metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊7Q8, C42.146D10, C10.942- 1+4, C4.15(Q8×D5), C5⋊4(Q8⋊3Q8), C20⋊Q8.13C2, C20.47(C2×Q8), C4⋊C4.202D10, C42.C2.6D5, (C2×C20).84C23, Dic5.25(C2×Q8), C10.39(C22×Q8), (C4×C20).190C22, (C2×C10).229C24, (C4×Dic10).24C2, C4.Dic10.13C2, Dic5.66(C4○D4), Dic5⋊3Q8.11C2, C4⋊Dic5.378C22, Dic5.Q8.2C2, C22.250(C23×D5), (C2×Dic5).119C23, (C4×Dic5).145C22, C2.55(D4.10D10), (C2×Dic10).306C22, C10.D4.143C22, C2.22(C2×Q8×D5), C2.82(D5×C4○D4), C10.193(C2×C4○D4), (C2×C4).75(C22×D5), (C5×C42.C2).5C2, (C5×C4⋊C4).184C22, SmallGroup(320,1357)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊7Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, dad-1=a9, cbc-1=dbd-1=a10b, dcd-1=c-1 >
Subgroups: 566 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, C42.C2, C42.C2, C4⋊Q8, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, Q8⋊3Q8, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C4×Dic10, Dic5⋊3Q8, Dic5⋊3Q8, C20⋊Q8, C20⋊Q8, Dic5.Q8, C4.Dic10, C5×C42.C2, Dic10⋊7Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2- 1+4, C22×D5, Q8⋊3Q8, Q8×D5, C23×D5, C2×Q8×D5, D5×C4○D4, D4.10D10, Dic10⋊7Q8
►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)
(1 156 11 146)(2 155 12 145)(3 154 13 144)(4 153 14 143)(5 152 15 142)(6 151 16 141)(7 150 17 160)(8 149 18 159)(9 148 19 158)(10 147 20 157)(21 174 31 164)(22 173 32 163)(23 172 33 162)(24 171 34 161)(25 170 35 180)(26 169 36 179)(27 168 37 178)(28 167 38 177)(29 166 39 176)(30 165 40 175)(41 131 51 121)(42 130 52 140)(43 129 53 139)(44 128 54 138)(45 127 55 137)(46 126 56 136)(47 125 57 135)(48 124 58 134)(49 123 59 133)(50 122 60 132)(61 81 71 91)(62 100 72 90)(63 99 73 89)(64 98 74 88)(65 97 75 87)(66 96 76 86)(67 95 77 85)(68 94 78 84)(69 93 79 83)(70 92 80 82)(101 188 111 198)(102 187 112 197)(103 186 113 196)(104 185 114 195)(105 184 115 194)(106 183 116 193)(107 182 117 192)(108 181 118 191)(109 200 119 190)(110 199 120 189)(201 278 211 268)(202 277 212 267)(203 276 213 266)(204 275 214 265)(205 274 215 264)(206 273 216 263)(207 272 217 262)(208 271 218 261)(209 270 219 280)(210 269 220 279)(221 253 231 243)(222 252 232 242)(223 251 233 241)(224 250 234 260)(225 249 235 259)(226 248 236 258)(227 247 237 257)(228 246 238 256)(229 245 239 255)(230 244 240 254)(281 303 291 313)(282 302 292 312)(283 301 293 311)(284 320 294 310)(285 319 295 309)(286 318 296 308)(287 317 297 307)(288 316 298 306)(289 315 299 305)(290 314 300 304)
(1 95 261 55)(2 96 262 56)(3 97 263 57)(4 98 264 58)(5 99 265 59)(6 100 266 60)(7 81 267 41)(8 82 268 42)(9 83 269 43)(10 84 270 44)(11 85 271 45)(12 86 272 46)(13 87 273 47)(14 88 274 48)(15 89 275 49)(16 90 276 50)(17 91 277 51)(18 92 278 52)(19 93 279 53)(20 94 280 54)(21 295 226 184)(22 296 227 185)(23 297 228 186)(24 298 229 187)(25 299 230 188)(26 300 231 189)(27 281 232 190)(28 282 233 191)(29 283 234 192)(30 284 235 193)(31 285 236 194)(32 286 237 195)(33 287 238 196)(34 288 239 197)(35 289 240 198)(36 290 221 199)(37 291 222 200)(38 292 223 181)(39 293 224 182)(40 294 225 183)(61 202 121 150)(62 203 122 151)(63 204 123 152)(64 205 124 153)(65 206 125 154)(66 207 126 155)(67 208 127 156)(68 209 128 157)(69 210 129 158)(70 211 130 159)(71 212 131 160)(72 213 132 141)(73 214 133 142)(74 215 134 143)(75 216 135 144)(76 217 136 145)(77 218 137 146)(78 219 138 147)(79 220 139 148)(80 201 140 149)(101 170 315 244)(102 171 316 245)(103 172 317 246)(104 173 318 247)(105 174 319 248)(106 175 320 249)(107 176 301 250)(108 177 302 251)(109 178 303 252)(110 179 304 253)(111 180 305 254)(112 161 306 255)(113 162 307 256)(114 163 308 257)(115 164 309 258)(116 165 310 259)(117 166 311 260)(118 167 312 241)(119 168 313 242)(120 169 314 243)
(1 317 261 103)(2 306 262 112)(3 315 263 101)(4 304 264 110)(5 313 265 119)(6 302 266 108)(7 311 267 117)(8 320 268 106)(9 309 269 115)(10 318 270 104)(11 307 271 113)(12 316 272 102)(13 305 273 111)(14 314 274 120)(15 303 275 109)(16 312 276 118)(17 301 277 107)(18 310 278 116)(19 319 279 105)(20 308 280 114)(21 139 226 79)(22 128 227 68)(23 137 228 77)(24 126 229 66)(25 135 230 75)(26 124 231 64)(27 133 232 73)(28 122 233 62)(29 131 234 71)(30 140 235 80)(31 129 236 69)(32 138 237 78)(33 127 238 67)(34 136 239 76)(35 125 240 65)(36 134 221 74)(37 123 222 63)(38 132 223 72)(39 121 224 61)(40 130 225 70)(41 260 81 166)(42 249 82 175)(43 258 83 164)(44 247 84 173)(45 256 85 162)(46 245 86 171)(47 254 87 180)(48 243 88 169)(49 252 89 178)(50 241 90 167)(51 250 91 176)(52 259 92 165)(53 248 93 174)(54 257 94 163)(55 246 95 172)(56 255 96 161)(57 244 97 170)(58 253 98 179)(59 242 99 168)(60 251 100 177)(141 292 213 181)(142 281 214 190)(143 290 215 199)(144 299 216 188)(145 288 217 197)(146 297 218 186)(147 286 219 195)(148 295 220 184)(149 284 201 193)(150 293 202 182)(151 282 203 191)(152 291 204 200)(153 300 205 189)(154 289 206 198)(155 298 207 187)(156 287 208 196)(157 296 209 185)(158 285 210 194)(159 294 211 183)(160 283 212 192)
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), (1,156,11,146)(2,155,12,145)(3,154,13,144)(4,153,14,143)(5,152,15,142)(6,151,16,141)(7,150,17,160)(8,149,18,159)(9,148,19,158)(10,147,20,157)(21,174,31,164)(22,173,32,163)(23,172,33,162)(24,171,34,161)(25,170,35,180)(26,169,36,179)(27,168,37,178)(28,167,38,177)(29,166,39,176)(30,165,40,175)(41,131,51,121)(42,130,52,140)(43,129,53,139)(44,128,54,138)(45,127,55,137)(46,126,56,136)(47,125,57,135)(48,124,58,134)(49,123,59,133)(50,122,60,132)(61,81,71,91)(62,100,72,90)(63,99,73,89)(64,98,74,88)(65,97,75,87)(66,96,76,86)(67,95,77,85)(68,94,78,84)(69,93,79,83)(70,92,80,82)(101,188,111,198)(102,187,112,197)(103,186,113,196)(104,185,114,195)(105,184,115,194)(106,183,116,193)(107,182,117,192)(108,181,118,191)(109,200,119,190)(110,199,120,189)(201,278,211,268)(202,277,212,267)(203,276,213,266)(204,275,214,265)(205,274,215,264)(206,273,216,263)(207,272,217,262)(208,271,218,261)(209,270,219,280)(210,269,220,279)(221,253,231,243)(222,252,232,242)(223,251,233,241)(224,250,234,260)(225,249,235,259)(226,248,236,258)(227,247,237,257)(228,246,238,256)(229,245,239,255)(230,244,240,254)(281,303,291,313)(282,302,292,312)(283,301,293,311)(284,320,294,310)(285,319,295,309)(286,318,296,308)(287,317,297,307)(288,316,298,306)(289,315,299,305)(290,314,300,304), (1,95,261,55)(2,96,262,56)(3,97,263,57)(4,98,264,58)(5,99,265,59)(6,100,266,60)(7,81,267,41)(8,82,268,42)(9,83,269,43)(10,84,270,44)(11,85,271,45)(12,86,272,46)(13,87,273,47)(14,88,274,48)(15,89,275,49)(16,90,276,50)(17,91,277,51)(18,92,278,52)(19,93,279,53)(20,94,280,54)(21,295,226,184)(22,296,227,185)(23,297,228,186)(24,298,229,187)(25,299,230,188)(26,300,231,189)(27,281,232,190)(28,282,233,191)(29,283,234,192)(30,284,235,193)(31,285,236,194)(32,286,237,195)(33,287,238,196)(34,288,239,197)(35,289,240,198)(36,290,221,199)(37,291,222,200)(38,292,223,181)(39,293,224,182)(40,294,225,183)(61,202,121,150)(62,203,122,151)(63,204,123,152)(64,205,124,153)(65,206,125,154)(66,207,126,155)(67,208,127,156)(68,209,128,157)(69,210,129,158)(70,211,130,159)(71,212,131,160)(72,213,132,141)(73,214,133,142)(74,215,134,143)(75,216,135,144)(76,217,136,145)(77,218,137,146)(78,219,138,147)(79,220,139,148)(80,201,140,149)(101,170,315,244)(102,171,316,245)(103,172,317,246)(104,173,318,247)(105,174,319,248)(106,175,320,249)(107,176,301,250)(108,177,302,251)(109,178,303,252)(110,179,304,253)(111,180,305,254)(112,161,306,255)(113,162,307,256)(114,163,308,257)(115,164,309,258)(116,165,310,259)(117,166,311,260)(118,167,312,241)(119,168,313,242)(120,169,314,243), (1,317,261,103)(2,306,262,112)(3,315,263,101)(4,304,264,110)(5,313,265,119)(6,302,266,108)(7,311,267,117)(8,320,268,106)(9,309,269,115)(10,318,270,104)(11,307,271,113)(12,316,272,102)(13,305,273,111)(14,314,274,120)(15,303,275,109)(16,312,276,118)(17,301,277,107)(18,310,278,116)(19,319,279,105)(20,308,280,114)(21,139,226,79)(22,128,227,68)(23,137,228,77)(24,126,229,66)(25,135,230,75)(26,124,231,64)(27,133,232,73)(28,122,233,62)(29,131,234,71)(30,140,235,80)(31,129,236,69)(32,138,237,78)(33,127,238,67)(34,136,239,76)(35,125,240,65)(36,134,221,74)(37,123,222,63)(38,132,223,72)(39,121,224,61)(40,130,225,70)(41,260,81,166)(42,249,82,175)(43,258,83,164)(44,247,84,173)(45,256,85,162)(46,245,86,171)(47,254,87,180)(48,243,88,169)(49,252,89,178)(50,241,90,167)(51,250,91,176)(52,259,92,165)(53,248,93,174)(54,257,94,163)(55,246,95,172)(56,255,96,161)(57,244,97,170)(58,253,98,179)(59,242,99,168)(60,251,100,177)(141,292,213,181)(142,281,214,190)(143,290,215,199)(144,299,216,188)(145,288,217,197)(146,297,218,186)(147,286,219,195)(148,295,220,184)(149,284,201,193)(150,293,202,182)(151,282,203,191)(152,291,204,200)(153,300,205,189)(154,289,206,198)(155,298,207,187)(156,287,208,196)(157,296,209,185)(158,285,210,194)(159,294,211,183)(160,283,212,192)>;
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), (1,156,11,146)(2,155,12,145)(3,154,13,144)(4,153,14,143)(5,152,15,142)(6,151,16,141)(7,150,17,160)(8,149,18,159)(9,148,19,158)(10,147,20,157)(21,174,31,164)(22,173,32,163)(23,172,33,162)(24,171,34,161)(25,170,35,180)(26,169,36,179)(27,168,37,178)(28,167,38,177)(29,166,39,176)(30,165,40,175)(41,131,51,121)(42,130,52,140)(43,129,53,139)(44,128,54,138)(45,127,55,137)(46,126,56,136)(47,125,57,135)(48,124,58,134)(49,123,59,133)(50,122,60,132)(61,81,71,91)(62,100,72,90)(63,99,73,89)(64,98,74,88)(65,97,75,87)(66,96,76,86)(67,95,77,85)(68,94,78,84)(69,93,79,83)(70,92,80,82)(101,188,111,198)(102,187,112,197)(103,186,113,196)(104,185,114,195)(105,184,115,194)(106,183,116,193)(107,182,117,192)(108,181,118,191)(109,200,119,190)(110,199,120,189)(201,278,211,268)(202,277,212,267)(203,276,213,266)(204,275,214,265)(205,274,215,264)(206,273,216,263)(207,272,217,262)(208,271,218,261)(209,270,219,280)(210,269,220,279)(221,253,231,243)(222,252,232,242)(223,251,233,241)(224,250,234,260)(225,249,235,259)(226,248,236,258)(227,247,237,257)(228,246,238,256)(229,245,239,255)(230,244,240,254)(281,303,291,313)(282,302,292,312)(283,301,293,311)(284,320,294,310)(285,319,295,309)(286,318,296,308)(287,317,297,307)(288,316,298,306)(289,315,299,305)(290,314,300,304), (1,95,261,55)(2,96,262,56)(3,97,263,57)(4,98,264,58)(5,99,265,59)(6,100,266,60)(7,81,267,41)(8,82,268,42)(9,83,269,43)(10,84,270,44)(11,85,271,45)(12,86,272,46)(13,87,273,47)(14,88,274,48)(15,89,275,49)(16,90,276,50)(17,91,277,51)(18,92,278,52)(19,93,279,53)(20,94,280,54)(21,295,226,184)(22,296,227,185)(23,297,228,186)(24,298,229,187)(25,299,230,188)(26,300,231,189)(27,281,232,190)(28,282,233,191)(29,283,234,192)(30,284,235,193)(31,285,236,194)(32,286,237,195)(33,287,238,196)(34,288,239,197)(35,289,240,198)(36,290,221,199)(37,291,222,200)(38,292,223,181)(39,293,224,182)(40,294,225,183)(61,202,121,150)(62,203,122,151)(63,204,123,152)(64,205,124,153)(65,206,125,154)(66,207,126,155)(67,208,127,156)(68,209,128,157)(69,210,129,158)(70,211,130,159)(71,212,131,160)(72,213,132,141)(73,214,133,142)(74,215,134,143)(75,216,135,144)(76,217,136,145)(77,218,137,146)(78,219,138,147)(79,220,139,148)(80,201,140,149)(101,170,315,244)(102,171,316,245)(103,172,317,246)(104,173,318,247)(105,174,319,248)(106,175,320,249)(107,176,301,250)(108,177,302,251)(109,178,303,252)(110,179,304,253)(111,180,305,254)(112,161,306,255)(113,162,307,256)(114,163,308,257)(115,164,309,258)(116,165,310,259)(117,166,311,260)(118,167,312,241)(119,168,313,242)(120,169,314,243), (1,317,261,103)(2,306,262,112)(3,315,263,101)(4,304,264,110)(5,313,265,119)(6,302,266,108)(7,311,267,117)(8,320,268,106)(9,309,269,115)(10,318,270,104)(11,307,271,113)(12,316,272,102)(13,305,273,111)(14,314,274,120)(15,303,275,109)(16,312,276,118)(17,301,277,107)(18,310,278,116)(19,319,279,105)(20,308,280,114)(21,139,226,79)(22,128,227,68)(23,137,228,77)(24,126,229,66)(25,135,230,75)(26,124,231,64)(27,133,232,73)(28,122,233,62)(29,131,234,71)(30,140,235,80)(31,129,236,69)(32,138,237,78)(33,127,238,67)(34,136,239,76)(35,125,240,65)(36,134,221,74)(37,123,222,63)(38,132,223,72)(39,121,224,61)(40,130,225,70)(41,260,81,166)(42,249,82,175)(43,258,83,164)(44,247,84,173)(45,256,85,162)(46,245,86,171)(47,254,87,180)(48,243,88,169)(49,252,89,178)(50,241,90,167)(51,250,91,176)(52,259,92,165)(53,248,93,174)(54,257,94,163)(55,246,95,172)(56,255,96,161)(57,244,97,170)(58,253,98,179)(59,242,99,168)(60,251,100,177)(141,292,213,181)(142,281,214,190)(143,290,215,199)(144,299,216,188)(145,288,217,197)(146,297,218,186)(147,286,219,195)(148,295,220,184)(149,284,201,193)(150,293,202,182)(151,282,203,191)(152,291,204,200)(153,300,205,189)(154,289,206,198)(155,298,207,187)(156,287,208,196)(157,296,209,185)(158,285,210,194)(159,294,211,183)(160,283,212,192) );
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)], [(1,156,11,146),(2,155,12,145),(3,154,13,144),(4,153,14,143),(5,152,15,142),(6,151,16,141),(7,150,17,160),(8,149,18,159),(9,148,19,158),(10,147,20,157),(21,174,31,164),(22,173,32,163),(23,172,33,162),(24,171,34,161),(25,170,35,180),(26,169,36,179),(27,168,37,178),(28,167,38,177),(29,166,39,176),(30,165,40,175),(41,131,51,121),(42,130,52,140),(43,129,53,139),(44,128,54,138),(45,127,55,137),(46,126,56,136),(47,125,57,135),(48,124,58,134),(49,123,59,133),(50,122,60,132),(61,81,71,91),(62,100,72,90),(63,99,73,89),(64,98,74,88),(65,97,75,87),(66,96,76,86),(67,95,77,85),(68,94,78,84),(69,93,79,83),(70,92,80,82),(101,188,111,198),(102,187,112,197),(103,186,113,196),(104,185,114,195),(105,184,115,194),(106,183,116,193),(107,182,117,192),(108,181,118,191),(109,200,119,190),(110,199,120,189),(201,278,211,268),(202,277,212,267),(203,276,213,266),(204,275,214,265),(205,274,215,264),(206,273,216,263),(207,272,217,262),(208,271,218,261),(209,270,219,280),(210,269,220,279),(221,253,231,243),(222,252,232,242),(223,251,233,241),(224,250,234,260),(225,249,235,259),(226,248,236,258),(227,247,237,257),(228,246,238,256),(229,245,239,255),(230,244,240,254),(281,303,291,313),(282,302,292,312),(283,301,293,311),(284,320,294,310),(285,319,295,309),(286,318,296,308),(287,317,297,307),(288,316,298,306),(289,315,299,305),(290,314,300,304)], [(1,95,261,55),(2,96,262,56),(3,97,263,57),(4,98,264,58),(5,99,265,59),(6,100,266,60),(7,81,267,41),(8,82,268,42),(9,83,269,43),(10,84,270,44),(11,85,271,45),(12,86,272,46),(13,87,273,47),(14,88,274,48),(15,89,275,49),(16,90,276,50),(17,91,277,51),(18,92,278,52),(19,93,279,53),(20,94,280,54),(21,295,226,184),(22,296,227,185),(23,297,228,186),(24,298,229,187),(25,299,230,188),(26,300,231,189),(27,281,232,190),(28,282,233,191),(29,283,234,192),(30,284,235,193),(31,285,236,194),(32,286,237,195),(33,287,238,196),(34,288,239,197),(35,289,240,198),(36,290,221,199),(37,291,222,200),(38,292,223,181),(39,293,224,182),(40,294,225,183),(61,202,121,150),(62,203,122,151),(63,204,123,152),(64,205,124,153),(65,206,125,154),(66,207,126,155),(67,208,127,156),(68,209,128,157),(69,210,129,158),(70,211,130,159),(71,212,131,160),(72,213,132,141),(73,214,133,142),(74,215,134,143),(75,216,135,144),(76,217,136,145),(77,218,137,146),(78,219,138,147),(79,220,139,148),(80,201,140,149),(101,170,315,244),(102,171,316,245),(103,172,317,246),(104,173,318,247),(105,174,319,248),(106,175,320,249),(107,176,301,250),(108,177,302,251),(109,178,303,252),(110,179,304,253),(111,180,305,254),(112,161,306,255),(113,162,307,256),(114,163,308,257),(115,164,309,258),(116,165,310,259),(117,166,311,260),(118,167,312,241),(119,168,313,242),(120,169,314,243)], [(1,317,261,103),(2,306,262,112),(3,315,263,101),(4,304,264,110),(5,313,265,119),(6,302,266,108),(7,311,267,117),(8,320,268,106),(9,309,269,115),(10,318,270,104),(11,307,271,113),(12,316,272,102),(13,305,273,111),(14,314,274,120),(15,303,275,109),(16,312,276,118),(17,301,277,107),(18,310,278,116),(19,319,279,105),(20,308,280,114),(21,139,226,79),(22,128,227,68),(23,137,228,77),(24,126,229,66),(25,135,230,75),(26,124,231,64),(27,133,232,73),(28,122,233,62),(29,131,234,71),(30,140,235,80),(31,129,236,69),(32,138,237,78),(33,127,238,67),(34,136,239,76),(35,125,240,65),(36,134,221,74),(37,123,222,63),(38,132,223,72),(39,121,224,61),(40,130,225,70),(41,260,81,166),(42,249,82,175),(43,258,83,164),(44,247,84,173),(45,256,85,162),(46,245,86,171),(47,254,87,180),(48,243,88,169),(49,252,89,178),(50,241,90,167),(51,250,91,176),(52,259,92,165),(53,248,93,174),(54,257,94,163),(55,246,95,172),(56,255,96,161),(57,244,97,170),(58,253,98,179),(59,242,99,168),(60,251,100,177),(141,292,213,181),(142,281,214,190),(143,290,215,199),(144,299,216,188),(145,288,217,197),(146,297,218,186),(147,286,219,195),(148,295,220,184),(149,284,201,193),(150,293,202,182),(151,282,203,191),(152,291,204,200),(153,300,205,189),(154,289,206,198),(155,298,207,187),(156,287,208,196),(157,296,209,185),(158,285,210,194),(159,294,211,183),(160,283,212,192)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | 2- 1+4 | Q8×D5 | D5×C4○D4 | D4.10D10 |
| kernel | Dic10⋊7Q8 | C4×Dic10 | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | C5×C42.C2 | Dic10 | C42.C2 | Dic5 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 4 | 3 | 4 | 1 | 1 | 4 | 2 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊7Q8 ►in GL6(𝔽41)
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 31 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 39 | 28 | 0 | 0 | 0 | 0 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 37 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 37 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 35 |
| 0 | 0 | 0 | 0 | 35 | 39 |
G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,32,0,0,0,0,0,0,0,31,31,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[39,13,0,0,0,0,28,2,0,0,0,0,0,0,38,2,0,0,0,0,37,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,38,2,0,0,0,0,37,3,0,0,0,0,0,0,2,35,0,0,0,0,35,39] >;
Dic10⋊7Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_7Q_8 % in TeX
G:=Group("Dic10:7Q8"); // GroupNames label
G:=SmallGroup(320,1357);
// by ID
G=gap.SmallGroup(320,1357);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,297,192,12550]);
// 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,d*a*d^-1=a^9,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations