metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6Dic10, C42.124D10, C10.652- 1+4, (C5×Q8)⋊7Q8, C5⋊3(Q8⋊3Q8), (C4×Q8).12D5, C20.46(C2×Q8), C4⋊C4.293D10, (Q8×C20).13C2, (C2×Q8).198D10, C20⋊2Q8.26C2, (Q8×Dic5).12C2, C4.19(C2×Dic10), C20.333(C4○D4), C10.17(C22×Q8), (C2×C10).114C24, (C2×C20).168C23, (C4×C20).166C22, C4.49(Q8⋊2D5), (C4×Dic10).23C2, C4⋊Dic5.43C22, C4.Dic10.12C2, (Q8×C10).214C22, (C4×Dic5).91C22, (C2×Dic5).52C23, C2.19(C22×Dic10), C22.139(C23×D5), C2.22(D4.10D10), (C2×Dic10).250C22, C10.D4.115C22, C10.110(C2×C4○D4), C2.10(C2×Q8⋊2D5), (C5×C4⋊C4).342C22, (C2×C4).733(C22×D5), SmallGroup(320,1242)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊6Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 550 in 200 conjugacy classes, 115 normal (18 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Q8⋊3Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×Dic10, C20⋊2Q8, C4.Dic10, Q8×Dic5, Q8×C20, Q8⋊6Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2- 1+4, Dic10, C22×D5, Q8⋊3Q8, C2×Dic10, Q8⋊2D5, C23×D5, C22×Dic10, C2×Q8⋊2D5, D4.10D10, Q8⋊6Dic10
►Regular action on 320 points
Generators in S320
(1 64 155 226)(2 65 156 227)(3 66 157 228)(4 67 158 229)(5 68 159 230)(6 69 160 231)(7 70 141 232)(8 71 142 233)(9 72 143 234)(10 73 144 235)(11 74 145 236)(12 75 146 237)(13 76 147 238)(14 77 148 239)(15 78 149 240)(16 79 150 221)(17 80 151 222)(18 61 152 223)(19 62 153 224)(20 63 154 225)(21 267 111 250)(22 268 112 251)(23 269 113 252)(24 270 114 253)(25 271 115 254)(26 272 116 255)(27 273 117 256)(28 274 118 257)(29 275 119 258)(30 276 120 259)(31 277 101 260)(32 278 102 241)(33 279 103 242)(34 280 104 243)(35 261 105 244)(36 262 106 245)(37 263 107 246)(38 264 108 247)(39 265 109 248)(40 266 110 249)(41 139 306 165)(42 140 307 166)(43 121 308 167)(44 122 309 168)(45 123 310 169)(46 124 311 170)(47 125 312 171)(48 126 313 172)(49 127 314 173)(50 128 315 174)(51 129 316 175)(52 130 317 176)(53 131 318 177)(54 132 319 178)(55 133 320 179)(56 134 301 180)(57 135 302 161)(58 136 303 162)(59 137 304 163)(60 138 305 164)(81 189 283 220)(82 190 284 201)(83 191 285 202)(84 192 286 203)(85 193 287 204)(86 194 288 205)(87 195 289 206)(88 196 290 207)(89 197 291 208)(90 198 292 209)(91 199 293 210)(92 200 294 211)(93 181 295 212)(94 182 296 213)(95 183 297 214)(96 184 298 215)(97 185 299 216)(98 186 300 217)(99 187 281 218)(100 188 282 219)
(1 272 155 255)(2 273 156 256)(3 274 157 257)(4 275 158 258)(5 276 159 259)(6 277 160 260)(7 278 141 241)(8 279 142 242)(9 280 143 243)(10 261 144 244)(11 262 145 245)(12 263 146 246)(13 264 147 247)(14 265 148 248)(15 266 149 249)(16 267 150 250)(17 268 151 251)(18 269 152 252)(19 270 153 253)(20 271 154 254)(21 221 111 79)(22 222 112 80)(23 223 113 61)(24 224 114 62)(25 225 115 63)(26 226 116 64)(27 227 117 65)(28 228 118 66)(29 229 119 67)(30 230 120 68)(31 231 101 69)(32 232 102 70)(33 233 103 71)(34 234 104 72)(35 235 105 73)(36 236 106 74)(37 237 107 75)(38 238 108 76)(39 239 109 77)(40 240 110 78)(41 83 306 285)(42 84 307 286)(43 85 308 287)(44 86 309 288)(45 87 310 289)(46 88 311 290)(47 89 312 291)(48 90 313 292)(49 91 314 293)(50 92 315 294)(51 93 316 295)(52 94 317 296)(53 95 318 297)(54 96 319 298)(55 97 320 299)(56 98 301 300)(57 99 302 281)(58 100 303 282)(59 81 304 283)(60 82 305 284)(121 204 167 193)(122 205 168 194)(123 206 169 195)(124 207 170 196)(125 208 171 197)(126 209 172 198)(127 210 173 199)(128 211 174 200)(129 212 175 181)(130 213 176 182)(131 214 177 183)(132 215 178 184)(133 216 179 185)(134 217 180 186)(135 218 161 187)(136 219 162 188)(137 220 163 189)(138 201 164 190)(139 202 165 191)(140 203 166 192)
(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)
(1 122 11 132)(2 121 12 131)(3 140 13 130)(4 139 14 129)(5 138 15 128)(6 137 16 127)(7 136 17 126)(8 135 18 125)(9 134 19 124)(10 133 20 123)(21 91 31 81)(22 90 32 100)(23 89 33 99)(24 88 34 98)(25 87 35 97)(26 86 36 96)(27 85 37 95)(28 84 38 94)(29 83 39 93)(30 82 40 92)(41 239 51 229)(42 238 52 228)(43 237 53 227)(44 236 54 226)(45 235 55 225)(46 234 56 224)(47 233 57 223)(48 232 58 222)(49 231 59 221)(50 230 60 240)(61 312 71 302)(62 311 72 301)(63 310 73 320)(64 309 74 319)(65 308 75 318)(66 307 76 317)(67 306 77 316)(68 305 78 315)(69 304 79 314)(70 303 80 313)(101 283 111 293)(102 282 112 292)(103 281 113 291)(104 300 114 290)(105 299 115 289)(106 298 116 288)(107 297 117 287)(108 296 118 286)(109 295 119 285)(110 294 120 284)(141 162 151 172)(142 161 152 171)(143 180 153 170)(144 179 154 169)(145 178 155 168)(146 177 156 167)(147 176 157 166)(148 175 158 165)(149 174 159 164)(150 173 160 163)(181 275 191 265)(182 274 192 264)(183 273 193 263)(184 272 194 262)(185 271 195 261)(186 270 196 280)(187 269 197 279)(188 268 198 278)(189 267 199 277)(190 266 200 276)(201 249 211 259)(202 248 212 258)(203 247 213 257)(204 246 214 256)(205 245 215 255)(206 244 216 254)(207 243 217 253)(208 242 218 252)(209 241 219 251)(210 260 220 250)
G:=sub<Sym(320)| (1,64,155,226)(2,65,156,227)(3,66,157,228)(4,67,158,229)(5,68,159,230)(6,69,160,231)(7,70,141,232)(8,71,142,233)(9,72,143,234)(10,73,144,235)(11,74,145,236)(12,75,146,237)(13,76,147,238)(14,77,148,239)(15,78,149,240)(16,79,150,221)(17,80,151,222)(18,61,152,223)(19,62,153,224)(20,63,154,225)(21,267,111,250)(22,268,112,251)(23,269,113,252)(24,270,114,253)(25,271,115,254)(26,272,116,255)(27,273,117,256)(28,274,118,257)(29,275,119,258)(30,276,120,259)(31,277,101,260)(32,278,102,241)(33,279,103,242)(34,280,104,243)(35,261,105,244)(36,262,106,245)(37,263,107,246)(38,264,108,247)(39,265,109,248)(40,266,110,249)(41,139,306,165)(42,140,307,166)(43,121,308,167)(44,122,309,168)(45,123,310,169)(46,124,311,170)(47,125,312,171)(48,126,313,172)(49,127,314,173)(50,128,315,174)(51,129,316,175)(52,130,317,176)(53,131,318,177)(54,132,319,178)(55,133,320,179)(56,134,301,180)(57,135,302,161)(58,136,303,162)(59,137,304,163)(60,138,305,164)(81,189,283,220)(82,190,284,201)(83,191,285,202)(84,192,286,203)(85,193,287,204)(86,194,288,205)(87,195,289,206)(88,196,290,207)(89,197,291,208)(90,198,292,209)(91,199,293,210)(92,200,294,211)(93,181,295,212)(94,182,296,213)(95,183,297,214)(96,184,298,215)(97,185,299,216)(98,186,300,217)(99,187,281,218)(100,188,282,219), (1,272,155,255)(2,273,156,256)(3,274,157,257)(4,275,158,258)(5,276,159,259)(6,277,160,260)(7,278,141,241)(8,279,142,242)(9,280,143,243)(10,261,144,244)(11,262,145,245)(12,263,146,246)(13,264,147,247)(14,265,148,248)(15,266,149,249)(16,267,150,250)(17,268,151,251)(18,269,152,252)(19,270,153,253)(20,271,154,254)(21,221,111,79)(22,222,112,80)(23,223,113,61)(24,224,114,62)(25,225,115,63)(26,226,116,64)(27,227,117,65)(28,228,118,66)(29,229,119,67)(30,230,120,68)(31,231,101,69)(32,232,102,70)(33,233,103,71)(34,234,104,72)(35,235,105,73)(36,236,106,74)(37,237,107,75)(38,238,108,76)(39,239,109,77)(40,240,110,78)(41,83,306,285)(42,84,307,286)(43,85,308,287)(44,86,309,288)(45,87,310,289)(46,88,311,290)(47,89,312,291)(48,90,313,292)(49,91,314,293)(50,92,315,294)(51,93,316,295)(52,94,317,296)(53,95,318,297)(54,96,319,298)(55,97,320,299)(56,98,301,300)(57,99,302,281)(58,100,303,282)(59,81,304,283)(60,82,305,284)(121,204,167,193)(122,205,168,194)(123,206,169,195)(124,207,170,196)(125,208,171,197)(126,209,172,198)(127,210,173,199)(128,211,174,200)(129,212,175,181)(130,213,176,182)(131,214,177,183)(132,215,178,184)(133,216,179,185)(134,217,180,186)(135,218,161,187)(136,219,162,188)(137,220,163,189)(138,201,164,190)(139,202,165,191)(140,203,166,192), (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), (1,122,11,132)(2,121,12,131)(3,140,13,130)(4,139,14,129)(5,138,15,128)(6,137,16,127)(7,136,17,126)(8,135,18,125)(9,134,19,124)(10,133,20,123)(21,91,31,81)(22,90,32,100)(23,89,33,99)(24,88,34,98)(25,87,35,97)(26,86,36,96)(27,85,37,95)(28,84,38,94)(29,83,39,93)(30,82,40,92)(41,239,51,229)(42,238,52,228)(43,237,53,227)(44,236,54,226)(45,235,55,225)(46,234,56,224)(47,233,57,223)(48,232,58,222)(49,231,59,221)(50,230,60,240)(61,312,71,302)(62,311,72,301)(63,310,73,320)(64,309,74,319)(65,308,75,318)(66,307,76,317)(67,306,77,316)(68,305,78,315)(69,304,79,314)(70,303,80,313)(101,283,111,293)(102,282,112,292)(103,281,113,291)(104,300,114,290)(105,299,115,289)(106,298,116,288)(107,297,117,287)(108,296,118,286)(109,295,119,285)(110,294,120,284)(141,162,151,172)(142,161,152,171)(143,180,153,170)(144,179,154,169)(145,178,155,168)(146,177,156,167)(147,176,157,166)(148,175,158,165)(149,174,159,164)(150,173,160,163)(181,275,191,265)(182,274,192,264)(183,273,193,263)(184,272,194,262)(185,271,195,261)(186,270,196,280)(187,269,197,279)(188,268,198,278)(189,267,199,277)(190,266,200,276)(201,249,211,259)(202,248,212,258)(203,247,213,257)(204,246,214,256)(205,245,215,255)(206,244,216,254)(207,243,217,253)(208,242,218,252)(209,241,219,251)(210,260,220,250)>;
G:=Group( (1,64,155,226)(2,65,156,227)(3,66,157,228)(4,67,158,229)(5,68,159,230)(6,69,160,231)(7,70,141,232)(8,71,142,233)(9,72,143,234)(10,73,144,235)(11,74,145,236)(12,75,146,237)(13,76,147,238)(14,77,148,239)(15,78,149,240)(16,79,150,221)(17,80,151,222)(18,61,152,223)(19,62,153,224)(20,63,154,225)(21,267,111,250)(22,268,112,251)(23,269,113,252)(24,270,114,253)(25,271,115,254)(26,272,116,255)(27,273,117,256)(28,274,118,257)(29,275,119,258)(30,276,120,259)(31,277,101,260)(32,278,102,241)(33,279,103,242)(34,280,104,243)(35,261,105,244)(36,262,106,245)(37,263,107,246)(38,264,108,247)(39,265,109,248)(40,266,110,249)(41,139,306,165)(42,140,307,166)(43,121,308,167)(44,122,309,168)(45,123,310,169)(46,124,311,170)(47,125,312,171)(48,126,313,172)(49,127,314,173)(50,128,315,174)(51,129,316,175)(52,130,317,176)(53,131,318,177)(54,132,319,178)(55,133,320,179)(56,134,301,180)(57,135,302,161)(58,136,303,162)(59,137,304,163)(60,138,305,164)(81,189,283,220)(82,190,284,201)(83,191,285,202)(84,192,286,203)(85,193,287,204)(86,194,288,205)(87,195,289,206)(88,196,290,207)(89,197,291,208)(90,198,292,209)(91,199,293,210)(92,200,294,211)(93,181,295,212)(94,182,296,213)(95,183,297,214)(96,184,298,215)(97,185,299,216)(98,186,300,217)(99,187,281,218)(100,188,282,219), (1,272,155,255)(2,273,156,256)(3,274,157,257)(4,275,158,258)(5,276,159,259)(6,277,160,260)(7,278,141,241)(8,279,142,242)(9,280,143,243)(10,261,144,244)(11,262,145,245)(12,263,146,246)(13,264,147,247)(14,265,148,248)(15,266,149,249)(16,267,150,250)(17,268,151,251)(18,269,152,252)(19,270,153,253)(20,271,154,254)(21,221,111,79)(22,222,112,80)(23,223,113,61)(24,224,114,62)(25,225,115,63)(26,226,116,64)(27,227,117,65)(28,228,118,66)(29,229,119,67)(30,230,120,68)(31,231,101,69)(32,232,102,70)(33,233,103,71)(34,234,104,72)(35,235,105,73)(36,236,106,74)(37,237,107,75)(38,238,108,76)(39,239,109,77)(40,240,110,78)(41,83,306,285)(42,84,307,286)(43,85,308,287)(44,86,309,288)(45,87,310,289)(46,88,311,290)(47,89,312,291)(48,90,313,292)(49,91,314,293)(50,92,315,294)(51,93,316,295)(52,94,317,296)(53,95,318,297)(54,96,319,298)(55,97,320,299)(56,98,301,300)(57,99,302,281)(58,100,303,282)(59,81,304,283)(60,82,305,284)(121,204,167,193)(122,205,168,194)(123,206,169,195)(124,207,170,196)(125,208,171,197)(126,209,172,198)(127,210,173,199)(128,211,174,200)(129,212,175,181)(130,213,176,182)(131,214,177,183)(132,215,178,184)(133,216,179,185)(134,217,180,186)(135,218,161,187)(136,219,162,188)(137,220,163,189)(138,201,164,190)(139,202,165,191)(140,203,166,192), (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), (1,122,11,132)(2,121,12,131)(3,140,13,130)(4,139,14,129)(5,138,15,128)(6,137,16,127)(7,136,17,126)(8,135,18,125)(9,134,19,124)(10,133,20,123)(21,91,31,81)(22,90,32,100)(23,89,33,99)(24,88,34,98)(25,87,35,97)(26,86,36,96)(27,85,37,95)(28,84,38,94)(29,83,39,93)(30,82,40,92)(41,239,51,229)(42,238,52,228)(43,237,53,227)(44,236,54,226)(45,235,55,225)(46,234,56,224)(47,233,57,223)(48,232,58,222)(49,231,59,221)(50,230,60,240)(61,312,71,302)(62,311,72,301)(63,310,73,320)(64,309,74,319)(65,308,75,318)(66,307,76,317)(67,306,77,316)(68,305,78,315)(69,304,79,314)(70,303,80,313)(101,283,111,293)(102,282,112,292)(103,281,113,291)(104,300,114,290)(105,299,115,289)(106,298,116,288)(107,297,117,287)(108,296,118,286)(109,295,119,285)(110,294,120,284)(141,162,151,172)(142,161,152,171)(143,180,153,170)(144,179,154,169)(145,178,155,168)(146,177,156,167)(147,176,157,166)(148,175,158,165)(149,174,159,164)(150,173,160,163)(181,275,191,265)(182,274,192,264)(183,273,193,263)(184,272,194,262)(185,271,195,261)(186,270,196,280)(187,269,197,279)(188,268,198,278)(189,267,199,277)(190,266,200,276)(201,249,211,259)(202,248,212,258)(203,247,213,257)(204,246,214,256)(205,245,215,255)(206,244,216,254)(207,243,217,253)(208,242,218,252)(209,241,219,251)(210,260,220,250) );
G=PermutationGroup([[(1,64,155,226),(2,65,156,227),(3,66,157,228),(4,67,158,229),(5,68,159,230),(6,69,160,231),(7,70,141,232),(8,71,142,233),(9,72,143,234),(10,73,144,235),(11,74,145,236),(12,75,146,237),(13,76,147,238),(14,77,148,239),(15,78,149,240),(16,79,150,221),(17,80,151,222),(18,61,152,223),(19,62,153,224),(20,63,154,225),(21,267,111,250),(22,268,112,251),(23,269,113,252),(24,270,114,253),(25,271,115,254),(26,272,116,255),(27,273,117,256),(28,274,118,257),(29,275,119,258),(30,276,120,259),(31,277,101,260),(32,278,102,241),(33,279,103,242),(34,280,104,243),(35,261,105,244),(36,262,106,245),(37,263,107,246),(38,264,108,247),(39,265,109,248),(40,266,110,249),(41,139,306,165),(42,140,307,166),(43,121,308,167),(44,122,309,168),(45,123,310,169),(46,124,311,170),(47,125,312,171),(48,126,313,172),(49,127,314,173),(50,128,315,174),(51,129,316,175),(52,130,317,176),(53,131,318,177),(54,132,319,178),(55,133,320,179),(56,134,301,180),(57,135,302,161),(58,136,303,162),(59,137,304,163),(60,138,305,164),(81,189,283,220),(82,190,284,201),(83,191,285,202),(84,192,286,203),(85,193,287,204),(86,194,288,205),(87,195,289,206),(88,196,290,207),(89,197,291,208),(90,198,292,209),(91,199,293,210),(92,200,294,211),(93,181,295,212),(94,182,296,213),(95,183,297,214),(96,184,298,215),(97,185,299,216),(98,186,300,217),(99,187,281,218),(100,188,282,219)], [(1,272,155,255),(2,273,156,256),(3,274,157,257),(4,275,158,258),(5,276,159,259),(6,277,160,260),(7,278,141,241),(8,279,142,242),(9,280,143,243),(10,261,144,244),(11,262,145,245),(12,263,146,246),(13,264,147,247),(14,265,148,248),(15,266,149,249),(16,267,150,250),(17,268,151,251),(18,269,152,252),(19,270,153,253),(20,271,154,254),(21,221,111,79),(22,222,112,80),(23,223,113,61),(24,224,114,62),(25,225,115,63),(26,226,116,64),(27,227,117,65),(28,228,118,66),(29,229,119,67),(30,230,120,68),(31,231,101,69),(32,232,102,70),(33,233,103,71),(34,234,104,72),(35,235,105,73),(36,236,106,74),(37,237,107,75),(38,238,108,76),(39,239,109,77),(40,240,110,78),(41,83,306,285),(42,84,307,286),(43,85,308,287),(44,86,309,288),(45,87,310,289),(46,88,311,290),(47,89,312,291),(48,90,313,292),(49,91,314,293),(50,92,315,294),(51,93,316,295),(52,94,317,296),(53,95,318,297),(54,96,319,298),(55,97,320,299),(56,98,301,300),(57,99,302,281),(58,100,303,282),(59,81,304,283),(60,82,305,284),(121,204,167,193),(122,205,168,194),(123,206,169,195),(124,207,170,196),(125,208,171,197),(126,209,172,198),(127,210,173,199),(128,211,174,200),(129,212,175,181),(130,213,176,182),(131,214,177,183),(132,215,178,184),(133,216,179,185),(134,217,180,186),(135,218,161,187),(136,219,162,188),(137,220,163,189),(138,201,164,190),(139,202,165,191),(140,203,166,192)], [(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)], [(1,122,11,132),(2,121,12,131),(3,140,13,130),(4,139,14,129),(5,138,15,128),(6,137,16,127),(7,136,17,126),(8,135,18,125),(9,134,19,124),(10,133,20,123),(21,91,31,81),(22,90,32,100),(23,89,33,99),(24,88,34,98),(25,87,35,97),(26,86,36,96),(27,85,37,95),(28,84,38,94),(29,83,39,93),(30,82,40,92),(41,239,51,229),(42,238,52,228),(43,237,53,227),(44,236,54,226),(45,235,55,225),(46,234,56,224),(47,233,57,223),(48,232,58,222),(49,231,59,221),(50,230,60,240),(61,312,71,302),(62,311,72,301),(63,310,73,320),(64,309,74,319),(65,308,75,318),(66,307,76,317),(67,306,77,316),(68,305,78,315),(69,304,79,314),(70,303,80,313),(101,283,111,293),(102,282,112,292),(103,281,113,291),(104,300,114,290),(105,299,115,289),(106,298,116,288),(107,297,117,287),(108,296,118,286),(109,295,119,285),(110,294,120,284),(141,162,151,172),(142,161,152,171),(143,180,153,170),(144,179,154,169),(145,178,155,168),(146,177,156,167),(147,176,157,166),(148,175,158,165),(149,174,159,164),(150,173,160,163),(181,275,191,265),(182,274,192,264),(183,273,193,263),(184,272,194,262),(185,271,195,261),(186,270,196,280),(187,269,197,279),(188,268,198,278),(189,267,199,277),(190,266,200,276),(201,249,211,259),(202,248,212,258),(203,247,213,257),(204,246,214,256),(205,245,215,255),(206,244,216,254),(207,243,217,253),(208,242,218,252),(209,241,219,251),(210,260,220,250)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | Dic10 | 2- 1+4 | Q8⋊2D5 | D4.10D10 |
| kernel | Q8⋊6Dic10 | C4×Dic10 | C20⋊2Q8 | C4.Dic10 | Q8×Dic5 | Q8×C20 | C5×Q8 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of Q8⋊6Dic10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 40 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 32 |
| 27 | 2 | 0 | 0 |
| 25 | 11 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 22 | 13 | 0 | 0 |
| 10 | 19 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 32 | 32 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,40,0,0,2,40],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,18,32],[27,25,0,0,2,11,0,0,0,0,40,0,0,0,0,40],[22,10,0,0,13,19,0,0,0,0,9,32,0,0,18,32] >;
Q8⋊6Dic10 in GAP, Magma, Sage, TeX
Q_8\rtimes_6{\rm Dic}_{10} % in TeX
G:=Group("Q8:6Dic10"); // GroupNames label
G:=SmallGroup(320,1242);
// by ID
G=gap.SmallGroup(320,1242);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=c^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations