metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.3Q8, C42.28D10, C4⋊C8.9D5, C5⋊3(Q8.Q8), C4.42(Q8×D5), (C2×C4).37D20, C40⋊5C4.8C2, C40⋊6C4.8C2, (C2×C20).243D4, (C2×C8).127D10, C20.101(C2×Q8), C10.11(C4○D8), (C4×C20).55C22, (C2×C40).21C22, (C4×Dic10).8C2, C20.6Q8.5C2, C20.285(C4○D4), (C2×C20).750C23, C20.44D4.2C2, C22.113(C2×D20), C10.29(C22⋊Q8), C4⋊Dic5.17C22, C4.109(D4⋊2D5), C2.13(D40⋊7C2), C2.10(D10⋊2Q8), C2.16(C8.D10), C10.13(C8.C22), (C2×Dic10).219C22, (C5×C4⋊C8).14C2, (C2×C10).133(C2×D4), (C2×C4).695(C22×D5), SmallGroup(320,456)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.3Q8
G = < a,b,c,d | a20=1, b2=c4=a10, d2=a15c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a15b, bd=db, dcd-1=c3 >
Subgroups: 326 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C40, Dic10, Dic10, C2×Dic5, C2×C20, Q8.Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C20.44D4, C40⋊6C4, C40⋊5C4, C5×C4⋊C8, C4×Dic10, C20.6Q8, Dic10.3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8.C22, D20, C22×D5, Q8.Q8, C2×D20, D4⋊2D5, Q8×D5, D10⋊2Q8, D40⋊7C2, C8.D10, Dic10.3Q8
►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)
(1 244 11 254)(2 243 12 253)(3 242 13 252)(4 241 14 251)(5 260 15 250)(6 259 16 249)(7 258 17 248)(8 257 18 247)(9 256 19 246)(10 255 20 245)(21 304 31 314)(22 303 32 313)(23 302 33 312)(24 301 34 311)(25 320 35 310)(26 319 36 309)(27 318 37 308)(28 317 38 307)(29 316 39 306)(30 315 40 305)(41 191 51 181)(42 190 52 200)(43 189 53 199)(44 188 54 198)(45 187 55 197)(46 186 56 196)(47 185 57 195)(48 184 58 194)(49 183 59 193)(50 182 60 192)(61 129 71 139)(62 128 72 138)(63 127 73 137)(64 126 74 136)(65 125 75 135)(66 124 76 134)(67 123 77 133)(68 122 78 132)(69 121 79 131)(70 140 80 130)(81 233 91 223)(82 232 92 222)(83 231 93 221)(84 230 94 240)(85 229 95 239)(86 228 96 238)(87 227 97 237)(88 226 98 236)(89 225 99 235)(90 224 100 234)(101 293 111 283)(102 292 112 282)(103 291 113 281)(104 290 114 300)(105 289 115 299)(106 288 116 298)(107 287 117 297)(108 286 118 296)(109 285 119 295)(110 284 120 294)(141 272 151 262)(142 271 152 261)(143 270 153 280)(144 269 154 279)(145 268 155 278)(146 267 156 277)(147 266 157 276)(148 265 158 275)(149 264 159 274)(150 263 160 273)(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)
(1 38 60 131 11 28 50 121)(2 39 41 132 12 29 51 122)(3 40 42 133 13 30 52 123)(4 21 43 134 14 31 53 124)(5 22 44 135 15 32 54 125)(6 23 45 136 16 33 55 126)(7 24 46 137 17 34 56 127)(8 25 47 138 18 35 57 128)(9 26 48 139 19 36 58 129)(10 27 49 140 20 37 59 130)(61 241 319 199 71 251 309 189)(62 242 320 200 72 252 310 190)(63 243 301 181 73 253 311 191)(64 244 302 182 74 254 312 192)(65 245 303 183 75 255 313 193)(66 246 304 184 76 256 314 194)(67 247 305 185 77 257 315 195)(68 248 306 186 78 258 316 196)(69 249 307 187 79 259 317 197)(70 250 308 188 80 260 318 198)(81 170 284 143 91 180 294 153)(82 171 285 144 92 161 295 154)(83 172 286 145 93 162 296 155)(84 173 287 146 94 163 297 156)(85 174 288 147 95 164 298 157)(86 175 289 148 96 165 299 158)(87 176 290 149 97 166 300 159)(88 177 291 150 98 167 281 160)(89 178 292 151 99 168 282 141)(90 179 293 152 100 169 283 142)(101 266 234 212 111 276 224 202)(102 267 235 213 112 277 225 203)(103 268 236 214 113 278 226 204)(104 269 237 215 114 279 227 205)(105 270 238 216 115 280 228 206)(106 271 239 217 116 261 229 207)(107 272 240 218 117 262 230 208)(108 273 221 219 118 263 231 209)(109 274 222 220 119 264 232 210)(110 275 223 201 120 265 233 211)
(1 230 55 102)(2 229 56 101)(3 228 57 120)(4 227 58 119)(5 226 59 118)(6 225 60 117)(7 224 41 116)(8 223 42 115)(9 222 43 114)(10 221 44 113)(11 240 45 112)(12 239 46 111)(13 238 47 110)(14 237 48 109)(15 236 49 108)(16 235 50 107)(17 234 51 106)(18 233 52 105)(19 232 53 104)(20 231 54 103)(21 269 129 210)(22 268 130 209)(23 267 131 208)(24 266 132 207)(25 265 133 206)(26 264 134 205)(27 263 135 204)(28 262 136 203)(29 261 137 202)(30 280 138 201)(31 279 139 220)(32 278 140 219)(33 277 121 218)(34 276 122 217)(35 275 123 216)(36 274 124 215)(37 273 125 214)(38 272 126 213)(39 271 127 212)(40 270 128 211)(61 176 314 144)(62 175 315 143)(63 174 316 142)(64 173 317 141)(65 172 318 160)(66 171 319 159)(67 170 320 158)(68 169 301 157)(69 168 302 156)(70 167 303 155)(71 166 304 154)(72 165 305 153)(73 164 306 152)(74 163 307 151)(75 162 308 150)(76 161 309 149)(77 180 310 148)(78 179 311 147)(79 178 312 146)(80 177 313 145)(81 190 299 257)(82 189 300 256)(83 188 281 255)(84 187 282 254)(85 186 283 253)(86 185 284 252)(87 184 285 251)(88 183 286 250)(89 182 287 249)(90 181 288 248)(91 200 289 247)(92 199 290 246)(93 198 291 245)(94 197 292 244)(95 196 293 243)(96 195 294 242)(97 194 295 241)(98 193 296 260)(99 192 297 259)(100 191 298 258)
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), (1,244,11,254)(2,243,12,253)(3,242,13,252)(4,241,14,251)(5,260,15,250)(6,259,16,249)(7,258,17,248)(8,257,18,247)(9,256,19,246)(10,255,20,245)(21,304,31,314)(22,303,32,313)(23,302,33,312)(24,301,34,311)(25,320,35,310)(26,319,36,309)(27,318,37,308)(28,317,38,307)(29,316,39,306)(30,315,40,305)(41,191,51,181)(42,190,52,200)(43,189,53,199)(44,188,54,198)(45,187,55,197)(46,186,56,196)(47,185,57,195)(48,184,58,194)(49,183,59,193)(50,182,60,192)(61,129,71,139)(62,128,72,138)(63,127,73,137)(64,126,74,136)(65,125,75,135)(66,124,76,134)(67,123,77,133)(68,122,78,132)(69,121,79,131)(70,140,80,130)(81,233,91,223)(82,232,92,222)(83,231,93,221)(84,230,94,240)(85,229,95,239)(86,228,96,238)(87,227,97,237)(88,226,98,236)(89,225,99,235)(90,224,100,234)(101,293,111,283)(102,292,112,282)(103,291,113,281)(104,290,114,300)(105,289,115,299)(106,288,116,298)(107,287,117,297)(108,286,118,296)(109,285,119,295)(110,284,120,294)(141,272,151,262)(142,271,152,261)(143,270,153,280)(144,269,154,279)(145,268,155,278)(146,267,156,277)(147,266,157,276)(148,265,158,275)(149,264,159,274)(150,263,160,273)(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), (1,38,60,131,11,28,50,121)(2,39,41,132,12,29,51,122)(3,40,42,133,13,30,52,123)(4,21,43,134,14,31,53,124)(5,22,44,135,15,32,54,125)(6,23,45,136,16,33,55,126)(7,24,46,137,17,34,56,127)(8,25,47,138,18,35,57,128)(9,26,48,139,19,36,58,129)(10,27,49,140,20,37,59,130)(61,241,319,199,71,251,309,189)(62,242,320,200,72,252,310,190)(63,243,301,181,73,253,311,191)(64,244,302,182,74,254,312,192)(65,245,303,183,75,255,313,193)(66,246,304,184,76,256,314,194)(67,247,305,185,77,257,315,195)(68,248,306,186,78,258,316,196)(69,249,307,187,79,259,317,197)(70,250,308,188,80,260,318,198)(81,170,284,143,91,180,294,153)(82,171,285,144,92,161,295,154)(83,172,286,145,93,162,296,155)(84,173,287,146,94,163,297,156)(85,174,288,147,95,164,298,157)(86,175,289,148,96,165,299,158)(87,176,290,149,97,166,300,159)(88,177,291,150,98,167,281,160)(89,178,292,151,99,168,282,141)(90,179,293,152,100,169,283,142)(101,266,234,212,111,276,224,202)(102,267,235,213,112,277,225,203)(103,268,236,214,113,278,226,204)(104,269,237,215,114,279,227,205)(105,270,238,216,115,280,228,206)(106,271,239,217,116,261,229,207)(107,272,240,218,117,262,230,208)(108,273,221,219,118,263,231,209)(109,274,222,220,119,264,232,210)(110,275,223,201,120,265,233,211), (1,230,55,102)(2,229,56,101)(3,228,57,120)(4,227,58,119)(5,226,59,118)(6,225,60,117)(7,224,41,116)(8,223,42,115)(9,222,43,114)(10,221,44,113)(11,240,45,112)(12,239,46,111)(13,238,47,110)(14,237,48,109)(15,236,49,108)(16,235,50,107)(17,234,51,106)(18,233,52,105)(19,232,53,104)(20,231,54,103)(21,269,129,210)(22,268,130,209)(23,267,131,208)(24,266,132,207)(25,265,133,206)(26,264,134,205)(27,263,135,204)(28,262,136,203)(29,261,137,202)(30,280,138,201)(31,279,139,220)(32,278,140,219)(33,277,121,218)(34,276,122,217)(35,275,123,216)(36,274,124,215)(37,273,125,214)(38,272,126,213)(39,271,127,212)(40,270,128,211)(61,176,314,144)(62,175,315,143)(63,174,316,142)(64,173,317,141)(65,172,318,160)(66,171,319,159)(67,170,320,158)(68,169,301,157)(69,168,302,156)(70,167,303,155)(71,166,304,154)(72,165,305,153)(73,164,306,152)(74,163,307,151)(75,162,308,150)(76,161,309,149)(77,180,310,148)(78,179,311,147)(79,178,312,146)(80,177,313,145)(81,190,299,257)(82,189,300,256)(83,188,281,255)(84,187,282,254)(85,186,283,253)(86,185,284,252)(87,184,285,251)(88,183,286,250)(89,182,287,249)(90,181,288,248)(91,200,289,247)(92,199,290,246)(93,198,291,245)(94,197,292,244)(95,196,293,243)(96,195,294,242)(97,194,295,241)(98,193,296,260)(99,192,297,259)(100,191,298,258)>;
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), (1,244,11,254)(2,243,12,253)(3,242,13,252)(4,241,14,251)(5,260,15,250)(6,259,16,249)(7,258,17,248)(8,257,18,247)(9,256,19,246)(10,255,20,245)(21,304,31,314)(22,303,32,313)(23,302,33,312)(24,301,34,311)(25,320,35,310)(26,319,36,309)(27,318,37,308)(28,317,38,307)(29,316,39,306)(30,315,40,305)(41,191,51,181)(42,190,52,200)(43,189,53,199)(44,188,54,198)(45,187,55,197)(46,186,56,196)(47,185,57,195)(48,184,58,194)(49,183,59,193)(50,182,60,192)(61,129,71,139)(62,128,72,138)(63,127,73,137)(64,126,74,136)(65,125,75,135)(66,124,76,134)(67,123,77,133)(68,122,78,132)(69,121,79,131)(70,140,80,130)(81,233,91,223)(82,232,92,222)(83,231,93,221)(84,230,94,240)(85,229,95,239)(86,228,96,238)(87,227,97,237)(88,226,98,236)(89,225,99,235)(90,224,100,234)(101,293,111,283)(102,292,112,282)(103,291,113,281)(104,290,114,300)(105,289,115,299)(106,288,116,298)(107,287,117,297)(108,286,118,296)(109,285,119,295)(110,284,120,294)(141,272,151,262)(142,271,152,261)(143,270,153,280)(144,269,154,279)(145,268,155,278)(146,267,156,277)(147,266,157,276)(148,265,158,275)(149,264,159,274)(150,263,160,273)(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), (1,38,60,131,11,28,50,121)(2,39,41,132,12,29,51,122)(3,40,42,133,13,30,52,123)(4,21,43,134,14,31,53,124)(5,22,44,135,15,32,54,125)(6,23,45,136,16,33,55,126)(7,24,46,137,17,34,56,127)(8,25,47,138,18,35,57,128)(9,26,48,139,19,36,58,129)(10,27,49,140,20,37,59,130)(61,241,319,199,71,251,309,189)(62,242,320,200,72,252,310,190)(63,243,301,181,73,253,311,191)(64,244,302,182,74,254,312,192)(65,245,303,183,75,255,313,193)(66,246,304,184,76,256,314,194)(67,247,305,185,77,257,315,195)(68,248,306,186,78,258,316,196)(69,249,307,187,79,259,317,197)(70,250,308,188,80,260,318,198)(81,170,284,143,91,180,294,153)(82,171,285,144,92,161,295,154)(83,172,286,145,93,162,296,155)(84,173,287,146,94,163,297,156)(85,174,288,147,95,164,298,157)(86,175,289,148,96,165,299,158)(87,176,290,149,97,166,300,159)(88,177,291,150,98,167,281,160)(89,178,292,151,99,168,282,141)(90,179,293,152,100,169,283,142)(101,266,234,212,111,276,224,202)(102,267,235,213,112,277,225,203)(103,268,236,214,113,278,226,204)(104,269,237,215,114,279,227,205)(105,270,238,216,115,280,228,206)(106,271,239,217,116,261,229,207)(107,272,240,218,117,262,230,208)(108,273,221,219,118,263,231,209)(109,274,222,220,119,264,232,210)(110,275,223,201,120,265,233,211), (1,230,55,102)(2,229,56,101)(3,228,57,120)(4,227,58,119)(5,226,59,118)(6,225,60,117)(7,224,41,116)(8,223,42,115)(9,222,43,114)(10,221,44,113)(11,240,45,112)(12,239,46,111)(13,238,47,110)(14,237,48,109)(15,236,49,108)(16,235,50,107)(17,234,51,106)(18,233,52,105)(19,232,53,104)(20,231,54,103)(21,269,129,210)(22,268,130,209)(23,267,131,208)(24,266,132,207)(25,265,133,206)(26,264,134,205)(27,263,135,204)(28,262,136,203)(29,261,137,202)(30,280,138,201)(31,279,139,220)(32,278,140,219)(33,277,121,218)(34,276,122,217)(35,275,123,216)(36,274,124,215)(37,273,125,214)(38,272,126,213)(39,271,127,212)(40,270,128,211)(61,176,314,144)(62,175,315,143)(63,174,316,142)(64,173,317,141)(65,172,318,160)(66,171,319,159)(67,170,320,158)(68,169,301,157)(69,168,302,156)(70,167,303,155)(71,166,304,154)(72,165,305,153)(73,164,306,152)(74,163,307,151)(75,162,308,150)(76,161,309,149)(77,180,310,148)(78,179,311,147)(79,178,312,146)(80,177,313,145)(81,190,299,257)(82,189,300,256)(83,188,281,255)(84,187,282,254)(85,186,283,253)(86,185,284,252)(87,184,285,251)(88,183,286,250)(89,182,287,249)(90,181,288,248)(91,200,289,247)(92,199,290,246)(93,198,291,245)(94,197,292,244)(95,196,293,243)(96,195,294,242)(97,194,295,241)(98,193,296,260)(99,192,297,259)(100,191,298,258) );
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)], [(1,244,11,254),(2,243,12,253),(3,242,13,252),(4,241,14,251),(5,260,15,250),(6,259,16,249),(7,258,17,248),(8,257,18,247),(9,256,19,246),(10,255,20,245),(21,304,31,314),(22,303,32,313),(23,302,33,312),(24,301,34,311),(25,320,35,310),(26,319,36,309),(27,318,37,308),(28,317,38,307),(29,316,39,306),(30,315,40,305),(41,191,51,181),(42,190,52,200),(43,189,53,199),(44,188,54,198),(45,187,55,197),(46,186,56,196),(47,185,57,195),(48,184,58,194),(49,183,59,193),(50,182,60,192),(61,129,71,139),(62,128,72,138),(63,127,73,137),(64,126,74,136),(65,125,75,135),(66,124,76,134),(67,123,77,133),(68,122,78,132),(69,121,79,131),(70,140,80,130),(81,233,91,223),(82,232,92,222),(83,231,93,221),(84,230,94,240),(85,229,95,239),(86,228,96,238),(87,227,97,237),(88,226,98,236),(89,225,99,235),(90,224,100,234),(101,293,111,283),(102,292,112,282),(103,291,113,281),(104,290,114,300),(105,289,115,299),(106,288,116,298),(107,287,117,297),(108,286,118,296),(109,285,119,295),(110,284,120,294),(141,272,151,262),(142,271,152,261),(143,270,153,280),(144,269,154,279),(145,268,155,278),(146,267,156,277),(147,266,157,276),(148,265,158,275),(149,264,159,274),(150,263,160,273),(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)], [(1,38,60,131,11,28,50,121),(2,39,41,132,12,29,51,122),(3,40,42,133,13,30,52,123),(4,21,43,134,14,31,53,124),(5,22,44,135,15,32,54,125),(6,23,45,136,16,33,55,126),(7,24,46,137,17,34,56,127),(8,25,47,138,18,35,57,128),(9,26,48,139,19,36,58,129),(10,27,49,140,20,37,59,130),(61,241,319,199,71,251,309,189),(62,242,320,200,72,252,310,190),(63,243,301,181,73,253,311,191),(64,244,302,182,74,254,312,192),(65,245,303,183,75,255,313,193),(66,246,304,184,76,256,314,194),(67,247,305,185,77,257,315,195),(68,248,306,186,78,258,316,196),(69,249,307,187,79,259,317,197),(70,250,308,188,80,260,318,198),(81,170,284,143,91,180,294,153),(82,171,285,144,92,161,295,154),(83,172,286,145,93,162,296,155),(84,173,287,146,94,163,297,156),(85,174,288,147,95,164,298,157),(86,175,289,148,96,165,299,158),(87,176,290,149,97,166,300,159),(88,177,291,150,98,167,281,160),(89,178,292,151,99,168,282,141),(90,179,293,152,100,169,283,142),(101,266,234,212,111,276,224,202),(102,267,235,213,112,277,225,203),(103,268,236,214,113,278,226,204),(104,269,237,215,114,279,227,205),(105,270,238,216,115,280,228,206),(106,271,239,217,116,261,229,207),(107,272,240,218,117,262,230,208),(108,273,221,219,118,263,231,209),(109,274,222,220,119,264,232,210),(110,275,223,201,120,265,233,211)], [(1,230,55,102),(2,229,56,101),(3,228,57,120),(4,227,58,119),(5,226,59,118),(6,225,60,117),(7,224,41,116),(8,223,42,115),(9,222,43,114),(10,221,44,113),(11,240,45,112),(12,239,46,111),(13,238,47,110),(14,237,48,109),(15,236,49,108),(16,235,50,107),(17,234,51,106),(18,233,52,105),(19,232,53,104),(20,231,54,103),(21,269,129,210),(22,268,130,209),(23,267,131,208),(24,266,132,207),(25,265,133,206),(26,264,134,205),(27,263,135,204),(28,262,136,203),(29,261,137,202),(30,280,138,201),(31,279,139,220),(32,278,140,219),(33,277,121,218),(34,276,122,217),(35,275,123,216),(36,274,124,215),(37,273,125,214),(38,272,126,213),(39,271,127,212),(40,270,128,211),(61,176,314,144),(62,175,315,143),(63,174,316,142),(64,173,317,141),(65,172,318,160),(66,171,319,159),(67,170,320,158),(68,169,301,157),(69,168,302,156),(70,167,303,155),(71,166,304,154),(72,165,305,153),(73,164,306,152),(74,163,307,151),(75,162,308,150),(76,161,309,149),(77,180,310,148),(78,179,311,147),(79,178,312,146),(80,177,313,145),(81,190,299,257),(82,189,300,256),(83,188,281,255),(84,187,282,254),(85,186,283,253),(86,185,284,252),(87,184,285,251),(88,183,286,250),(89,182,287,249),(90,181,288,248),(91,200,289,247),(92,199,290,246),(93,198,291,245),(94,197,292,244),(95,196,293,243),(96,195,294,242),(97,194,295,241),(98,193,296,260),(99,192,297,259),(100,191,298,258)]])
59 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 | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| 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 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D40⋊7C2 | C8.C22 | D4⋊2D5 | Q8×D5 | C8.D10 |
| kernel | Dic10.3Q8 | C20.44D4 | C40⋊6C4 | C40⋊5C4 | C5×C4⋊C8 | C4×Dic10 | C20.6Q8 | Dic10 | C2×C20 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C2 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of Dic10.3Q8 ►in GL4(𝔽41) generated by
| 28 | 39 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 21 | 0 | 0 |
| 18 | 21 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 26 | 10 | 0 | 0 |
| 31 | 4 | 0 | 0 |
| 0 | 0 | 19 | 5 |
| 0 | 0 | 26 | 22 |
| 16 | 25 | 0 | 0 |
| 39 | 25 | 0 | 0 |
| 0 | 0 | 30 | 30 |
| 0 | 0 | 26 | 11 |
G:=sub<GL(4,GF(41))| [28,2,0,0,39,16,0,0,0,0,1,0,0,0,0,1],[20,18,0,0,21,21,0,0,0,0,1,0,0,0,0,1],[26,31,0,0,10,4,0,0,0,0,19,26,0,0,5,22],[16,39,0,0,25,25,0,0,0,0,30,26,0,0,30,11] >;
Dic10.3Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._3Q_8 % in TeX
G:=Group("Dic10.3Q8"); // GroupNames label
G:=SmallGroup(320,456);
// by ID
G=gap.SmallGroup(320,456);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=1,b^2=c^4=a^10,d^2=a^15*c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations