metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊4Q8, C20.14SD16, C42.42D10, C4⋊C8.10D5, C4.47(Q8×D5), C5⋊2(Q8⋊Q8), C40⋊6C4.9C2, (C2×C8).134D10, (C2×C4).138D20, (C2×C20).127D4, C4.8(C40⋊C2), C20.106(C2×Q8), (C4×C20).77C22, C20⋊2Q8.13C2, C10.13(C2×SD16), C20.290(C4○D4), (C2×C40).141C22, (C2×C20).761C23, (C4×Dic10).13C2, C20.44D4.5C2, C22.124(C2×D20), C10.34(C22⋊Q8), C4⋊Dic5.22C22, C4.114(D4⋊2D5), C2.15(D10⋊2Q8), C2.21(C8.D10), C10.18(C8.C22), (C2×Dic10).223C22, (C5×C4⋊C8).15C2, C2.16(C2×C40⋊C2), (C2×C10).144(C2×D4), (C2×C4).706(C22×D5), SmallGroup(320,478)
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=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >
Subgroups: 374 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, C2×C8, C2×Q8, Dic5, C20, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C40, Dic10, Dic10, C2×Dic5, C2×C20, Q8⋊Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×Dic10, C20.44D4, C40⋊6C4, C5×C4⋊C8, C4×Dic10, C20⋊2Q8, Dic10⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8.C22, D20, C22×D5, Q8⋊Q8, C40⋊C2, C2×D20, D4⋊2D5, Q8×D5, D10⋊2Q8, C2×C40⋊C2, C8.D10, 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)
(1 248 11 258)(2 247 12 257)(3 246 13 256)(4 245 14 255)(5 244 15 254)(6 243 16 253)(7 242 17 252)(8 241 18 251)(9 260 19 250)(10 259 20 249)(21 125 31 135)(22 124 32 134)(23 123 33 133)(24 122 34 132)(25 121 35 131)(26 140 36 130)(27 139 37 129)(28 138 38 128)(29 137 39 127)(30 136 40 126)(41 113 51 103)(42 112 52 102)(43 111 53 101)(44 110 54 120)(45 109 55 119)(46 108 56 118)(47 107 57 117)(48 106 58 116)(49 105 59 115)(50 104 60 114)(61 317 71 307)(62 316 72 306)(63 315 73 305)(64 314 74 304)(65 313 75 303)(66 312 76 302)(67 311 77 301)(68 310 78 320)(69 309 79 319)(70 308 80 318)(81 266 91 276)(82 265 92 275)(83 264 93 274)(84 263 94 273)(85 262 95 272)(86 261 96 271)(87 280 97 270)(88 279 98 269)(89 278 99 268)(90 277 100 267)(141 229 151 239)(142 228 152 238)(143 227 153 237)(144 226 154 236)(145 225 155 235)(146 224 156 234)(147 223 157 233)(148 222 158 232)(149 221 159 231)(150 240 160 230)(161 291 171 281)(162 290 172 300)(163 289 173 299)(164 288 174 298)(165 287 175 297)(166 286 176 296)(167 285 177 295)(168 284 178 294)(169 283 179 293)(170 282 180 292)(181 218 191 208)(182 217 192 207)(183 216 193 206)(184 215 194 205)(185 214 195 204)(186 213 196 203)(187 212 197 202)(188 211 198 201)(189 210 199 220)(190 209 200 219)
(1 171 210 79)(2 170 211 78)(3 169 212 77)(4 168 213 76)(5 167 214 75)(6 166 215 74)(7 165 216 73)(8 164 217 72)(9 163 218 71)(10 162 219 70)(11 161 220 69)(12 180 201 68)(13 179 202 67)(14 178 203 66)(15 177 204 65)(16 176 205 64)(17 175 206 63)(18 174 207 62)(19 173 208 61)(20 172 209 80)(21 160 274 108)(22 159 275 107)(23 158 276 106)(24 157 277 105)(25 156 278 104)(26 155 279 103)(27 154 280 102)(28 153 261 101)(29 152 262 120)(30 151 263 119)(31 150 264 118)(32 149 265 117)(33 148 266 116)(34 147 267 115)(35 146 268 114)(36 145 269 113)(37 144 270 112)(38 143 271 111)(39 142 272 110)(40 141 273 109)(41 135 235 93)(42 134 236 92)(43 133 237 91)(44 132 238 90)(45 131 239 89)(46 130 240 88)(47 129 221 87)(48 128 222 86)(49 127 223 85)(50 126 224 84)(51 125 225 83)(52 124 226 82)(53 123 227 81)(54 122 228 100)(55 121 229 99)(56 140 230 98)(57 139 231 97)(58 138 232 96)(59 137 233 95)(60 136 234 94)(181 312 250 294)(182 311 251 293)(183 310 252 292)(184 309 253 291)(185 308 254 290)(186 307 255 289)(187 306 256 288)(188 305 257 287)(189 304 258 286)(190 303 259 285)(191 302 260 284)(192 301 241 283)(193 320 242 282)(194 319 243 281)(195 318 244 300)(196 317 245 299)(197 316 246 298)(198 315 247 297)(199 314 248 296)(200 313 249 295)
(1 264 210 31)(2 265 211 32)(3 266 212 33)(4 267 213 34)(5 268 214 35)(6 269 215 36)(7 270 216 37)(8 271 217 38)(9 272 218 39)(10 273 219 40)(11 274 220 21)(12 275 201 22)(13 276 202 23)(14 277 203 24)(15 278 204 25)(16 279 205 26)(17 280 206 27)(18 261 207 28)(19 262 208 29)(20 263 209 30)(41 296 235 314)(42 297 236 315)(43 298 237 316)(44 299 238 317)(45 300 239 318)(46 281 240 319)(47 282 221 320)(48 283 222 301)(49 284 223 302)(50 285 224 303)(51 286 225 304)(52 287 226 305)(53 288 227 306)(54 289 228 307)(55 290 229 308)(56 291 230 309)(57 292 231 310)(58 293 232 311)(59 294 233 312)(60 295 234 313)(61 120 173 152)(62 101 174 153)(63 102 175 154)(64 103 176 155)(65 104 177 156)(66 105 178 157)(67 106 179 158)(68 107 180 159)(69 108 161 160)(70 109 162 141)(71 110 163 142)(72 111 164 143)(73 112 165 144)(74 113 166 145)(75 114 167 146)(76 115 168 147)(77 116 169 148)(78 117 170 149)(79 118 171 150)(80 119 172 151)(81 187 123 256)(82 188 124 257)(83 189 125 258)(84 190 126 259)(85 191 127 260)(86 192 128 241)(87 193 129 242)(88 194 130 243)(89 195 131 244)(90 196 132 245)(91 197 133 246)(92 198 134 247)(93 199 135 248)(94 200 136 249)(95 181 137 250)(96 182 138 251)(97 183 139 252)(98 184 140 253)(99 185 121 254)(100 186 122 255)
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), (1,248,11,258)(2,247,12,257)(3,246,13,256)(4,245,14,255)(5,244,15,254)(6,243,16,253)(7,242,17,252)(8,241,18,251)(9,260,19,250)(10,259,20,249)(21,125,31,135)(22,124,32,134)(23,123,33,133)(24,122,34,132)(25,121,35,131)(26,140,36,130)(27,139,37,129)(28,138,38,128)(29,137,39,127)(30,136,40,126)(41,113,51,103)(42,112,52,102)(43,111,53,101)(44,110,54,120)(45,109,55,119)(46,108,56,118)(47,107,57,117)(48,106,58,116)(49,105,59,115)(50,104,60,114)(61,317,71,307)(62,316,72,306)(63,315,73,305)(64,314,74,304)(65,313,75,303)(66,312,76,302)(67,311,77,301)(68,310,78,320)(69,309,79,319)(70,308,80,318)(81,266,91,276)(82,265,92,275)(83,264,93,274)(84,263,94,273)(85,262,95,272)(86,261,96,271)(87,280,97,270)(88,279,98,269)(89,278,99,268)(90,277,100,267)(141,229,151,239)(142,228,152,238)(143,227,153,237)(144,226,154,236)(145,225,155,235)(146,224,156,234)(147,223,157,233)(148,222,158,232)(149,221,159,231)(150,240,160,230)(161,291,171,281)(162,290,172,300)(163,289,173,299)(164,288,174,298)(165,287,175,297)(166,286,176,296)(167,285,177,295)(168,284,178,294)(169,283,179,293)(170,282,180,292)(181,218,191,208)(182,217,192,207)(183,216,193,206)(184,215,194,205)(185,214,195,204)(186,213,196,203)(187,212,197,202)(188,211,198,201)(189,210,199,220)(190,209,200,219), (1,171,210,79)(2,170,211,78)(3,169,212,77)(4,168,213,76)(5,167,214,75)(6,166,215,74)(7,165,216,73)(8,164,217,72)(9,163,218,71)(10,162,219,70)(11,161,220,69)(12,180,201,68)(13,179,202,67)(14,178,203,66)(15,177,204,65)(16,176,205,64)(17,175,206,63)(18,174,207,62)(19,173,208,61)(20,172,209,80)(21,160,274,108)(22,159,275,107)(23,158,276,106)(24,157,277,105)(25,156,278,104)(26,155,279,103)(27,154,280,102)(28,153,261,101)(29,152,262,120)(30,151,263,119)(31,150,264,118)(32,149,265,117)(33,148,266,116)(34,147,267,115)(35,146,268,114)(36,145,269,113)(37,144,270,112)(38,143,271,111)(39,142,272,110)(40,141,273,109)(41,135,235,93)(42,134,236,92)(43,133,237,91)(44,132,238,90)(45,131,239,89)(46,130,240,88)(47,129,221,87)(48,128,222,86)(49,127,223,85)(50,126,224,84)(51,125,225,83)(52,124,226,82)(53,123,227,81)(54,122,228,100)(55,121,229,99)(56,140,230,98)(57,139,231,97)(58,138,232,96)(59,137,233,95)(60,136,234,94)(181,312,250,294)(182,311,251,293)(183,310,252,292)(184,309,253,291)(185,308,254,290)(186,307,255,289)(187,306,256,288)(188,305,257,287)(189,304,258,286)(190,303,259,285)(191,302,260,284)(192,301,241,283)(193,320,242,282)(194,319,243,281)(195,318,244,300)(196,317,245,299)(197,316,246,298)(198,315,247,297)(199,314,248,296)(200,313,249,295), (1,264,210,31)(2,265,211,32)(3,266,212,33)(4,267,213,34)(5,268,214,35)(6,269,215,36)(7,270,216,37)(8,271,217,38)(9,272,218,39)(10,273,219,40)(11,274,220,21)(12,275,201,22)(13,276,202,23)(14,277,203,24)(15,278,204,25)(16,279,205,26)(17,280,206,27)(18,261,207,28)(19,262,208,29)(20,263,209,30)(41,296,235,314)(42,297,236,315)(43,298,237,316)(44,299,238,317)(45,300,239,318)(46,281,240,319)(47,282,221,320)(48,283,222,301)(49,284,223,302)(50,285,224,303)(51,286,225,304)(52,287,226,305)(53,288,227,306)(54,289,228,307)(55,290,229,308)(56,291,230,309)(57,292,231,310)(58,293,232,311)(59,294,233,312)(60,295,234,313)(61,120,173,152)(62,101,174,153)(63,102,175,154)(64,103,176,155)(65,104,177,156)(66,105,178,157)(67,106,179,158)(68,107,180,159)(69,108,161,160)(70,109,162,141)(71,110,163,142)(72,111,164,143)(73,112,165,144)(74,113,166,145)(75,114,167,146)(76,115,168,147)(77,116,169,148)(78,117,170,149)(79,118,171,150)(80,119,172,151)(81,187,123,256)(82,188,124,257)(83,189,125,258)(84,190,126,259)(85,191,127,260)(86,192,128,241)(87,193,129,242)(88,194,130,243)(89,195,131,244)(90,196,132,245)(91,197,133,246)(92,198,134,247)(93,199,135,248)(94,200,136,249)(95,181,137,250)(96,182,138,251)(97,183,139,252)(98,184,140,253)(99,185,121,254)(100,186,122,255)>;
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), (1,248,11,258)(2,247,12,257)(3,246,13,256)(4,245,14,255)(5,244,15,254)(6,243,16,253)(7,242,17,252)(8,241,18,251)(9,260,19,250)(10,259,20,249)(21,125,31,135)(22,124,32,134)(23,123,33,133)(24,122,34,132)(25,121,35,131)(26,140,36,130)(27,139,37,129)(28,138,38,128)(29,137,39,127)(30,136,40,126)(41,113,51,103)(42,112,52,102)(43,111,53,101)(44,110,54,120)(45,109,55,119)(46,108,56,118)(47,107,57,117)(48,106,58,116)(49,105,59,115)(50,104,60,114)(61,317,71,307)(62,316,72,306)(63,315,73,305)(64,314,74,304)(65,313,75,303)(66,312,76,302)(67,311,77,301)(68,310,78,320)(69,309,79,319)(70,308,80,318)(81,266,91,276)(82,265,92,275)(83,264,93,274)(84,263,94,273)(85,262,95,272)(86,261,96,271)(87,280,97,270)(88,279,98,269)(89,278,99,268)(90,277,100,267)(141,229,151,239)(142,228,152,238)(143,227,153,237)(144,226,154,236)(145,225,155,235)(146,224,156,234)(147,223,157,233)(148,222,158,232)(149,221,159,231)(150,240,160,230)(161,291,171,281)(162,290,172,300)(163,289,173,299)(164,288,174,298)(165,287,175,297)(166,286,176,296)(167,285,177,295)(168,284,178,294)(169,283,179,293)(170,282,180,292)(181,218,191,208)(182,217,192,207)(183,216,193,206)(184,215,194,205)(185,214,195,204)(186,213,196,203)(187,212,197,202)(188,211,198,201)(189,210,199,220)(190,209,200,219), (1,171,210,79)(2,170,211,78)(3,169,212,77)(4,168,213,76)(5,167,214,75)(6,166,215,74)(7,165,216,73)(8,164,217,72)(9,163,218,71)(10,162,219,70)(11,161,220,69)(12,180,201,68)(13,179,202,67)(14,178,203,66)(15,177,204,65)(16,176,205,64)(17,175,206,63)(18,174,207,62)(19,173,208,61)(20,172,209,80)(21,160,274,108)(22,159,275,107)(23,158,276,106)(24,157,277,105)(25,156,278,104)(26,155,279,103)(27,154,280,102)(28,153,261,101)(29,152,262,120)(30,151,263,119)(31,150,264,118)(32,149,265,117)(33,148,266,116)(34,147,267,115)(35,146,268,114)(36,145,269,113)(37,144,270,112)(38,143,271,111)(39,142,272,110)(40,141,273,109)(41,135,235,93)(42,134,236,92)(43,133,237,91)(44,132,238,90)(45,131,239,89)(46,130,240,88)(47,129,221,87)(48,128,222,86)(49,127,223,85)(50,126,224,84)(51,125,225,83)(52,124,226,82)(53,123,227,81)(54,122,228,100)(55,121,229,99)(56,140,230,98)(57,139,231,97)(58,138,232,96)(59,137,233,95)(60,136,234,94)(181,312,250,294)(182,311,251,293)(183,310,252,292)(184,309,253,291)(185,308,254,290)(186,307,255,289)(187,306,256,288)(188,305,257,287)(189,304,258,286)(190,303,259,285)(191,302,260,284)(192,301,241,283)(193,320,242,282)(194,319,243,281)(195,318,244,300)(196,317,245,299)(197,316,246,298)(198,315,247,297)(199,314,248,296)(200,313,249,295), (1,264,210,31)(2,265,211,32)(3,266,212,33)(4,267,213,34)(5,268,214,35)(6,269,215,36)(7,270,216,37)(8,271,217,38)(9,272,218,39)(10,273,219,40)(11,274,220,21)(12,275,201,22)(13,276,202,23)(14,277,203,24)(15,278,204,25)(16,279,205,26)(17,280,206,27)(18,261,207,28)(19,262,208,29)(20,263,209,30)(41,296,235,314)(42,297,236,315)(43,298,237,316)(44,299,238,317)(45,300,239,318)(46,281,240,319)(47,282,221,320)(48,283,222,301)(49,284,223,302)(50,285,224,303)(51,286,225,304)(52,287,226,305)(53,288,227,306)(54,289,228,307)(55,290,229,308)(56,291,230,309)(57,292,231,310)(58,293,232,311)(59,294,233,312)(60,295,234,313)(61,120,173,152)(62,101,174,153)(63,102,175,154)(64,103,176,155)(65,104,177,156)(66,105,178,157)(67,106,179,158)(68,107,180,159)(69,108,161,160)(70,109,162,141)(71,110,163,142)(72,111,164,143)(73,112,165,144)(74,113,166,145)(75,114,167,146)(76,115,168,147)(77,116,169,148)(78,117,170,149)(79,118,171,150)(80,119,172,151)(81,187,123,256)(82,188,124,257)(83,189,125,258)(84,190,126,259)(85,191,127,260)(86,192,128,241)(87,193,129,242)(88,194,130,243)(89,195,131,244)(90,196,132,245)(91,197,133,246)(92,198,134,247)(93,199,135,248)(94,200,136,249)(95,181,137,250)(96,182,138,251)(97,183,139,252)(98,184,140,253)(99,185,121,254)(100,186,122,255) );
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)], [(1,248,11,258),(2,247,12,257),(3,246,13,256),(4,245,14,255),(5,244,15,254),(6,243,16,253),(7,242,17,252),(8,241,18,251),(9,260,19,250),(10,259,20,249),(21,125,31,135),(22,124,32,134),(23,123,33,133),(24,122,34,132),(25,121,35,131),(26,140,36,130),(27,139,37,129),(28,138,38,128),(29,137,39,127),(30,136,40,126),(41,113,51,103),(42,112,52,102),(43,111,53,101),(44,110,54,120),(45,109,55,119),(46,108,56,118),(47,107,57,117),(48,106,58,116),(49,105,59,115),(50,104,60,114),(61,317,71,307),(62,316,72,306),(63,315,73,305),(64,314,74,304),(65,313,75,303),(66,312,76,302),(67,311,77,301),(68,310,78,320),(69,309,79,319),(70,308,80,318),(81,266,91,276),(82,265,92,275),(83,264,93,274),(84,263,94,273),(85,262,95,272),(86,261,96,271),(87,280,97,270),(88,279,98,269),(89,278,99,268),(90,277,100,267),(141,229,151,239),(142,228,152,238),(143,227,153,237),(144,226,154,236),(145,225,155,235),(146,224,156,234),(147,223,157,233),(148,222,158,232),(149,221,159,231),(150,240,160,230),(161,291,171,281),(162,290,172,300),(163,289,173,299),(164,288,174,298),(165,287,175,297),(166,286,176,296),(167,285,177,295),(168,284,178,294),(169,283,179,293),(170,282,180,292),(181,218,191,208),(182,217,192,207),(183,216,193,206),(184,215,194,205),(185,214,195,204),(186,213,196,203),(187,212,197,202),(188,211,198,201),(189,210,199,220),(190,209,200,219)], [(1,171,210,79),(2,170,211,78),(3,169,212,77),(4,168,213,76),(5,167,214,75),(6,166,215,74),(7,165,216,73),(8,164,217,72),(9,163,218,71),(10,162,219,70),(11,161,220,69),(12,180,201,68),(13,179,202,67),(14,178,203,66),(15,177,204,65),(16,176,205,64),(17,175,206,63),(18,174,207,62),(19,173,208,61),(20,172,209,80),(21,160,274,108),(22,159,275,107),(23,158,276,106),(24,157,277,105),(25,156,278,104),(26,155,279,103),(27,154,280,102),(28,153,261,101),(29,152,262,120),(30,151,263,119),(31,150,264,118),(32,149,265,117),(33,148,266,116),(34,147,267,115),(35,146,268,114),(36,145,269,113),(37,144,270,112),(38,143,271,111),(39,142,272,110),(40,141,273,109),(41,135,235,93),(42,134,236,92),(43,133,237,91),(44,132,238,90),(45,131,239,89),(46,130,240,88),(47,129,221,87),(48,128,222,86),(49,127,223,85),(50,126,224,84),(51,125,225,83),(52,124,226,82),(53,123,227,81),(54,122,228,100),(55,121,229,99),(56,140,230,98),(57,139,231,97),(58,138,232,96),(59,137,233,95),(60,136,234,94),(181,312,250,294),(182,311,251,293),(183,310,252,292),(184,309,253,291),(185,308,254,290),(186,307,255,289),(187,306,256,288),(188,305,257,287),(189,304,258,286),(190,303,259,285),(191,302,260,284),(192,301,241,283),(193,320,242,282),(194,319,243,281),(195,318,244,300),(196,317,245,299),(197,316,246,298),(198,315,247,297),(199,314,248,296),(200,313,249,295)], [(1,264,210,31),(2,265,211,32),(3,266,212,33),(4,267,213,34),(5,268,214,35),(6,269,215,36),(7,270,216,37),(8,271,217,38),(9,272,218,39),(10,273,219,40),(11,274,220,21),(12,275,201,22),(13,276,202,23),(14,277,203,24),(15,278,204,25),(16,279,205,26),(17,280,206,27),(18,261,207,28),(19,262,208,29),(20,263,209,30),(41,296,235,314),(42,297,236,315),(43,298,237,316),(44,299,238,317),(45,300,239,318),(46,281,240,319),(47,282,221,320),(48,283,222,301),(49,284,223,302),(50,285,224,303),(51,286,225,304),(52,287,226,305),(53,288,227,306),(54,289,228,307),(55,290,229,308),(56,291,230,309),(57,292,231,310),(58,293,232,311),(59,294,233,312),(60,295,234,313),(61,120,173,152),(62,101,174,153),(63,102,175,154),(64,103,176,155),(65,104,177,156),(66,105,178,157),(67,106,179,158),(68,107,180,159),(69,108,161,160),(70,109,162,141),(71,110,163,142),(72,111,164,143),(73,112,165,144),(74,113,166,145),(75,114,167,146),(76,115,168,147),(77,116,169,148),(78,117,170,149),(79,118,171,150),(80,119,172,151),(81,187,123,256),(82,188,124,257),(83,189,125,258),(84,190,126,259),(85,191,127,260),(86,192,128,241),(87,193,129,242),(88,194,130,243),(89,195,131,244),(90,196,132,245),(91,197,133,246),(92,198,134,247),(93,199,135,248),(94,200,136,249),(95,181,137,250),(96,182,138,251),(97,183,139,252),(98,184,140,253),(99,185,121,254),(100,186,122,255)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D20 | C40⋊C2 | C8.C22 | D4⋊2D5 | Q8×D5 | C8.D10 |
| kernel | Dic10⋊4Q8 | C20.44D4 | C40⋊6C4 | C5×C4⋊C8 | C4×Dic10 | C20⋊2Q8 | Dic10 | C2×C20 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of Dic10⋊4Q8 ►in GL6(𝔽41)
| 1 | 40 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 9 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 21 | 8 | 0 | 0 | 0 | 0 |
| 27 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 12 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 0 | 0 | 39 | 28 |
| 38 | 8 | 0 | 0 | 0 | 0 |
| 40 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 28 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 23 |
| 0 | 0 | 0 | 0 | 18 | 35 |
G:=sub<GL(6,GF(41))| [1,8,0,0,0,0,40,34,0,0,0,0,0,0,20,1,0,0,0,0,9,21,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,27,0,0,0,0,8,20,0,0,0,0,0,0,39,3,0,0,0,0,12,2,0,0,0,0,0,0,13,39,0,0,0,0,2,28],[38,40,0,0,0,0,8,3,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,39,13,0,0,0,0,28,2],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,18,0,0,0,0,23,35] >;
Dic10⋊4Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_4Q_8 % in TeX
G:=Group("Dic10:4Q8"); // GroupNames label
G:=SmallGroup(320,478);
// by ID
G=gap.SmallGroup(320,478);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations