metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊10Q8, C42.121D10, C10.642- (1+4), C4.49(Q8×D5), C5⋊1(Q8⋊3Q8), (C4×Q8).10D5, C4⋊C4.321D10, (Q8×C20).11C2, C20.107(C2×Q8), C4.17(C4○D20), (C2×Q8).175D10, C20⋊2Q8.25C2, Dic5.22(C2×Q8), C20.115(C4○D4), C10.27(C22×Q8), (C2×C10).111C24, (C2×C20).589C23, (C4×C20).164C22, (C4×Dic10).21C2, Dic5⋊Q8.9C2, C4⋊Dic5.42C22, C20.6Q8.10C2, Dic5⋊3Q8.10C2, Dic5.Q8.1C2, (Q8×C10).211C22, (C4×Dic5).88C22, (C2×Dic5).50C23, C22.136(C23×D5), C2.21(D4.10D10), (C2×Dic10).152C22, C10.D4.114C22, C2.11(C2×Q8×D5), C2.59(C2×C4○D20), C10.52(C2×C4○D4), (C5×C4⋊C4).339C22, (C2×C4).166(C22×D5), SmallGroup(320,1239)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 550 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C4 [×4], C4 [×15], C22, C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C10 [×3], C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×5], C2×C10, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic10 [×4], Dic10 [×4], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], Q8⋊3Q8, C4×Dic5 [×6], C10.D4 [×14], C4⋊Dic5, C4⋊Dic5 [×4], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], Q8×C10, C4×Dic10, C4×Dic10 [×2], C20⋊2Q8, C20.6Q8 [×2], Dic5⋊3Q8 [×2], Dic5.Q8 [×4], Dic5⋊Q8 [×2], Q8×C20, Dic10⋊10Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C22×D5 [×7], Q8⋊3Q8, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, C2×Q8×D5, D4.10D10, Dic10⋊10Q8
Generators and relations
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd-1=c-1 >
►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)
(1 316 11 306)(2 315 12 305)(3 314 13 304)(4 313 14 303)(5 312 15 302)(6 311 16 301)(7 310 17 320)(8 309 18 319)(9 308 19 318)(10 307 20 317)(21 187 31 197)(22 186 32 196)(23 185 33 195)(24 184 34 194)(25 183 35 193)(26 182 36 192)(27 181 37 191)(28 200 38 190)(29 199 39 189)(30 198 40 188)(41 99 51 89)(42 98 52 88)(43 97 53 87)(44 96 54 86)(45 95 55 85)(46 94 56 84)(47 93 57 83)(48 92 58 82)(49 91 59 81)(50 90 60 100)(61 293 71 283)(62 292 72 282)(63 291 73 281)(64 290 74 300)(65 289 75 299)(66 288 76 298)(67 287 77 297)(68 286 78 296)(69 285 79 295)(70 284 80 294)(101 154 111 144)(102 153 112 143)(103 152 113 142)(104 151 114 141)(105 150 115 160)(106 149 116 159)(107 148 117 158)(108 147 118 157)(109 146 119 156)(110 145 120 155)(121 265 131 275)(122 264 132 274)(123 263 133 273)(124 262 134 272)(125 261 135 271)(126 280 136 270)(127 279 137 269)(128 278 138 268)(129 277 139 267)(130 276 140 266)(161 209 171 219)(162 208 172 218)(163 207 173 217)(164 206 174 216)(165 205 175 215)(166 204 176 214)(167 203 177 213)(168 202 178 212)(169 201 179 211)(170 220 180 210)(221 242 231 252)(222 241 232 251)(223 260 233 250)(224 259 234 249)(225 258 235 248)(226 257 236 247)(227 256 237 246)(228 255 238 245)(229 254 239 244)(230 253 240 243)
(1 121 158 259)(2 122 159 260)(3 123 160 241)(4 124 141 242)(5 125 142 243)(6 126 143 244)(7 127 144 245)(8 128 145 246)(9 129 146 247)(10 130 147 248)(11 131 148 249)(12 132 149 250)(13 133 150 251)(14 134 151 252)(15 135 152 253)(16 136 153 254)(17 137 154 255)(18 138 155 256)(19 139 156 257)(20 140 157 258)(21 70 212 92)(22 71 213 93)(23 72 214 94)(24 73 215 95)(25 74 216 96)(26 75 217 97)(27 76 218 98)(28 77 219 99)(29 78 220 100)(30 79 201 81)(31 80 202 82)(32 61 203 83)(33 62 204 84)(34 63 205 85)(35 64 206 86)(36 65 207 87)(37 66 208 88)(38 67 209 89)(39 68 210 90)(40 69 211 91)(41 200 287 161)(42 181 288 162)(43 182 289 163)(44 183 290 164)(45 184 291 165)(46 185 292 166)(47 186 293 167)(48 187 294 168)(49 188 295 169)(50 189 296 170)(51 190 297 171)(52 191 298 172)(53 192 299 173)(54 193 300 174)(55 194 281 175)(56 195 282 176)(57 196 283 177)(58 197 284 178)(59 198 285 179)(60 199 286 180)(101 238 310 269)(102 239 311 270)(103 240 312 271)(104 221 313 272)(105 222 314 273)(106 223 315 274)(107 224 316 275)(108 225 317 276)(109 226 318 277)(110 227 319 278)(111 228 320 279)(112 229 301 280)(113 230 302 261)(114 231 303 262)(115 232 304 263)(116 233 305 264)(117 234 306 265)(118 235 307 266)(119 236 308 267)(120 237 309 268)
(1 43 158 289)(2 44 159 290)(3 45 160 291)(4 46 141 292)(5 47 142 293)(6 48 143 294)(7 49 144 295)(8 50 145 296)(9 51 146 297)(10 52 147 298)(11 53 148 299)(12 54 149 300)(13 55 150 281)(14 56 151 282)(15 57 152 283)(16 58 153 284)(17 59 154 285)(18 60 155 286)(19 41 156 287)(20 42 157 288)(21 270 212 239)(22 271 213 240)(23 272 214 221)(24 273 215 222)(25 274 216 223)(26 275 217 224)(27 276 218 225)(28 277 219 226)(29 278 220 227)(30 279 201 228)(31 280 202 229)(32 261 203 230)(33 262 204 231)(34 263 205 232)(35 264 206 233)(36 265 207 234)(37 266 208 235)(38 267 209 236)(39 268 210 237)(40 269 211 238)(61 302 83 113)(62 303 84 114)(63 304 85 115)(64 305 86 116)(65 306 87 117)(66 307 88 118)(67 308 89 119)(68 309 90 120)(69 310 91 101)(70 311 92 102)(71 312 93 103)(72 313 94 104)(73 314 95 105)(74 315 96 106)(75 316 97 107)(76 317 98 108)(77 318 99 109)(78 319 100 110)(79 320 81 111)(80 301 82 112)(121 163 259 182)(122 164 260 183)(123 165 241 184)(124 166 242 185)(125 167 243 186)(126 168 244 187)(127 169 245 188)(128 170 246 189)(129 171 247 190)(130 172 248 191)(131 173 249 192)(132 174 250 193)(133 175 251 194)(134 176 252 195)(135 177 253 196)(136 178 254 197)(137 179 255 198)(138 180 256 199)(139 161 257 200)(140 162 258 181)
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), (1,316,11,306)(2,315,12,305)(3,314,13,304)(4,313,14,303)(5,312,15,302)(6,311,16,301)(7,310,17,320)(8,309,18,319)(9,308,19,318)(10,307,20,317)(21,187,31,197)(22,186,32,196)(23,185,33,195)(24,184,34,194)(25,183,35,193)(26,182,36,192)(27,181,37,191)(28,200,38,190)(29,199,39,189)(30,198,40,188)(41,99,51,89)(42,98,52,88)(43,97,53,87)(44,96,54,86)(45,95,55,85)(46,94,56,84)(47,93,57,83)(48,92,58,82)(49,91,59,81)(50,90,60,100)(61,293,71,283)(62,292,72,282)(63,291,73,281)(64,290,74,300)(65,289,75,299)(66,288,76,298)(67,287,77,297)(68,286,78,296)(69,285,79,295)(70,284,80,294)(101,154,111,144)(102,153,112,143)(103,152,113,142)(104,151,114,141)(105,150,115,160)(106,149,116,159)(107,148,117,158)(108,147,118,157)(109,146,119,156)(110,145,120,155)(121,265,131,275)(122,264,132,274)(123,263,133,273)(124,262,134,272)(125,261,135,271)(126,280,136,270)(127,279,137,269)(128,278,138,268)(129,277,139,267)(130,276,140,266)(161,209,171,219)(162,208,172,218)(163,207,173,217)(164,206,174,216)(165,205,175,215)(166,204,176,214)(167,203,177,213)(168,202,178,212)(169,201,179,211)(170,220,180,210)(221,242,231,252)(222,241,232,251)(223,260,233,250)(224,259,234,249)(225,258,235,248)(226,257,236,247)(227,256,237,246)(228,255,238,245)(229,254,239,244)(230,253,240,243), (1,121,158,259)(2,122,159,260)(3,123,160,241)(4,124,141,242)(5,125,142,243)(6,126,143,244)(7,127,144,245)(8,128,145,246)(9,129,146,247)(10,130,147,248)(11,131,148,249)(12,132,149,250)(13,133,150,251)(14,134,151,252)(15,135,152,253)(16,136,153,254)(17,137,154,255)(18,138,155,256)(19,139,156,257)(20,140,157,258)(21,70,212,92)(22,71,213,93)(23,72,214,94)(24,73,215,95)(25,74,216,96)(26,75,217,97)(27,76,218,98)(28,77,219,99)(29,78,220,100)(30,79,201,81)(31,80,202,82)(32,61,203,83)(33,62,204,84)(34,63,205,85)(35,64,206,86)(36,65,207,87)(37,66,208,88)(38,67,209,89)(39,68,210,90)(40,69,211,91)(41,200,287,161)(42,181,288,162)(43,182,289,163)(44,183,290,164)(45,184,291,165)(46,185,292,166)(47,186,293,167)(48,187,294,168)(49,188,295,169)(50,189,296,170)(51,190,297,171)(52,191,298,172)(53,192,299,173)(54,193,300,174)(55,194,281,175)(56,195,282,176)(57,196,283,177)(58,197,284,178)(59,198,285,179)(60,199,286,180)(101,238,310,269)(102,239,311,270)(103,240,312,271)(104,221,313,272)(105,222,314,273)(106,223,315,274)(107,224,316,275)(108,225,317,276)(109,226,318,277)(110,227,319,278)(111,228,320,279)(112,229,301,280)(113,230,302,261)(114,231,303,262)(115,232,304,263)(116,233,305,264)(117,234,306,265)(118,235,307,266)(119,236,308,267)(120,237,309,268), (1,43,158,289)(2,44,159,290)(3,45,160,291)(4,46,141,292)(5,47,142,293)(6,48,143,294)(7,49,144,295)(8,50,145,296)(9,51,146,297)(10,52,147,298)(11,53,148,299)(12,54,149,300)(13,55,150,281)(14,56,151,282)(15,57,152,283)(16,58,153,284)(17,59,154,285)(18,60,155,286)(19,41,156,287)(20,42,157,288)(21,270,212,239)(22,271,213,240)(23,272,214,221)(24,273,215,222)(25,274,216,223)(26,275,217,224)(27,276,218,225)(28,277,219,226)(29,278,220,227)(30,279,201,228)(31,280,202,229)(32,261,203,230)(33,262,204,231)(34,263,205,232)(35,264,206,233)(36,265,207,234)(37,266,208,235)(38,267,209,236)(39,268,210,237)(40,269,211,238)(61,302,83,113)(62,303,84,114)(63,304,85,115)(64,305,86,116)(65,306,87,117)(66,307,88,118)(67,308,89,119)(68,309,90,120)(69,310,91,101)(70,311,92,102)(71,312,93,103)(72,313,94,104)(73,314,95,105)(74,315,96,106)(75,316,97,107)(76,317,98,108)(77,318,99,109)(78,319,100,110)(79,320,81,111)(80,301,82,112)(121,163,259,182)(122,164,260,183)(123,165,241,184)(124,166,242,185)(125,167,243,186)(126,168,244,187)(127,169,245,188)(128,170,246,189)(129,171,247,190)(130,172,248,191)(131,173,249,192)(132,174,250,193)(133,175,251,194)(134,176,252,195)(135,177,253,196)(136,178,254,197)(137,179,255,198)(138,180,256,199)(139,161,257,200)(140,162,258,181)>;
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), (1,316,11,306)(2,315,12,305)(3,314,13,304)(4,313,14,303)(5,312,15,302)(6,311,16,301)(7,310,17,320)(8,309,18,319)(9,308,19,318)(10,307,20,317)(21,187,31,197)(22,186,32,196)(23,185,33,195)(24,184,34,194)(25,183,35,193)(26,182,36,192)(27,181,37,191)(28,200,38,190)(29,199,39,189)(30,198,40,188)(41,99,51,89)(42,98,52,88)(43,97,53,87)(44,96,54,86)(45,95,55,85)(46,94,56,84)(47,93,57,83)(48,92,58,82)(49,91,59,81)(50,90,60,100)(61,293,71,283)(62,292,72,282)(63,291,73,281)(64,290,74,300)(65,289,75,299)(66,288,76,298)(67,287,77,297)(68,286,78,296)(69,285,79,295)(70,284,80,294)(101,154,111,144)(102,153,112,143)(103,152,113,142)(104,151,114,141)(105,150,115,160)(106,149,116,159)(107,148,117,158)(108,147,118,157)(109,146,119,156)(110,145,120,155)(121,265,131,275)(122,264,132,274)(123,263,133,273)(124,262,134,272)(125,261,135,271)(126,280,136,270)(127,279,137,269)(128,278,138,268)(129,277,139,267)(130,276,140,266)(161,209,171,219)(162,208,172,218)(163,207,173,217)(164,206,174,216)(165,205,175,215)(166,204,176,214)(167,203,177,213)(168,202,178,212)(169,201,179,211)(170,220,180,210)(221,242,231,252)(222,241,232,251)(223,260,233,250)(224,259,234,249)(225,258,235,248)(226,257,236,247)(227,256,237,246)(228,255,238,245)(229,254,239,244)(230,253,240,243), (1,121,158,259)(2,122,159,260)(3,123,160,241)(4,124,141,242)(5,125,142,243)(6,126,143,244)(7,127,144,245)(8,128,145,246)(9,129,146,247)(10,130,147,248)(11,131,148,249)(12,132,149,250)(13,133,150,251)(14,134,151,252)(15,135,152,253)(16,136,153,254)(17,137,154,255)(18,138,155,256)(19,139,156,257)(20,140,157,258)(21,70,212,92)(22,71,213,93)(23,72,214,94)(24,73,215,95)(25,74,216,96)(26,75,217,97)(27,76,218,98)(28,77,219,99)(29,78,220,100)(30,79,201,81)(31,80,202,82)(32,61,203,83)(33,62,204,84)(34,63,205,85)(35,64,206,86)(36,65,207,87)(37,66,208,88)(38,67,209,89)(39,68,210,90)(40,69,211,91)(41,200,287,161)(42,181,288,162)(43,182,289,163)(44,183,290,164)(45,184,291,165)(46,185,292,166)(47,186,293,167)(48,187,294,168)(49,188,295,169)(50,189,296,170)(51,190,297,171)(52,191,298,172)(53,192,299,173)(54,193,300,174)(55,194,281,175)(56,195,282,176)(57,196,283,177)(58,197,284,178)(59,198,285,179)(60,199,286,180)(101,238,310,269)(102,239,311,270)(103,240,312,271)(104,221,313,272)(105,222,314,273)(106,223,315,274)(107,224,316,275)(108,225,317,276)(109,226,318,277)(110,227,319,278)(111,228,320,279)(112,229,301,280)(113,230,302,261)(114,231,303,262)(115,232,304,263)(116,233,305,264)(117,234,306,265)(118,235,307,266)(119,236,308,267)(120,237,309,268), (1,43,158,289)(2,44,159,290)(3,45,160,291)(4,46,141,292)(5,47,142,293)(6,48,143,294)(7,49,144,295)(8,50,145,296)(9,51,146,297)(10,52,147,298)(11,53,148,299)(12,54,149,300)(13,55,150,281)(14,56,151,282)(15,57,152,283)(16,58,153,284)(17,59,154,285)(18,60,155,286)(19,41,156,287)(20,42,157,288)(21,270,212,239)(22,271,213,240)(23,272,214,221)(24,273,215,222)(25,274,216,223)(26,275,217,224)(27,276,218,225)(28,277,219,226)(29,278,220,227)(30,279,201,228)(31,280,202,229)(32,261,203,230)(33,262,204,231)(34,263,205,232)(35,264,206,233)(36,265,207,234)(37,266,208,235)(38,267,209,236)(39,268,210,237)(40,269,211,238)(61,302,83,113)(62,303,84,114)(63,304,85,115)(64,305,86,116)(65,306,87,117)(66,307,88,118)(67,308,89,119)(68,309,90,120)(69,310,91,101)(70,311,92,102)(71,312,93,103)(72,313,94,104)(73,314,95,105)(74,315,96,106)(75,316,97,107)(76,317,98,108)(77,318,99,109)(78,319,100,110)(79,320,81,111)(80,301,82,112)(121,163,259,182)(122,164,260,183)(123,165,241,184)(124,166,242,185)(125,167,243,186)(126,168,244,187)(127,169,245,188)(128,170,246,189)(129,171,247,190)(130,172,248,191)(131,173,249,192)(132,174,250,193)(133,175,251,194)(134,176,252,195)(135,177,253,196)(136,178,254,197)(137,179,255,198)(138,180,256,199)(139,161,257,200)(140,162,258,181) );
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)], [(1,316,11,306),(2,315,12,305),(3,314,13,304),(4,313,14,303),(5,312,15,302),(6,311,16,301),(7,310,17,320),(8,309,18,319),(9,308,19,318),(10,307,20,317),(21,187,31,197),(22,186,32,196),(23,185,33,195),(24,184,34,194),(25,183,35,193),(26,182,36,192),(27,181,37,191),(28,200,38,190),(29,199,39,189),(30,198,40,188),(41,99,51,89),(42,98,52,88),(43,97,53,87),(44,96,54,86),(45,95,55,85),(46,94,56,84),(47,93,57,83),(48,92,58,82),(49,91,59,81),(50,90,60,100),(61,293,71,283),(62,292,72,282),(63,291,73,281),(64,290,74,300),(65,289,75,299),(66,288,76,298),(67,287,77,297),(68,286,78,296),(69,285,79,295),(70,284,80,294),(101,154,111,144),(102,153,112,143),(103,152,113,142),(104,151,114,141),(105,150,115,160),(106,149,116,159),(107,148,117,158),(108,147,118,157),(109,146,119,156),(110,145,120,155),(121,265,131,275),(122,264,132,274),(123,263,133,273),(124,262,134,272),(125,261,135,271),(126,280,136,270),(127,279,137,269),(128,278,138,268),(129,277,139,267),(130,276,140,266),(161,209,171,219),(162,208,172,218),(163,207,173,217),(164,206,174,216),(165,205,175,215),(166,204,176,214),(167,203,177,213),(168,202,178,212),(169,201,179,211),(170,220,180,210),(221,242,231,252),(222,241,232,251),(223,260,233,250),(224,259,234,249),(225,258,235,248),(226,257,236,247),(227,256,237,246),(228,255,238,245),(229,254,239,244),(230,253,240,243)], [(1,121,158,259),(2,122,159,260),(3,123,160,241),(4,124,141,242),(5,125,142,243),(6,126,143,244),(7,127,144,245),(8,128,145,246),(9,129,146,247),(10,130,147,248),(11,131,148,249),(12,132,149,250),(13,133,150,251),(14,134,151,252),(15,135,152,253),(16,136,153,254),(17,137,154,255),(18,138,155,256),(19,139,156,257),(20,140,157,258),(21,70,212,92),(22,71,213,93),(23,72,214,94),(24,73,215,95),(25,74,216,96),(26,75,217,97),(27,76,218,98),(28,77,219,99),(29,78,220,100),(30,79,201,81),(31,80,202,82),(32,61,203,83),(33,62,204,84),(34,63,205,85),(35,64,206,86),(36,65,207,87),(37,66,208,88),(38,67,209,89),(39,68,210,90),(40,69,211,91),(41,200,287,161),(42,181,288,162),(43,182,289,163),(44,183,290,164),(45,184,291,165),(46,185,292,166),(47,186,293,167),(48,187,294,168),(49,188,295,169),(50,189,296,170),(51,190,297,171),(52,191,298,172),(53,192,299,173),(54,193,300,174),(55,194,281,175),(56,195,282,176),(57,196,283,177),(58,197,284,178),(59,198,285,179),(60,199,286,180),(101,238,310,269),(102,239,311,270),(103,240,312,271),(104,221,313,272),(105,222,314,273),(106,223,315,274),(107,224,316,275),(108,225,317,276),(109,226,318,277),(110,227,319,278),(111,228,320,279),(112,229,301,280),(113,230,302,261),(114,231,303,262),(115,232,304,263),(116,233,305,264),(117,234,306,265),(118,235,307,266),(119,236,308,267),(120,237,309,268)], [(1,43,158,289),(2,44,159,290),(3,45,160,291),(4,46,141,292),(5,47,142,293),(6,48,143,294),(7,49,144,295),(8,50,145,296),(9,51,146,297),(10,52,147,298),(11,53,148,299),(12,54,149,300),(13,55,150,281),(14,56,151,282),(15,57,152,283),(16,58,153,284),(17,59,154,285),(18,60,155,286),(19,41,156,287),(20,42,157,288),(21,270,212,239),(22,271,213,240),(23,272,214,221),(24,273,215,222),(25,274,216,223),(26,275,217,224),(27,276,218,225),(28,277,219,226),(29,278,220,227),(30,279,201,228),(31,280,202,229),(32,261,203,230),(33,262,204,231),(34,263,205,232),(35,264,206,233),(36,265,207,234),(37,266,208,235),(38,267,209,236),(39,268,210,237),(40,269,211,238),(61,302,83,113),(62,303,84,114),(63,304,85,115),(64,305,86,116),(65,306,87,117),(66,307,88,118),(67,308,89,119),(68,309,90,120),(69,310,91,101),(70,311,92,102),(71,312,93,103),(72,313,94,104),(73,314,95,105),(74,315,96,106),(75,316,97,107),(76,317,98,108),(77,318,99,109),(78,319,100,110),(79,320,81,111),(80,301,82,112),(121,163,259,182),(122,164,260,183),(123,165,241,184),(124,166,242,185),(125,167,243,186),(126,168,244,187),(127,169,245,188),(128,170,246,189),(129,171,247,190),(130,172,248,191),(131,173,249,192),(132,174,250,193),(133,175,251,194),(134,176,252,195),(135,177,253,196),(136,178,254,197),(137,179,255,198),(138,180,256,199),(139,161,257,200),(140,162,258,181)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 25 | 2 | 0 | 0 |
| 39 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 35 | 0 | 0 |
| 20 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 6 | 0 | 0 |
| 35 | 23 | 0 | 0 |
| 0 | 0 | 32 | 23 |
| 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 7 | 13 |
| 0 | 0 | 34 | 34 |
G:=sub<GL(4,GF(41))| [25,39,0,0,2,13,0,0,0,0,1,0,0,0,0,1],[23,20,0,0,35,18,0,0,0,0,1,0,0,0,0,1],[18,35,0,0,6,23,0,0,0,0,32,0,0,0,23,9],[40,0,0,0,0,40,0,0,0,0,7,34,0,0,13,34] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | Q8×D5 | D4.10D10 |
| kernel | Dic10⋊10Q8 | C4×Dic10 | C20⋊2Q8 | C20.6Q8 | Dic5⋊3Q8 | Dic5.Q8 | Dic5⋊Q8 | Q8×C20 | Dic10 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{10}Q_8 % in TeX
G:=Group("Dic10:10Q8"); // GroupNames label
G:=SmallGroup(320,1239);
// by ID
G=gap.SmallGroup(320,1239);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,185,192,12550]);
// 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,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀