263)(68,210,158,264)(69,211,159,265)(70,212,160,266)(71,213,141,267)(72,214,142,268)(73,215,143,269)(74,216,144,270)(75,217,145,271)(76,218,146,272)(77,219,147,273)(78,220,148,274)(79,201,149,275)(80,202,150,276)(81,296,179,256)(82,297,180,257)(83,298,161,258)(84,299,162,259)(85,300,163,260)(86,281,164,241)(87,282,165,242)(88,283,166,243)(89,284,167,244)(90,285,168,245)(91,286,169,246)(92,287,170,247)(93,288,171,248)(94,289,172,249)(95,290,173,250)(96,291,174,251)(97,292,175,252)(98,293,176,253)(99,294,177,254)(100,295,178,255)>;
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), (1,303,11,313)(2,302,12,312)(3,301,13,311)(4,320,14,310)(5,319,15,309)(6,318,16,308)(7,317,17,307)(8,316,18,306)(9,315,19,305)(10,314,20,304)(21,119,31,109)(22,118,32,108)(23,117,33,107)(24,116,34,106)(25,115,35,105)(26,114,36,104)(27,113,37,103)(28,112,38,102)(29,111,39,101)(30,110,40,120)(41,130,51,140)(42,129,52,139)(43,128,53,138)(44,127,54,137)(45,126,55,136)(46,125,56,135)(47,124,57,134)(48,123,58,133)(49,122,59,132)(50,121,60,131)(61,243,71,253)(62,242,72,252)(63,241,73,251)(64,260,74,250)(65,259,75,249)(66,258,76,248)(67,257,77,247)(68,256,78,246)(69,255,79,245)(70,254,80,244)(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)(141,293,151,283)(142,292,152,282)(143,291,153,281)(144,290,154,300)(145,289,155,299)(146,288,156,298)(147,287,157,297)(148,286,158,296)(149,285,159,295)(150,284,160,294)(161,272,171,262)(162,271,172,261)(163,270,173,280)(164,269,174,279)(165,268,175,278)(166,267,176,277)(167,266,177,276)(168,265,178,275)(169,264,179,274)(170,263,180,273)(181,222,191,232)(182,221,192,231)(183,240,193,230)(184,239,194,229)(185,238,195,228)(186,237,196,227)(187,236,197,226)(188,235,198,225)(189,234,199,224)(190,233,200,223), (1,297,120,257)(2,288,101,248)(3,299,102,259)(4,290,103,250)(5,281,104,241)(6,292,105,252)(7,283,106,243)(8,294,107,254)(9,285,108,245)(10,296,109,256)(11,287,110,247)(12,298,111,258)(13,289,112,249)(14,300,113,260)(15,291,114,251)(16,282,115,242)(17,293,116,253)(18,284,117,244)(19,295,118,255)(20,286,119,246)(21,73,314,143)(22,64,315,154)(23,75,316,145)(24,66,317,156)(25,77,318,147)(26,68,319,158)(27,79,320,149)(28,70,301,160)(29,61,302,151)(30,72,303,142)(31,63,304,153)(32,74,305,144)(33,65,306,155)(34,76,307,146)(35,67,308,157)(36,78,309,148)(37,69,310,159)(38,80,311,150)(39,71,312,141)(40,62,313,152)(41,178,225,100)(42,169,226,91)(43,180,227,82)(44,171,228,93)(45,162,229,84)(46,173,230,95)(47,164,231,86)(48,175,232,97)(49,166,233,88)(50,177,234,99)(51,168,235,90)(52,179,236,81)(53,170,237,92)(54,161,238,83)(55,172,239,94)(56,163,240,85)(57,174,221,96)(58,165,222,87)(59,176,223,98)(60,167,224,89)(121,271,199,217)(122,262,200,208)(123,273,181,219)(124,264,182,210)(125,275,183,201)(126,266,184,212)(127,277,185,203)(128,268,186,214)(129,279,187,205)(130,270,188,216)(131,261,189,207)(132,272,190,218)(133,263,191,209)(134,274,192,220)(135,265,193,211)(136,276,194,202)(137,267,195,213)(138,278,196,204)(139,269,197,215)(140,280,198,206), (1,227,120,43)(2,228,101,44)(3,229,102,45)(4,230,103,46)(5,231,104,47)(6,232,105,48)(7,233,106,49)(8,234,107,50)(9,235,108,51)(10,236,109,52)(11,237,110,53)(12,238,111,54)(13,239,112,55)(14,240,113,56)(15,221,114,57)(16,222,115,58)(17,223,116,59)(18,224,117,60)(19,225,118,41)(20,226,119,42)(21,139,314,197)(22,140,315,198)(23,121,316,199)(24,122,317,200)(25,123,318,181)(26,124,319,182)(27,125,320,183)(28,126,301,184)(29,127,302,185)(30,128,303,186)(31,129,304,187)(32,130,305,188)(33,131,306,189)(34,132,307,190)(35,133,308,191)(36,134,309,192)(37,135,310,193)(38,136,311,194)(39,137,312,195)(40,138,313,196)(61,203,151,277)(62,204,152,278)(63,205,153,279)(64,206,154,280)(65,207,155,261)(66,208,156,262)(67,209,157,263)(68,210,158,264)(69,211,159,265)(70,212,160,266)(71,213,141,267)(72,214,142,268)(73,215,143,269)(74,216,144,270)(75,217,145,271)(76,218,146,272)(77,219,147,273)(78,220,148,274)(79,201,149,275)(80,202,150,276)(81,296,179,256)(82,297,180,257)(83,298,161,258)(84,299,162,259)(85,300,163,260)(86,281,164,241)(87,282,165,242)(88,283,166,243)(89,284,167,244)(90,285,168,245)(91,286,169,246)(92,287,170,247)(93,288,171,248)(94,289,172,249)(95,290,173,250)(96,291,174,251)(97,292,175,252)(98,293,176,253)(99,294,177,254)(100,295,178,255) );
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)], [(1,303,11,313),(2,302,12,312),(3,301,13,311),(4,320,14,310),(5,319,15,309),(6,318,16,308),(7,317,17,307),(8,316,18,306),(9,315,19,305),(10,314,20,304),(21,119,31,109),(22,118,32,108),(23,117,33,107),(24,116,34,106),(25,115,35,105),(26,114,36,104),(27,113,37,103),(28,112,38,102),(29,111,39,101),(30,110,40,120),(41,130,51,140),(42,129,52,139),(43,128,53,138),(44,127,54,137),(45,126,55,136),(46,125,56,135),(47,124,57,134),(48,123,58,133),(49,122,59,132),(50,121,60,131),(61,243,71,253),(62,242,72,252),(63,241,73,251),(64,260,74,250),(65,259,75,249),(66,258,76,248),(67,257,77,247),(68,256,78,246),(69,255,79,245),(70,254,80,244),(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),(141,293,151,283),(142,292,152,282),(143,291,153,281),(144,290,154,300),(145,289,155,299),(146,288,156,298),(147,287,157,297),(148,286,158,296),(149,285,159,295),(150,284,160,294),(161,272,171,262),(162,271,172,261),(163,270,173,280),(164,269,174,279),(165,268,175,278),(166,267,176,277),(167,266,177,276),(168,265,178,275),(169,264,179,274),(170,263,180,273),(181,222,191,232),(182,221,192,231),(183,240,193,230),(184,239,194,229),(185,238,195,228),(186,237,196,227),(187,236,197,226),(188,235,198,225),(189,234,199,224),(190,233,200,223)], [(1,297,120,257),(2,288,101,248),(3,299,102,259),(4,290,103,250),(5,281,104,241),(6,292,105,252),(7,283,106,243),(8,294,107,254),(9,285,108,245),(10,296,109,256),(11,287,110,247),(12,298,111,258),(13,289,112,249),(14,300,113,260),(15,291,114,251),(16,282,115,242),(17,293,116,253),(18,284,117,244),(19,295,118,255),(20,286,119,246),(21,73,314,143),(22,64,315,154),(23,75,316,145),(24,66,317,156),(25,77,318,147),(26,68,319,158),(27,79,320,149),(28,70,301,160),(29,61,302,151),(30,72,303,142),(31,63,304,153),(32,74,305,144),(33,65,306,155),(34,76,307,146),(35,67,308,157),(36,78,309,148),(37,69,310,159),(38,80,311,150),(39,71,312,141),(40,62,313,152),(41,178,225,100),(42,169,226,91),(43,180,227,82),(44,171,228,93),(45,162,229,84),(46,173,230,95),(47,164,231,86),(48,175,232,97),(49,166,233,88),(50,177,234,99),(51,168,235,90),(52,179,236,81),(53,170,237,92),(54,161,238,83),(55,172,239,94),(56,163,240,85),(57,174,221,96),(58,165,222,87),(59,176,223,98),(60,167,224,89),(121,271,199,217),(122,262,200,208),(123,273,181,219),(124,264,182,210),(125,275,183,201),(126,266,184,212),(127,277,185,203),(128,268,186,214),(129,279,187,205),(130,270,188,216),(131,261,189,207),(132,272,190,218),(133,263,191,209),(134,274,192,220),(135,265,193,211),(136,276,194,202),(137,267,195,213),(138,278,196,204),(139,269,197,215),(140,280,198,206)], [(1,227,120,43),(2,228,101,44),(3,229,102,45),(4,230,103,46),(5,231,104,47),(6,232,105,48),(7,233,106,49),(8,234,107,50),(9,235,108,51),(10,236,109,52),(11,237,110,53),(12,238,111,54),(13,239,112,55),(14,240,113,56),(15,221,114,57),(16,222,115,58),(17,223,116,59),(18,224,117,60),(19,225,118,41),(20,226,119,42),(21,139,314,197),(22,140,315,198),(23,121,316,199),(24,122,317,200),(25,123,318,181),(26,124,319,182),(27,125,320,183),(28,126,301,184),(29,127,302,185),(30,128,303,186),(31,129,304,187),(32,130,305,188),(33,131,306,189),(34,132,307,190),(35,133,308,191),(36,134,309,192),(37,135,310,193),(38,136,311,194),(39,137,312,195),(40,138,313,196),(61,203,151,277),(62,204,152,278),(63,205,153,279),(64,206,154,280),(65,207,155,261),(66,208,156,262),(67,209,157,263),(68,210,158,264),(69,211,159,265),(70,212,160,266),(71,213,141,267),(72,214,142,268),(73,215,143,269),(74,216,144,270),(75,217,145,271),(76,218,146,272),(77,219,147,273),(78,220,148,274),(79,201,149,275),(80,202,150,276),(81,296,179,256),(82,297,180,257),(83,298,161,258),(84,299,162,259),(85,300,163,260),(86,281,164,241),(87,282,165,242),(88,283,166,243),(89,284,167,244),(90,285,168,245),(91,286,169,246),(92,287,170,247),(93,288,171,248),(94,289,172,249),(95,290,173,250),(96,291,174,251),(97,292,175,252),(98,293,176,253),(99,294,177,254),(100,295,178,255)]])
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 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | C8.C22 | D4.D5 | Q8×D5 | Q8⋊2D5 | C20.C23 |
| kernel | Dic10⋊6Q8 | C20⋊3C8 | C20.Q8 | 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⋊6Q8 ►in GL6(𝔽41)
| 40 | 32 | 0 | 0 | 0 | 0 |
| 23 | 1 | 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 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 39 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 26 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [40,23,0,0,0,0,32,1,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,39,0,0,0,0,0,32,0,0,0,0,0,0,32,11,0,0,0,0,30,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,30,0,0,0,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;
Dic10⋊6Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_6Q_8 % in TeX
G:=Group("Dic10:6Q8"); // GroupNames label
G:=SmallGroup(320,721);
// by ID
G=gap.SmallGroup(320,721);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,100,1123,297,136,12550]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations