metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.4Q8, C42.67D10, C4.9(Q8×D5), C5⋊7(Q8.Q8), C4⋊C4.72D10, C20.31(C2×Q8), (C2×C20).273D4, C20⋊3C8.20C2, C42.C2.2D5, C20.69(C4○D4), C10.107(C4○D8), (C2×C20).381C23, (C4×C20).111C22, C4.32(Q8⋊2D5), (C4×Dic10).16C2, C10.Q16.12C2, C10.D8.12C2, C10.73(C22⋊Q8), C20.Q8.13C2, C2.10(D10⋊3Q8), C4⋊Dic5.342C22, C2.20(D4.9D10), C2.26(D4.8D10), C10.121(C8.C22), (C2×Dic10).281C22, (C2×C10).512(C2×D4), (C2×C4).64(C5⋊D4), (C5×C42.C2).1C2, (C5×C4⋊C4).119C22, (C2×C4).479(C22×D5), C22.185(C2×C5⋊D4), (C2×C5⋊2C8).124C22, SmallGroup(320,690)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.4Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=dad-1=a-1, cac-1=a11, cbc-1=a5b, bd=db, dcd-1=a15c-1 >
Subgroups: 278 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, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, Q8.Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C20⋊3C8, C10.D8, C20.Q8, C10.Q16, C4×Dic10, C5×C42.C2, Dic10.4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8.C22, C5⋊D4, C22×D5, Q8.Q8, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D10⋊3Q8, D4.8D10, D4.9D10, Dic10.4Q8
►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)
(1 180 11 170)(2 179 12 169)(3 178 13 168)(4 177 14 167)(5 176 15 166)(6 175 16 165)(7 174 17 164)(8 173 18 163)(9 172 19 162)(10 171 20 161)(21 69 31 79)(22 68 32 78)(23 67 33 77)(24 66 34 76)(25 65 35 75)(26 64 36 74)(27 63 37 73)(28 62 38 72)(29 61 39 71)(30 80 40 70)(41 221 51 231)(42 240 52 230)(43 239 53 229)(44 238 54 228)(45 237 55 227)(46 236 56 226)(47 235 57 225)(48 234 58 224)(49 233 59 223)(50 232 60 222)(81 315 91 305)(82 314 92 304)(83 313 93 303)(84 312 94 302)(85 311 95 301)(86 310 96 320)(87 309 97 319)(88 308 98 318)(89 307 99 317)(90 306 100 316)(101 150 111 160)(102 149 112 159)(103 148 113 158)(104 147 114 157)(105 146 115 156)(106 145 116 155)(107 144 117 154)(108 143 118 153)(109 142 119 152)(110 141 120 151)(121 201 131 211)(122 220 132 210)(123 219 133 209)(124 218 134 208)(125 217 135 207)(126 216 136 206)(127 215 137 205)(128 214 138 204)(129 213 139 203)(130 212 140 202)(181 271 191 261)(182 270 192 280)(183 269 193 279)(184 268 194 278)(185 267 195 277)(186 266 196 276)(187 265 197 275)(188 264 198 274)(189 263 199 273)(190 262 200 272)(241 298 251 288)(242 297 252 287)(243 296 253 286)(244 295 254 285)(245 294 255 284)(246 293 256 283)(247 292 257 282)(248 291 258 281)(249 290 259 300)(250 289 260 299)
(1 304 128 102)(2 315 129 113)(3 306 130 104)(4 317 131 115)(5 308 132 106)(6 319 133 117)(7 310 134 108)(8 301 135 119)(9 312 136 110)(10 303 137 101)(11 314 138 112)(12 305 139 103)(13 316 140 114)(14 307 121 105)(15 318 122 116)(16 309 123 107)(17 320 124 118)(18 311 125 109)(19 302 126 120)(20 313 127 111)(21 271 236 296)(22 262 237 287)(23 273 238 298)(24 264 239 289)(25 275 240 300)(26 266 221 291)(27 277 222 282)(28 268 223 293)(29 279 224 284)(30 270 225 295)(31 261 226 286)(32 272 227 297)(33 263 228 288)(34 274 229 299)(35 265 230 290)(36 276 231 281)(37 267 232 292)(38 278 233 283)(39 269 234 294)(40 280 235 285)(41 243 74 181)(42 254 75 192)(43 245 76 183)(44 256 77 194)(45 247 78 185)(46 258 79 196)(47 249 80 187)(48 260 61 198)(49 251 62 189)(50 242 63 200)(51 253 64 191)(52 244 65 182)(53 255 66 193)(54 246 67 184)(55 257 68 195)(56 248 69 186)(57 259 70 197)(58 250 71 188)(59 241 72 199)(60 252 73 190)(81 218 148 164)(82 209 149 175)(83 220 150 166)(84 211 151 177)(85 202 152 168)(86 213 153 179)(87 204 154 170)(88 215 155 161)(89 206 156 172)(90 217 157 163)(91 208 158 174)(92 219 159 165)(93 210 160 176)(94 201 141 167)(95 212 142 178)(96 203 143 169)(97 214 144 180)(98 205 145 171)(99 216 146 162)(100 207 147 173)
(1 66 128 53)(2 65 129 52)(3 64 130 51)(4 63 131 50)(5 62 132 49)(6 61 133 48)(7 80 134 47)(8 79 135 46)(9 78 136 45)(10 77 137 44)(11 76 138 43)(12 75 139 42)(13 74 140 41)(14 73 121 60)(15 72 122 59)(16 71 123 58)(17 70 124 57)(18 69 125 56)(19 68 126 55)(20 67 127 54)(21 207 236 173)(22 206 237 172)(23 205 238 171)(24 204 239 170)(25 203 240 169)(26 202 221 168)(27 201 222 167)(28 220 223 166)(29 219 224 165)(30 218 225 164)(31 217 226 163)(32 216 227 162)(33 215 228 161)(34 214 229 180)(35 213 230 179)(36 212 231 178)(37 211 232 177)(38 210 233 176)(39 209 234 175)(40 208 235 174)(81 300 148 275)(82 299 149 274)(83 298 150 273)(84 297 151 272)(85 296 152 271)(86 295 153 270)(87 294 154 269)(88 293 155 268)(89 292 156 267)(90 291 157 266)(91 290 158 265)(92 289 159 264)(93 288 160 263)(94 287 141 262)(95 286 142 261)(96 285 143 280)(97 284 144 279)(98 283 145 278)(99 282 146 277)(100 281 147 276)(101 199 303 241)(102 198 304 260)(103 197 305 259)(104 196 306 258)(105 195 307 257)(106 194 308 256)(107 193 309 255)(108 192 310 254)(109 191 311 253)(110 190 312 252)(111 189 313 251)(112 188 314 250)(113 187 315 249)(114 186 316 248)(115 185 317 247)(116 184 318 246)(117 183 319 245)(118 182 320 244)(119 181 301 243)(120 200 302 242)
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), (1,180,11,170)(2,179,12,169)(3,178,13,168)(4,177,14,167)(5,176,15,166)(6,175,16,165)(7,174,17,164)(8,173,18,163)(9,172,19,162)(10,171,20,161)(21,69,31,79)(22,68,32,78)(23,67,33,77)(24,66,34,76)(25,65,35,75)(26,64,36,74)(27,63,37,73)(28,62,38,72)(29,61,39,71)(30,80,40,70)(41,221,51,231)(42,240,52,230)(43,239,53,229)(44,238,54,228)(45,237,55,227)(46,236,56,226)(47,235,57,225)(48,234,58,224)(49,233,59,223)(50,232,60,222)(81,315,91,305)(82,314,92,304)(83,313,93,303)(84,312,94,302)(85,311,95,301)(86,310,96,320)(87,309,97,319)(88,308,98,318)(89,307,99,317)(90,306,100,316)(101,150,111,160)(102,149,112,159)(103,148,113,158)(104,147,114,157)(105,146,115,156)(106,145,116,155)(107,144,117,154)(108,143,118,153)(109,142,119,152)(110,141,120,151)(121,201,131,211)(122,220,132,210)(123,219,133,209)(124,218,134,208)(125,217,135,207)(126,216,136,206)(127,215,137,205)(128,214,138,204)(129,213,139,203)(130,212,140,202)(181,271,191,261)(182,270,192,280)(183,269,193,279)(184,268,194,278)(185,267,195,277)(186,266,196,276)(187,265,197,275)(188,264,198,274)(189,263,199,273)(190,262,200,272)(241,298,251,288)(242,297,252,287)(243,296,253,286)(244,295,254,285)(245,294,255,284)(246,293,256,283)(247,292,257,282)(248,291,258,281)(249,290,259,300)(250,289,260,299), (1,304,128,102)(2,315,129,113)(3,306,130,104)(4,317,131,115)(5,308,132,106)(6,319,133,117)(7,310,134,108)(8,301,135,119)(9,312,136,110)(10,303,137,101)(11,314,138,112)(12,305,139,103)(13,316,140,114)(14,307,121,105)(15,318,122,116)(16,309,123,107)(17,320,124,118)(18,311,125,109)(19,302,126,120)(20,313,127,111)(21,271,236,296)(22,262,237,287)(23,273,238,298)(24,264,239,289)(25,275,240,300)(26,266,221,291)(27,277,222,282)(28,268,223,293)(29,279,224,284)(30,270,225,295)(31,261,226,286)(32,272,227,297)(33,263,228,288)(34,274,229,299)(35,265,230,290)(36,276,231,281)(37,267,232,292)(38,278,233,283)(39,269,234,294)(40,280,235,285)(41,243,74,181)(42,254,75,192)(43,245,76,183)(44,256,77,194)(45,247,78,185)(46,258,79,196)(47,249,80,187)(48,260,61,198)(49,251,62,189)(50,242,63,200)(51,253,64,191)(52,244,65,182)(53,255,66,193)(54,246,67,184)(55,257,68,195)(56,248,69,186)(57,259,70,197)(58,250,71,188)(59,241,72,199)(60,252,73,190)(81,218,148,164)(82,209,149,175)(83,220,150,166)(84,211,151,177)(85,202,152,168)(86,213,153,179)(87,204,154,170)(88,215,155,161)(89,206,156,172)(90,217,157,163)(91,208,158,174)(92,219,159,165)(93,210,160,176)(94,201,141,167)(95,212,142,178)(96,203,143,169)(97,214,144,180)(98,205,145,171)(99,216,146,162)(100,207,147,173), (1,66,128,53)(2,65,129,52)(3,64,130,51)(4,63,131,50)(5,62,132,49)(6,61,133,48)(7,80,134,47)(8,79,135,46)(9,78,136,45)(10,77,137,44)(11,76,138,43)(12,75,139,42)(13,74,140,41)(14,73,121,60)(15,72,122,59)(16,71,123,58)(17,70,124,57)(18,69,125,56)(19,68,126,55)(20,67,127,54)(21,207,236,173)(22,206,237,172)(23,205,238,171)(24,204,239,170)(25,203,240,169)(26,202,221,168)(27,201,222,167)(28,220,223,166)(29,219,224,165)(30,218,225,164)(31,217,226,163)(32,216,227,162)(33,215,228,161)(34,214,229,180)(35,213,230,179)(36,212,231,178)(37,211,232,177)(38,210,233,176)(39,209,234,175)(40,208,235,174)(81,300,148,275)(82,299,149,274)(83,298,150,273)(84,297,151,272)(85,296,152,271)(86,295,153,270)(87,294,154,269)(88,293,155,268)(89,292,156,267)(90,291,157,266)(91,290,158,265)(92,289,159,264)(93,288,160,263)(94,287,141,262)(95,286,142,261)(96,285,143,280)(97,284,144,279)(98,283,145,278)(99,282,146,277)(100,281,147,276)(101,199,303,241)(102,198,304,260)(103,197,305,259)(104,196,306,258)(105,195,307,257)(106,194,308,256)(107,193,309,255)(108,192,310,254)(109,191,311,253)(110,190,312,252)(111,189,313,251)(112,188,314,250)(113,187,315,249)(114,186,316,248)(115,185,317,247)(116,184,318,246)(117,183,319,245)(118,182,320,244)(119,181,301,243)(120,200,302,242)>;
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), (1,180,11,170)(2,179,12,169)(3,178,13,168)(4,177,14,167)(5,176,15,166)(6,175,16,165)(7,174,17,164)(8,173,18,163)(9,172,19,162)(10,171,20,161)(21,69,31,79)(22,68,32,78)(23,67,33,77)(24,66,34,76)(25,65,35,75)(26,64,36,74)(27,63,37,73)(28,62,38,72)(29,61,39,71)(30,80,40,70)(41,221,51,231)(42,240,52,230)(43,239,53,229)(44,238,54,228)(45,237,55,227)(46,236,56,226)(47,235,57,225)(48,234,58,224)(49,233,59,223)(50,232,60,222)(81,315,91,305)(82,314,92,304)(83,313,93,303)(84,312,94,302)(85,311,95,301)(86,310,96,320)(87,309,97,319)(88,308,98,318)(89,307,99,317)(90,306,100,316)(101,150,111,160)(102,149,112,159)(103,148,113,158)(104,147,114,157)(105,146,115,156)(106,145,116,155)(107,144,117,154)(108,143,118,153)(109,142,119,152)(110,141,120,151)(121,201,131,211)(122,220,132,210)(123,219,133,209)(124,218,134,208)(125,217,135,207)(126,216,136,206)(127,215,137,205)(128,214,138,204)(129,213,139,203)(130,212,140,202)(181,271,191,261)(182,270,192,280)(183,269,193,279)(184,268,194,278)(185,267,195,277)(186,266,196,276)(187,265,197,275)(188,264,198,274)(189,263,199,273)(190,262,200,272)(241,298,251,288)(242,297,252,287)(243,296,253,286)(244,295,254,285)(245,294,255,284)(246,293,256,283)(247,292,257,282)(248,291,258,281)(249,290,259,300)(250,289,260,299), (1,304,128,102)(2,315,129,113)(3,306,130,104)(4,317,131,115)(5,308,132,106)(6,319,133,117)(7,310,134,108)(8,301,135,119)(9,312,136,110)(10,303,137,101)(11,314,138,112)(12,305,139,103)(13,316,140,114)(14,307,121,105)(15,318,122,116)(16,309,123,107)(17,320,124,118)(18,311,125,109)(19,302,126,120)(20,313,127,111)(21,271,236,296)(22,262,237,287)(23,273,238,298)(24,264,239,289)(25,275,240,300)(26,266,221,291)(27,277,222,282)(28,268,223,293)(29,279,224,284)(30,270,225,295)(31,261,226,286)(32,272,227,297)(33,263,228,288)(34,274,229,299)(35,265,230,290)(36,276,231,281)(37,267,232,292)(38,278,233,283)(39,269,234,294)(40,280,235,285)(41,243,74,181)(42,254,75,192)(43,245,76,183)(44,256,77,194)(45,247,78,185)(46,258,79,196)(47,249,80,187)(48,260,61,198)(49,251,62,189)(50,242,63,200)(51,253,64,191)(52,244,65,182)(53,255,66,193)(54,246,67,184)(55,257,68,195)(56,248,69,186)(57,259,70,197)(58,250,71,188)(59,241,72,199)(60,252,73,190)(81,218,148,164)(82,209,149,175)(83,220,150,166)(84,211,151,177)(85,202,152,168)(86,213,153,179)(87,204,154,170)(88,215,155,161)(89,206,156,172)(90,217,157,163)(91,208,158,174)(92,219,159,165)(93,210,160,176)(94,201,141,167)(95,212,142,178)(96,203,143,169)(97,214,144,180)(98,205,145,171)(99,216,146,162)(100,207,147,173), (1,66,128,53)(2,65,129,52)(3,64,130,51)(4,63,131,50)(5,62,132,49)(6,61,133,48)(7,80,134,47)(8,79,135,46)(9,78,136,45)(10,77,137,44)(11,76,138,43)(12,75,139,42)(13,74,140,41)(14,73,121,60)(15,72,122,59)(16,71,123,58)(17,70,124,57)(18,69,125,56)(19,68,126,55)(20,67,127,54)(21,207,236,173)(22,206,237,172)(23,205,238,171)(24,204,239,170)(25,203,240,169)(26,202,221,168)(27,201,222,167)(28,220,223,166)(29,219,224,165)(30,218,225,164)(31,217,226,163)(32,216,227,162)(33,215,228,161)(34,214,229,180)(35,213,230,179)(36,212,231,178)(37,211,232,177)(38,210,233,176)(39,209,234,175)(40,208,235,174)(81,300,148,275)(82,299,149,274)(83,298,150,273)(84,297,151,272)(85,296,152,271)(86,295,153,270)(87,294,154,269)(88,293,155,268)(89,292,156,267)(90,291,157,266)(91,290,158,265)(92,289,159,264)(93,288,160,263)(94,287,141,262)(95,286,142,261)(96,285,143,280)(97,284,144,279)(98,283,145,278)(99,282,146,277)(100,281,147,276)(101,199,303,241)(102,198,304,260)(103,197,305,259)(104,196,306,258)(105,195,307,257)(106,194,308,256)(107,193,309,255)(108,192,310,254)(109,191,311,253)(110,190,312,252)(111,189,313,251)(112,188,314,250)(113,187,315,249)(114,186,316,248)(115,185,317,247)(116,184,318,246)(117,183,319,245)(118,182,320,244)(119,181,301,243)(120,200,302,242) );
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)], [(1,180,11,170),(2,179,12,169),(3,178,13,168),(4,177,14,167),(5,176,15,166),(6,175,16,165),(7,174,17,164),(8,173,18,163),(9,172,19,162),(10,171,20,161),(21,69,31,79),(22,68,32,78),(23,67,33,77),(24,66,34,76),(25,65,35,75),(26,64,36,74),(27,63,37,73),(28,62,38,72),(29,61,39,71),(30,80,40,70),(41,221,51,231),(42,240,52,230),(43,239,53,229),(44,238,54,228),(45,237,55,227),(46,236,56,226),(47,235,57,225),(48,234,58,224),(49,233,59,223),(50,232,60,222),(81,315,91,305),(82,314,92,304),(83,313,93,303),(84,312,94,302),(85,311,95,301),(86,310,96,320),(87,309,97,319),(88,308,98,318),(89,307,99,317),(90,306,100,316),(101,150,111,160),(102,149,112,159),(103,148,113,158),(104,147,114,157),(105,146,115,156),(106,145,116,155),(107,144,117,154),(108,143,118,153),(109,142,119,152),(110,141,120,151),(121,201,131,211),(122,220,132,210),(123,219,133,209),(124,218,134,208),(125,217,135,207),(126,216,136,206),(127,215,137,205),(128,214,138,204),(129,213,139,203),(130,212,140,202),(181,271,191,261),(182,270,192,280),(183,269,193,279),(184,268,194,278),(185,267,195,277),(186,266,196,276),(187,265,197,275),(188,264,198,274),(189,263,199,273),(190,262,200,272),(241,298,251,288),(242,297,252,287),(243,296,253,286),(244,295,254,285),(245,294,255,284),(246,293,256,283),(247,292,257,282),(248,291,258,281),(249,290,259,300),(250,289,260,299)], [(1,304,128,102),(2,315,129,113),(3,306,130,104),(4,317,131,115),(5,308,132,106),(6,319,133,117),(7,310,134,108),(8,301,135,119),(9,312,136,110),(10,303,137,101),(11,314,138,112),(12,305,139,103),(13,316,140,114),(14,307,121,105),(15,318,122,116),(16,309,123,107),(17,320,124,118),(18,311,125,109),(19,302,126,120),(20,313,127,111),(21,271,236,296),(22,262,237,287),(23,273,238,298),(24,264,239,289),(25,275,240,300),(26,266,221,291),(27,277,222,282),(28,268,223,293),(29,279,224,284),(30,270,225,295),(31,261,226,286),(32,272,227,297),(33,263,228,288),(34,274,229,299),(35,265,230,290),(36,276,231,281),(37,267,232,292),(38,278,233,283),(39,269,234,294),(40,280,235,285),(41,243,74,181),(42,254,75,192),(43,245,76,183),(44,256,77,194),(45,247,78,185),(46,258,79,196),(47,249,80,187),(48,260,61,198),(49,251,62,189),(50,242,63,200),(51,253,64,191),(52,244,65,182),(53,255,66,193),(54,246,67,184),(55,257,68,195),(56,248,69,186),(57,259,70,197),(58,250,71,188),(59,241,72,199),(60,252,73,190),(81,218,148,164),(82,209,149,175),(83,220,150,166),(84,211,151,177),(85,202,152,168),(86,213,153,179),(87,204,154,170),(88,215,155,161),(89,206,156,172),(90,217,157,163),(91,208,158,174),(92,219,159,165),(93,210,160,176),(94,201,141,167),(95,212,142,178),(96,203,143,169),(97,214,144,180),(98,205,145,171),(99,216,146,162),(100,207,147,173)], [(1,66,128,53),(2,65,129,52),(3,64,130,51),(4,63,131,50),(5,62,132,49),(6,61,133,48),(7,80,134,47),(8,79,135,46),(9,78,136,45),(10,77,137,44),(11,76,138,43),(12,75,139,42),(13,74,140,41),(14,73,121,60),(15,72,122,59),(16,71,123,58),(17,70,124,57),(18,69,125,56),(19,68,126,55),(20,67,127,54),(21,207,236,173),(22,206,237,172),(23,205,238,171),(24,204,239,170),(25,203,240,169),(26,202,221,168),(27,201,222,167),(28,220,223,166),(29,219,224,165),(30,218,225,164),(31,217,226,163),(32,216,227,162),(33,215,228,161),(34,214,229,180),(35,213,230,179),(36,212,231,178),(37,211,232,177),(38,210,233,176),(39,209,234,175),(40,208,235,174),(81,300,148,275),(82,299,149,274),(83,298,150,273),(84,297,151,272),(85,296,152,271),(86,295,153,270),(87,294,154,269),(88,293,155,268),(89,292,156,267),(90,291,157,266),(91,290,158,265),(92,289,159,264),(93,288,160,263),(94,287,141,262),(95,286,142,261),(96,285,143,280),(97,284,144,279),(98,283,145,278),(99,282,146,277),(100,281,147,276),(101,199,303,241),(102,198,304,260),(103,197,305,259),(104,196,306,258),(105,195,307,257),(106,194,308,256),(107,193,309,255),(108,192,310,254),(109,191,311,253),(110,190,312,252),(111,189,313,251),(112,188,314,250),(113,187,315,249),(114,186,316,248),(115,185,317,247),(116,184,318,246),(117,183,319,245),(118,182,320,244),(119,181,301,243),(120,200,302,242)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C5⋊D4 | C8.C22 | Q8×D5 | Q8⋊2D5 | D4.8D10 | D4.9D10 |
| kernel | Dic10.4Q8 | C20⋊3C8 | C10.D8 | C20.Q8 | C10.Q16 | C4×Dic10 | C5×C42.C2 | Dic10 | C2×C20 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C2×C4 | C10 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10.4Q8 ►in GL6(𝔽41)
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 6 | 0 | 0 | 0 | 0 |
| 21 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 16 | 0 | 0 |
| 0 | 0 | 2 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 15 | 34 | 0 | 0 | 0 | 0 |
| 32 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 16 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 16 | 0 | 0 |
| 0 | 0 | 2 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 6 |
| 0 | 0 | 0 | 0 | 6 | 39 |
G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,21,0,0,0,0,6,23,0,0,0,0,0,0,25,2,0,0,0,0,16,16,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[15,32,0,0,0,0,34,26,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[2,16,0,0,0,0,28,39,0,0,0,0,0,0,25,2,0,0,0,0,16,16,0,0,0,0,0,0,2,6,0,0,0,0,6,39] >;
Dic10.4Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._4Q_8 % in TeX
G:=Group("Dic10.4Q8"); // GroupNames label
G:=SmallGroup(320,690);
// by ID
G=gap.SmallGroup(320,690);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,254,219,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^15*c^-1>;
// generators/relations