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, (C2×Dic5).52C23, (C4×Dic5).91C22, 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
Subgroups: 550 in 200 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C4 [×8], C4 [×11], C22, C5, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], C10 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic5 [×8], C20 [×8], C20 [×3], C2×C10, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic10 [×6], C2×Dic5 [×8], C2×C20, C2×C20 [×6], C5×Q8 [×4], Q8⋊3Q8, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×12], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], Q8×C10, C4×Dic10 [×3], C20⋊2Q8 [×3], C4.Dic10 [×6], Q8×Dic5 [×2], Q8×C20, Q8⋊6Dic10
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), Dic10 [×4], C22×D5 [×7], Q8⋊3Q8, C2×Dic10 [×6], Q8⋊2D5 [×2], C23×D5, C22×Dic10, C2×Q8⋊2D5, D4.10D10, Q8⋊6Dic10
Generators and relations
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 >
►Regular action on 320 points
Generators in S320
(1 79 275 141)(2 80 276 142)(3 61 277 143)(4 62 278 144)(5 63 279 145)(6 64 280 146)(7 65 261 147)(8 66 262 148)(9 67 263 149)(10 68 264 150)(11 69 265 151)(12 70 266 152)(13 71 267 153)(14 72 268 154)(15 73 269 155)(16 74 270 156)(17 75 271 157)(18 76 272 158)(19 77 273 159)(20 78 274 160)(21 120 297 236)(22 101 298 237)(23 102 299 238)(24 103 300 239)(25 104 281 240)(26 105 282 221)(27 106 283 222)(28 107 284 223)(29 108 285 224)(30 109 286 225)(31 110 287 226)(32 111 288 227)(33 112 289 228)(34 113 290 229)(35 114 291 230)(36 115 292 231)(37 116 293 232)(38 117 294 233)(39 118 295 234)(40 119 296 235)(41 133 318 82)(42 134 319 83)(43 135 320 84)(44 136 301 85)(45 137 302 86)(46 138 303 87)(47 139 304 88)(48 140 305 89)(49 121 306 90)(50 122 307 91)(51 123 308 92)(52 124 309 93)(53 125 310 94)(54 126 311 95)(55 127 312 96)(56 128 313 97)(57 129 314 98)(58 130 315 99)(59 131 316 100)(60 132 317 81)(161 198 208 246)(162 199 209 247)(163 200 210 248)(164 181 211 249)(165 182 212 250)(166 183 213 251)(167 184 214 252)(168 185 215 253)(169 186 216 254)(170 187 217 255)(171 188 218 256)(172 189 219 257)(173 190 220 258)(174 191 201 259)(175 192 202 260)(176 193 203 241)(177 194 204 242)(178 195 205 243)(179 196 206 244)(180 197 207 245)
(1 25 275 281)(2 26 276 282)(3 27 277 283)(4 28 278 284)(5 29 279 285)(6 30 280 286)(7 31 261 287)(8 32 262 288)(9 33 263 289)(10 34 264 290)(11 35 265 291)(12 36 266 292)(13 37 267 293)(14 38 268 294)(15 39 269 295)(16 40 270 296)(17 21 271 297)(18 22 272 298)(19 23 273 299)(20 24 274 300)(41 161 318 208)(42 162 319 209)(43 163 320 210)(44 164 301 211)(45 165 302 212)(46 166 303 213)(47 167 304 214)(48 168 305 215)(49 169 306 216)(50 170 307 217)(51 171 308 218)(52 172 309 219)(53 173 310 220)(54 174 311 201)(55 175 312 202)(56 176 313 203)(57 177 314 204)(58 178 315 205)(59 179 316 206)(60 180 317 207)(61 222 143 106)(62 223 144 107)(63 224 145 108)(64 225 146 109)(65 226 147 110)(66 227 148 111)(67 228 149 112)(68 229 150 113)(69 230 151 114)(70 231 152 115)(71 232 153 116)(72 233 154 117)(73 234 155 118)(74 235 156 119)(75 236 157 120)(76 237 158 101)(77 238 159 102)(78 239 160 103)(79 240 141 104)(80 221 142 105)(81 197 132 245)(82 198 133 246)(83 199 134 247)(84 200 135 248)(85 181 136 249)(86 182 137 250)(87 183 138 251)(88 184 139 252)(89 185 140 253)(90 186 121 254)(91 187 122 255)(92 188 123 256)(93 189 124 257)(94 190 125 258)(95 191 126 259)(96 192 127 260)(97 193 128 241)(98 194 129 242)(99 195 130 243)(100 196 131 244)
(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)
(1 126 11 136)(2 125 12 135)(3 124 13 134)(4 123 14 133)(5 122 15 132)(6 121 16 131)(7 140 17 130)(8 139 18 129)(9 138 19 128)(10 137 20 127)(21 195 31 185)(22 194 32 184)(23 193 33 183)(24 192 34 182)(25 191 35 181)(26 190 36 200)(27 189 37 199)(28 188 38 198)(29 187 39 197)(30 186 40 196)(41 144 51 154)(42 143 52 153)(43 142 53 152)(44 141 54 151)(45 160 55 150)(46 159 56 149)(47 158 57 148)(48 157 58 147)(49 156 59 146)(50 155 60 145)(61 309 71 319)(62 308 72 318)(63 307 73 317)(64 306 74 316)(65 305 75 315)(66 304 76 314)(67 303 77 313)(68 302 78 312)(69 301 79 311)(70 320 80 310)(81 279 91 269)(82 278 92 268)(83 277 93 267)(84 276 94 266)(85 275 95 265)(86 274 96 264)(87 273 97 263)(88 272 98 262)(89 271 99 261)(90 270 100 280)(101 204 111 214)(102 203 112 213)(103 202 113 212)(104 201 114 211)(105 220 115 210)(106 219 116 209)(107 218 117 208)(108 217 118 207)(109 216 119 206)(110 215 120 205)(161 223 171 233)(162 222 172 232)(163 221 173 231)(164 240 174 230)(165 239 175 229)(166 238 176 228)(167 237 177 227)(168 236 178 226)(169 235 179 225)(170 234 180 224)(241 289 251 299)(242 288 252 298)(243 287 253 297)(244 286 254 296)(245 285 255 295)(246 284 256 294)(247 283 257 293)(248 282 258 292)(249 281 259 291)(250 300 260 290)
G:=sub<Sym(320)| (1,79,275,141)(2,80,276,142)(3,61,277,143)(4,62,278,144)(5,63,279,145)(6,64,280,146)(7,65,261,147)(8,66,262,148)(9,67,263,149)(10,68,264,150)(11,69,265,151)(12,70,266,152)(13,71,267,153)(14,72,268,154)(15,73,269,155)(16,74,270,156)(17,75,271,157)(18,76,272,158)(19,77,273,159)(20,78,274,160)(21,120,297,236)(22,101,298,237)(23,102,299,238)(24,103,300,239)(25,104,281,240)(26,105,282,221)(27,106,283,222)(28,107,284,223)(29,108,285,224)(30,109,286,225)(31,110,287,226)(32,111,288,227)(33,112,289,228)(34,113,290,229)(35,114,291,230)(36,115,292,231)(37,116,293,232)(38,117,294,233)(39,118,295,234)(40,119,296,235)(41,133,318,82)(42,134,319,83)(43,135,320,84)(44,136,301,85)(45,137,302,86)(46,138,303,87)(47,139,304,88)(48,140,305,89)(49,121,306,90)(50,122,307,91)(51,123,308,92)(52,124,309,93)(53,125,310,94)(54,126,311,95)(55,127,312,96)(56,128,313,97)(57,129,314,98)(58,130,315,99)(59,131,316,100)(60,132,317,81)(161,198,208,246)(162,199,209,247)(163,200,210,248)(164,181,211,249)(165,182,212,250)(166,183,213,251)(167,184,214,252)(168,185,215,253)(169,186,216,254)(170,187,217,255)(171,188,218,256)(172,189,219,257)(173,190,220,258)(174,191,201,259)(175,192,202,260)(176,193,203,241)(177,194,204,242)(178,195,205,243)(179,196,206,244)(180,197,207,245), (1,25,275,281)(2,26,276,282)(3,27,277,283)(4,28,278,284)(5,29,279,285)(6,30,280,286)(7,31,261,287)(8,32,262,288)(9,33,263,289)(10,34,264,290)(11,35,265,291)(12,36,266,292)(13,37,267,293)(14,38,268,294)(15,39,269,295)(16,40,270,296)(17,21,271,297)(18,22,272,298)(19,23,273,299)(20,24,274,300)(41,161,318,208)(42,162,319,209)(43,163,320,210)(44,164,301,211)(45,165,302,212)(46,166,303,213)(47,167,304,214)(48,168,305,215)(49,169,306,216)(50,170,307,217)(51,171,308,218)(52,172,309,219)(53,173,310,220)(54,174,311,201)(55,175,312,202)(56,176,313,203)(57,177,314,204)(58,178,315,205)(59,179,316,206)(60,180,317,207)(61,222,143,106)(62,223,144,107)(63,224,145,108)(64,225,146,109)(65,226,147,110)(66,227,148,111)(67,228,149,112)(68,229,150,113)(69,230,151,114)(70,231,152,115)(71,232,153,116)(72,233,154,117)(73,234,155,118)(74,235,156,119)(75,236,157,120)(76,237,158,101)(77,238,159,102)(78,239,160,103)(79,240,141,104)(80,221,142,105)(81,197,132,245)(82,198,133,246)(83,199,134,247)(84,200,135,248)(85,181,136,249)(86,182,137,250)(87,183,138,251)(88,184,139,252)(89,185,140,253)(90,186,121,254)(91,187,122,255)(92,188,123,256)(93,189,124,257)(94,190,125,258)(95,191,126,259)(96,192,127,260)(97,193,128,241)(98,194,129,242)(99,195,130,243)(100,196,131,244), (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), (1,126,11,136)(2,125,12,135)(3,124,13,134)(4,123,14,133)(5,122,15,132)(6,121,16,131)(7,140,17,130)(8,139,18,129)(9,138,19,128)(10,137,20,127)(21,195,31,185)(22,194,32,184)(23,193,33,183)(24,192,34,182)(25,191,35,181)(26,190,36,200)(27,189,37,199)(28,188,38,198)(29,187,39,197)(30,186,40,196)(41,144,51,154)(42,143,52,153)(43,142,53,152)(44,141,54,151)(45,160,55,150)(46,159,56,149)(47,158,57,148)(48,157,58,147)(49,156,59,146)(50,155,60,145)(61,309,71,319)(62,308,72,318)(63,307,73,317)(64,306,74,316)(65,305,75,315)(66,304,76,314)(67,303,77,313)(68,302,78,312)(69,301,79,311)(70,320,80,310)(81,279,91,269)(82,278,92,268)(83,277,93,267)(84,276,94,266)(85,275,95,265)(86,274,96,264)(87,273,97,263)(88,272,98,262)(89,271,99,261)(90,270,100,280)(101,204,111,214)(102,203,112,213)(103,202,113,212)(104,201,114,211)(105,220,115,210)(106,219,116,209)(107,218,117,208)(108,217,118,207)(109,216,119,206)(110,215,120,205)(161,223,171,233)(162,222,172,232)(163,221,173,231)(164,240,174,230)(165,239,175,229)(166,238,176,228)(167,237,177,227)(168,236,178,226)(169,235,179,225)(170,234,180,224)(241,289,251,299)(242,288,252,298)(243,287,253,297)(244,286,254,296)(245,285,255,295)(246,284,256,294)(247,283,257,293)(248,282,258,292)(249,281,259,291)(250,300,260,290)>;
G:=Group( (1,79,275,141)(2,80,276,142)(3,61,277,143)(4,62,278,144)(5,63,279,145)(6,64,280,146)(7,65,261,147)(8,66,262,148)(9,67,263,149)(10,68,264,150)(11,69,265,151)(12,70,266,152)(13,71,267,153)(14,72,268,154)(15,73,269,155)(16,74,270,156)(17,75,271,157)(18,76,272,158)(19,77,273,159)(20,78,274,160)(21,120,297,236)(22,101,298,237)(23,102,299,238)(24,103,300,239)(25,104,281,240)(26,105,282,221)(27,106,283,222)(28,107,284,223)(29,108,285,224)(30,109,286,225)(31,110,287,226)(32,111,288,227)(33,112,289,228)(34,113,290,229)(35,114,291,230)(36,115,292,231)(37,116,293,232)(38,117,294,233)(39,118,295,234)(40,119,296,235)(41,133,318,82)(42,134,319,83)(43,135,320,84)(44,136,301,85)(45,137,302,86)(46,138,303,87)(47,139,304,88)(48,140,305,89)(49,121,306,90)(50,122,307,91)(51,123,308,92)(52,124,309,93)(53,125,310,94)(54,126,311,95)(55,127,312,96)(56,128,313,97)(57,129,314,98)(58,130,315,99)(59,131,316,100)(60,132,317,81)(161,198,208,246)(162,199,209,247)(163,200,210,248)(164,181,211,249)(165,182,212,250)(166,183,213,251)(167,184,214,252)(168,185,215,253)(169,186,216,254)(170,187,217,255)(171,188,218,256)(172,189,219,257)(173,190,220,258)(174,191,201,259)(175,192,202,260)(176,193,203,241)(177,194,204,242)(178,195,205,243)(179,196,206,244)(180,197,207,245), (1,25,275,281)(2,26,276,282)(3,27,277,283)(4,28,278,284)(5,29,279,285)(6,30,280,286)(7,31,261,287)(8,32,262,288)(9,33,263,289)(10,34,264,290)(11,35,265,291)(12,36,266,292)(13,37,267,293)(14,38,268,294)(15,39,269,295)(16,40,270,296)(17,21,271,297)(18,22,272,298)(19,23,273,299)(20,24,274,300)(41,161,318,208)(42,162,319,209)(43,163,320,210)(44,164,301,211)(45,165,302,212)(46,166,303,213)(47,167,304,214)(48,168,305,215)(49,169,306,216)(50,170,307,217)(51,171,308,218)(52,172,309,219)(53,173,310,220)(54,174,311,201)(55,175,312,202)(56,176,313,203)(57,177,314,204)(58,178,315,205)(59,179,316,206)(60,180,317,207)(61,222,143,106)(62,223,144,107)(63,224,145,108)(64,225,146,109)(65,226,147,110)(66,227,148,111)(67,228,149,112)(68,229,150,113)(69,230,151,114)(70,231,152,115)(71,232,153,116)(72,233,154,117)(73,234,155,118)(74,235,156,119)(75,236,157,120)(76,237,158,101)(77,238,159,102)(78,239,160,103)(79,240,141,104)(80,221,142,105)(81,197,132,245)(82,198,133,246)(83,199,134,247)(84,200,135,248)(85,181,136,249)(86,182,137,250)(87,183,138,251)(88,184,139,252)(89,185,140,253)(90,186,121,254)(91,187,122,255)(92,188,123,256)(93,189,124,257)(94,190,125,258)(95,191,126,259)(96,192,127,260)(97,193,128,241)(98,194,129,242)(99,195,130,243)(100,196,131,244), (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), (1,126,11,136)(2,125,12,135)(3,124,13,134)(4,123,14,133)(5,122,15,132)(6,121,16,131)(7,140,17,130)(8,139,18,129)(9,138,19,128)(10,137,20,127)(21,195,31,185)(22,194,32,184)(23,193,33,183)(24,192,34,182)(25,191,35,181)(26,190,36,200)(27,189,37,199)(28,188,38,198)(29,187,39,197)(30,186,40,196)(41,144,51,154)(42,143,52,153)(43,142,53,152)(44,141,54,151)(45,160,55,150)(46,159,56,149)(47,158,57,148)(48,157,58,147)(49,156,59,146)(50,155,60,145)(61,309,71,319)(62,308,72,318)(63,307,73,317)(64,306,74,316)(65,305,75,315)(66,304,76,314)(67,303,77,313)(68,302,78,312)(69,301,79,311)(70,320,80,310)(81,279,91,269)(82,278,92,268)(83,277,93,267)(84,276,94,266)(85,275,95,265)(86,274,96,264)(87,273,97,263)(88,272,98,262)(89,271,99,261)(90,270,100,280)(101,204,111,214)(102,203,112,213)(103,202,113,212)(104,201,114,211)(105,220,115,210)(106,219,116,209)(107,218,117,208)(108,217,118,207)(109,216,119,206)(110,215,120,205)(161,223,171,233)(162,222,172,232)(163,221,173,231)(164,240,174,230)(165,239,175,229)(166,238,176,228)(167,237,177,227)(168,236,178,226)(169,235,179,225)(170,234,180,224)(241,289,251,299)(242,288,252,298)(243,287,253,297)(244,286,254,296)(245,285,255,295)(246,284,256,294)(247,283,257,293)(248,282,258,292)(249,281,259,291)(250,300,260,290) );
G=PermutationGroup([(1,79,275,141),(2,80,276,142),(3,61,277,143),(4,62,278,144),(5,63,279,145),(6,64,280,146),(7,65,261,147),(8,66,262,148),(9,67,263,149),(10,68,264,150),(11,69,265,151),(12,70,266,152),(13,71,267,153),(14,72,268,154),(15,73,269,155),(16,74,270,156),(17,75,271,157),(18,76,272,158),(19,77,273,159),(20,78,274,160),(21,120,297,236),(22,101,298,237),(23,102,299,238),(24,103,300,239),(25,104,281,240),(26,105,282,221),(27,106,283,222),(28,107,284,223),(29,108,285,224),(30,109,286,225),(31,110,287,226),(32,111,288,227),(33,112,289,228),(34,113,290,229),(35,114,291,230),(36,115,292,231),(37,116,293,232),(38,117,294,233),(39,118,295,234),(40,119,296,235),(41,133,318,82),(42,134,319,83),(43,135,320,84),(44,136,301,85),(45,137,302,86),(46,138,303,87),(47,139,304,88),(48,140,305,89),(49,121,306,90),(50,122,307,91),(51,123,308,92),(52,124,309,93),(53,125,310,94),(54,126,311,95),(55,127,312,96),(56,128,313,97),(57,129,314,98),(58,130,315,99),(59,131,316,100),(60,132,317,81),(161,198,208,246),(162,199,209,247),(163,200,210,248),(164,181,211,249),(165,182,212,250),(166,183,213,251),(167,184,214,252),(168,185,215,253),(169,186,216,254),(170,187,217,255),(171,188,218,256),(172,189,219,257),(173,190,220,258),(174,191,201,259),(175,192,202,260),(176,193,203,241),(177,194,204,242),(178,195,205,243),(179,196,206,244),(180,197,207,245)], [(1,25,275,281),(2,26,276,282),(3,27,277,283),(4,28,278,284),(5,29,279,285),(6,30,280,286),(7,31,261,287),(8,32,262,288),(9,33,263,289),(10,34,264,290),(11,35,265,291),(12,36,266,292),(13,37,267,293),(14,38,268,294),(15,39,269,295),(16,40,270,296),(17,21,271,297),(18,22,272,298),(19,23,273,299),(20,24,274,300),(41,161,318,208),(42,162,319,209),(43,163,320,210),(44,164,301,211),(45,165,302,212),(46,166,303,213),(47,167,304,214),(48,168,305,215),(49,169,306,216),(50,170,307,217),(51,171,308,218),(52,172,309,219),(53,173,310,220),(54,174,311,201),(55,175,312,202),(56,176,313,203),(57,177,314,204),(58,178,315,205),(59,179,316,206),(60,180,317,207),(61,222,143,106),(62,223,144,107),(63,224,145,108),(64,225,146,109),(65,226,147,110),(66,227,148,111),(67,228,149,112),(68,229,150,113),(69,230,151,114),(70,231,152,115),(71,232,153,116),(72,233,154,117),(73,234,155,118),(74,235,156,119),(75,236,157,120),(76,237,158,101),(77,238,159,102),(78,239,160,103),(79,240,141,104),(80,221,142,105),(81,197,132,245),(82,198,133,246),(83,199,134,247),(84,200,135,248),(85,181,136,249),(86,182,137,250),(87,183,138,251),(88,184,139,252),(89,185,140,253),(90,186,121,254),(91,187,122,255),(92,188,123,256),(93,189,124,257),(94,190,125,258),(95,191,126,259),(96,192,127,260),(97,193,128,241),(98,194,129,242),(99,195,130,243),(100,196,131,244)], [(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)], [(1,126,11,136),(2,125,12,135),(3,124,13,134),(4,123,14,133),(5,122,15,132),(6,121,16,131),(7,140,17,130),(8,139,18,129),(9,138,19,128),(10,137,20,127),(21,195,31,185),(22,194,32,184),(23,193,33,183),(24,192,34,182),(25,191,35,181),(26,190,36,200),(27,189,37,199),(28,188,38,198),(29,187,39,197),(30,186,40,196),(41,144,51,154),(42,143,52,153),(43,142,53,152),(44,141,54,151),(45,160,55,150),(46,159,56,149),(47,158,57,148),(48,157,58,147),(49,156,59,146),(50,155,60,145),(61,309,71,319),(62,308,72,318),(63,307,73,317),(64,306,74,316),(65,305,75,315),(66,304,76,314),(67,303,77,313),(68,302,78,312),(69,301,79,311),(70,320,80,310),(81,279,91,269),(82,278,92,268),(83,277,93,267),(84,276,94,266),(85,275,95,265),(86,274,96,264),(87,273,97,263),(88,272,98,262),(89,271,99,261),(90,270,100,280),(101,204,111,214),(102,203,112,213),(103,202,113,212),(104,201,114,211),(105,220,115,210),(106,219,116,209),(107,218,117,208),(108,217,118,207),(109,216,119,206),(110,215,120,205),(161,223,171,233),(162,222,172,232),(163,221,173,231),(164,240,174,230),(165,239,175,229),(166,238,176,228),(167,237,177,227),(168,236,178,226),(169,235,179,225),(170,234,180,224),(241,289,251,299),(242,288,252,298),(243,287,253,297),(244,286,254,296),(245,285,255,295),(246,284,256,294),(247,283,257,293),(248,282,258,292),(249,281,259,291),(250,300,260,290)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_6Dic_{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
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