direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×Dic10, C42.120D10, C10.1082+ (1+4), C5⋊1(Q82), (C5×Q8)⋊5Q8, (C4×Q8).9D5, C4.48(Q8×D5), C20⋊Q8.11C2, C20.44(C2×Q8), C4⋊C4.290D10, (Q8×C20).10C2, (C2×Q8).196D10, C20⋊2Q8.24C2, Dic5.21(C2×Q8), (Q8×Dic5).10C2, C4.17(C2×Dic10), C10.15(C22×Q8), (C4×C20).163C22, (C2×C10).110C24, (C2×C20).167C23, (C4×Dic10).20C2, C2.21(D4⋊8D10), C4⋊Dic5.201C22, (Q8×C10).210C22, (C4×Dic5).87C22, C2.17(C22×Dic10), C22.135(C23×D5), (C2×Dic5).220C23, (C2×Dic10).32C22, C10.D4.113C22, C2.10(C2×Q8×D5), (C5×C4⋊C4).338C22, (C2×C4).582(C22×D5), SmallGroup(320,1238)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 646 in 212 conjugacy classes, 123 normal (18 characteristic)
C1, C2 [×3], C4 [×8], C4 [×13], C22, C5, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×10], C10 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×15], C2×Q8, C2×Q8 [×7], Dic5 [×4], Dic5 [×6], C20 [×8], C20 [×3], C2×C10, C4×Q8, C4×Q8 [×5], C4⋊Q8 [×9], Dic10 [×4], Dic10 [×6], C2×Dic5 [×8], C2×C20, C2×C20 [×6], C5×Q8 [×4], Q82, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5 [×9], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×Dic10 [×6], Q8×C10, C4×Dic10 [×3], C20⋊2Q8 [×3], C20⋊Q8 [×6], Q8×Dic5 [×2], Q8×C20, Q8×Dic10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], D5, C2×Q8 [×12], C24, D10 [×7], C22×Q8 [×2], 2+ (1+4), Dic10 [×4], C22×D5 [×7], Q82, C2×Dic10 [×6], Q8×D5 [×2], C23×D5, C22×Dic10, C2×Q8×D5, D4⋊8D10, Q8×Dic10
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, bd=db, dcd-1=c-1 >
(1 51 305 110)(2 52 306 111)(3 53 307 112)(4 54 308 113)(5 55 309 114)(6 56 310 115)(7 57 311 116)(8 58 312 117)(9 59 313 118)(10 60 314 119)(11 41 315 120)(12 42 316 101)(13 43 317 102)(14 44 318 103)(15 45 319 104)(16 46 320 105)(17 47 301 106)(18 48 302 107)(19 49 303 108)(20 50 304 109)(21 233 202 148)(22 234 203 149)(23 235 204 150)(24 236 205 151)(25 237 206 152)(26 238 207 153)(27 239 208 154)(28 240 209 155)(29 221 210 156)(30 222 211 157)(31 223 212 158)(32 224 213 159)(33 225 214 160)(34 226 215 141)(35 227 216 142)(36 228 217 143)(37 229 218 144)(38 230 219 145)(39 231 220 146)(40 232 201 147)(61 125 193 252)(62 126 194 253)(63 127 195 254)(64 128 196 255)(65 129 197 256)(66 130 198 257)(67 131 199 258)(68 132 200 259)(69 133 181 260)(70 134 182 241)(71 135 183 242)(72 136 184 243)(73 137 185 244)(74 138 186 245)(75 139 187 246)(76 140 188 247)(77 121 189 248)(78 122 190 249)(79 123 191 250)(80 124 192 251)(81 172 273 298)(82 173 274 299)(83 174 275 300)(84 175 276 281)(85 176 277 282)(86 177 278 283)(87 178 279 284)(88 179 280 285)(89 180 261 286)(90 161 262 287)(91 162 263 288)(92 163 264 289)(93 164 265 290)(94 165 266 291)(95 166 267 292)(96 167 268 293)(97 168 269 294)(98 169 270 295)(99 170 271 296)(100 171 272 297)
(1 187 305 75)(2 188 306 76)(3 189 307 77)(4 190 308 78)(5 191 309 79)(6 192 310 80)(7 193 311 61)(8 194 312 62)(9 195 313 63)(10 196 314 64)(11 197 315 65)(12 198 316 66)(13 199 317 67)(14 200 318 68)(15 181 319 69)(16 182 320 70)(17 183 301 71)(18 184 302 72)(19 185 303 73)(20 186 304 74)(21 273 202 81)(22 274 203 82)(23 275 204 83)(24 276 205 84)(25 277 206 85)(26 278 207 86)(27 279 208 87)(28 280 209 88)(29 261 210 89)(30 262 211 90)(31 263 212 91)(32 264 213 92)(33 265 214 93)(34 266 215 94)(35 267 216 95)(36 268 217 96)(37 269 218 97)(38 270 219 98)(39 271 220 99)(40 272 201 100)(41 129 120 256)(42 130 101 257)(43 131 102 258)(44 132 103 259)(45 133 104 260)(46 134 105 241)(47 135 106 242)(48 136 107 243)(49 137 108 244)(50 138 109 245)(51 139 110 246)(52 140 111 247)(53 121 112 248)(54 122 113 249)(55 123 114 250)(56 124 115 251)(57 125 116 252)(58 126 117 253)(59 127 118 254)(60 128 119 255)(141 291 226 165)(142 292 227 166)(143 293 228 167)(144 294 229 168)(145 295 230 169)(146 296 231 170)(147 297 232 171)(148 298 233 172)(149 299 234 173)(150 300 235 174)(151 281 236 175)(152 282 237 176)(153 283 238 177)(154 284 239 178)(155 285 240 179)(156 286 221 180)(157 287 222 161)(158 288 223 162)(159 289 224 163)(160 290 225 164)
(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)
(1 283 11 293)(2 282 12 292)(3 281 13 291)(4 300 14 290)(5 299 15 289)(6 298 16 288)(7 297 17 287)(8 296 18 286)(9 295 19 285)(10 294 20 284)(21 134 31 124)(22 133 32 123)(23 132 33 122)(24 131 34 121)(25 130 35 140)(26 129 36 139)(27 128 37 138)(28 127 38 137)(29 126 39 136)(30 125 40 135)(41 96 51 86)(42 95 52 85)(43 94 53 84)(44 93 54 83)(45 92 55 82)(46 91 56 81)(47 90 57 100)(48 89 58 99)(49 88 59 98)(50 87 60 97)(61 147 71 157)(62 146 72 156)(63 145 73 155)(64 144 74 154)(65 143 75 153)(66 142 76 152)(67 141 77 151)(68 160 78 150)(69 159 79 149)(70 158 80 148)(101 267 111 277)(102 266 112 276)(103 265 113 275)(104 264 114 274)(105 263 115 273)(106 262 116 272)(107 261 117 271)(108 280 118 270)(109 279 119 269)(110 278 120 268)(161 311 171 301)(162 310 172 320)(163 309 173 319)(164 308 174 318)(165 307 175 317)(166 306 176 316)(167 305 177 315)(168 304 178 314)(169 303 179 313)(170 302 180 312)(181 224 191 234)(182 223 192 233)(183 222 193 232)(184 221 194 231)(185 240 195 230)(186 239 196 229)(187 238 197 228)(188 237 198 227)(189 236 199 226)(190 235 200 225)(201 242 211 252)(202 241 212 251)(203 260 213 250)(204 259 214 249)(205 258 215 248)(206 257 216 247)(207 256 217 246)(208 255 218 245)(209 254 219 244)(210 253 220 243)
G:=sub<Sym(320)| (1,51,305,110)(2,52,306,111)(3,53,307,112)(4,54,308,113)(5,55,309,114)(6,56,310,115)(7,57,311,116)(8,58,312,117)(9,59,313,118)(10,60,314,119)(11,41,315,120)(12,42,316,101)(13,43,317,102)(14,44,318,103)(15,45,319,104)(16,46,320,105)(17,47,301,106)(18,48,302,107)(19,49,303,108)(20,50,304,109)(21,233,202,148)(22,234,203,149)(23,235,204,150)(24,236,205,151)(25,237,206,152)(26,238,207,153)(27,239,208,154)(28,240,209,155)(29,221,210,156)(30,222,211,157)(31,223,212,158)(32,224,213,159)(33,225,214,160)(34,226,215,141)(35,227,216,142)(36,228,217,143)(37,229,218,144)(38,230,219,145)(39,231,220,146)(40,232,201,147)(61,125,193,252)(62,126,194,253)(63,127,195,254)(64,128,196,255)(65,129,197,256)(66,130,198,257)(67,131,199,258)(68,132,200,259)(69,133,181,260)(70,134,182,241)(71,135,183,242)(72,136,184,243)(73,137,185,244)(74,138,186,245)(75,139,187,246)(76,140,188,247)(77,121,189,248)(78,122,190,249)(79,123,191,250)(80,124,192,251)(81,172,273,298)(82,173,274,299)(83,174,275,300)(84,175,276,281)(85,176,277,282)(86,177,278,283)(87,178,279,284)(88,179,280,285)(89,180,261,286)(90,161,262,287)(91,162,263,288)(92,163,264,289)(93,164,265,290)(94,165,266,291)(95,166,267,292)(96,167,268,293)(97,168,269,294)(98,169,270,295)(99,170,271,296)(100,171,272,297), (1,187,305,75)(2,188,306,76)(3,189,307,77)(4,190,308,78)(5,191,309,79)(6,192,310,80)(7,193,311,61)(8,194,312,62)(9,195,313,63)(10,196,314,64)(11,197,315,65)(12,198,316,66)(13,199,317,67)(14,200,318,68)(15,181,319,69)(16,182,320,70)(17,183,301,71)(18,184,302,72)(19,185,303,73)(20,186,304,74)(21,273,202,81)(22,274,203,82)(23,275,204,83)(24,276,205,84)(25,277,206,85)(26,278,207,86)(27,279,208,87)(28,280,209,88)(29,261,210,89)(30,262,211,90)(31,263,212,91)(32,264,213,92)(33,265,214,93)(34,266,215,94)(35,267,216,95)(36,268,217,96)(37,269,218,97)(38,270,219,98)(39,271,220,99)(40,272,201,100)(41,129,120,256)(42,130,101,257)(43,131,102,258)(44,132,103,259)(45,133,104,260)(46,134,105,241)(47,135,106,242)(48,136,107,243)(49,137,108,244)(50,138,109,245)(51,139,110,246)(52,140,111,247)(53,121,112,248)(54,122,113,249)(55,123,114,250)(56,124,115,251)(57,125,116,252)(58,126,117,253)(59,127,118,254)(60,128,119,255)(141,291,226,165)(142,292,227,166)(143,293,228,167)(144,294,229,168)(145,295,230,169)(146,296,231,170)(147,297,232,171)(148,298,233,172)(149,299,234,173)(150,300,235,174)(151,281,236,175)(152,282,237,176)(153,283,238,177)(154,284,239,178)(155,285,240,179)(156,286,221,180)(157,287,222,161)(158,288,223,162)(159,289,224,163)(160,290,225,164), (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), (1,283,11,293)(2,282,12,292)(3,281,13,291)(4,300,14,290)(5,299,15,289)(6,298,16,288)(7,297,17,287)(8,296,18,286)(9,295,19,285)(10,294,20,284)(21,134,31,124)(22,133,32,123)(23,132,33,122)(24,131,34,121)(25,130,35,140)(26,129,36,139)(27,128,37,138)(28,127,38,137)(29,126,39,136)(30,125,40,135)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,147,71,157)(62,146,72,156)(63,145,73,155)(64,144,74,154)(65,143,75,153)(66,142,76,152)(67,141,77,151)(68,160,78,150)(69,159,79,149)(70,158,80,148)(101,267,111,277)(102,266,112,276)(103,265,113,275)(104,264,114,274)(105,263,115,273)(106,262,116,272)(107,261,117,271)(108,280,118,270)(109,279,119,269)(110,278,120,268)(161,311,171,301)(162,310,172,320)(163,309,173,319)(164,308,174,318)(165,307,175,317)(166,306,176,316)(167,305,177,315)(168,304,178,314)(169,303,179,313)(170,302,180,312)(181,224,191,234)(182,223,192,233)(183,222,193,232)(184,221,194,231)(185,240,195,230)(186,239,196,229)(187,238,197,228)(188,237,198,227)(189,236,199,226)(190,235,200,225)(201,242,211,252)(202,241,212,251)(203,260,213,250)(204,259,214,249)(205,258,215,248)(206,257,216,247)(207,256,217,246)(208,255,218,245)(209,254,219,244)(210,253,220,243)>;
G:=Group( (1,51,305,110)(2,52,306,111)(3,53,307,112)(4,54,308,113)(5,55,309,114)(6,56,310,115)(7,57,311,116)(8,58,312,117)(9,59,313,118)(10,60,314,119)(11,41,315,120)(12,42,316,101)(13,43,317,102)(14,44,318,103)(15,45,319,104)(16,46,320,105)(17,47,301,106)(18,48,302,107)(19,49,303,108)(20,50,304,109)(21,233,202,148)(22,234,203,149)(23,235,204,150)(24,236,205,151)(25,237,206,152)(26,238,207,153)(27,239,208,154)(28,240,209,155)(29,221,210,156)(30,222,211,157)(31,223,212,158)(32,224,213,159)(33,225,214,160)(34,226,215,141)(35,227,216,142)(36,228,217,143)(37,229,218,144)(38,230,219,145)(39,231,220,146)(40,232,201,147)(61,125,193,252)(62,126,194,253)(63,127,195,254)(64,128,196,255)(65,129,197,256)(66,130,198,257)(67,131,199,258)(68,132,200,259)(69,133,181,260)(70,134,182,241)(71,135,183,242)(72,136,184,243)(73,137,185,244)(74,138,186,245)(75,139,187,246)(76,140,188,247)(77,121,189,248)(78,122,190,249)(79,123,191,250)(80,124,192,251)(81,172,273,298)(82,173,274,299)(83,174,275,300)(84,175,276,281)(85,176,277,282)(86,177,278,283)(87,178,279,284)(88,179,280,285)(89,180,261,286)(90,161,262,287)(91,162,263,288)(92,163,264,289)(93,164,265,290)(94,165,266,291)(95,166,267,292)(96,167,268,293)(97,168,269,294)(98,169,270,295)(99,170,271,296)(100,171,272,297), (1,187,305,75)(2,188,306,76)(3,189,307,77)(4,190,308,78)(5,191,309,79)(6,192,310,80)(7,193,311,61)(8,194,312,62)(9,195,313,63)(10,196,314,64)(11,197,315,65)(12,198,316,66)(13,199,317,67)(14,200,318,68)(15,181,319,69)(16,182,320,70)(17,183,301,71)(18,184,302,72)(19,185,303,73)(20,186,304,74)(21,273,202,81)(22,274,203,82)(23,275,204,83)(24,276,205,84)(25,277,206,85)(26,278,207,86)(27,279,208,87)(28,280,209,88)(29,261,210,89)(30,262,211,90)(31,263,212,91)(32,264,213,92)(33,265,214,93)(34,266,215,94)(35,267,216,95)(36,268,217,96)(37,269,218,97)(38,270,219,98)(39,271,220,99)(40,272,201,100)(41,129,120,256)(42,130,101,257)(43,131,102,258)(44,132,103,259)(45,133,104,260)(46,134,105,241)(47,135,106,242)(48,136,107,243)(49,137,108,244)(50,138,109,245)(51,139,110,246)(52,140,111,247)(53,121,112,248)(54,122,113,249)(55,123,114,250)(56,124,115,251)(57,125,116,252)(58,126,117,253)(59,127,118,254)(60,128,119,255)(141,291,226,165)(142,292,227,166)(143,293,228,167)(144,294,229,168)(145,295,230,169)(146,296,231,170)(147,297,232,171)(148,298,233,172)(149,299,234,173)(150,300,235,174)(151,281,236,175)(152,282,237,176)(153,283,238,177)(154,284,239,178)(155,285,240,179)(156,286,221,180)(157,287,222,161)(158,288,223,162)(159,289,224,163)(160,290,225,164), (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), (1,283,11,293)(2,282,12,292)(3,281,13,291)(4,300,14,290)(5,299,15,289)(6,298,16,288)(7,297,17,287)(8,296,18,286)(9,295,19,285)(10,294,20,284)(21,134,31,124)(22,133,32,123)(23,132,33,122)(24,131,34,121)(25,130,35,140)(26,129,36,139)(27,128,37,138)(28,127,38,137)(29,126,39,136)(30,125,40,135)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,147,71,157)(62,146,72,156)(63,145,73,155)(64,144,74,154)(65,143,75,153)(66,142,76,152)(67,141,77,151)(68,160,78,150)(69,159,79,149)(70,158,80,148)(101,267,111,277)(102,266,112,276)(103,265,113,275)(104,264,114,274)(105,263,115,273)(106,262,116,272)(107,261,117,271)(108,280,118,270)(109,279,119,269)(110,278,120,268)(161,311,171,301)(162,310,172,320)(163,309,173,319)(164,308,174,318)(165,307,175,317)(166,306,176,316)(167,305,177,315)(168,304,178,314)(169,303,179,313)(170,302,180,312)(181,224,191,234)(182,223,192,233)(183,222,193,232)(184,221,194,231)(185,240,195,230)(186,239,196,229)(187,238,197,228)(188,237,198,227)(189,236,199,226)(190,235,200,225)(201,242,211,252)(202,241,212,251)(203,260,213,250)(204,259,214,249)(205,258,215,248)(206,257,216,247)(207,256,217,246)(208,255,218,245)(209,254,219,244)(210,253,220,243) );
G=PermutationGroup([(1,51,305,110),(2,52,306,111),(3,53,307,112),(4,54,308,113),(5,55,309,114),(6,56,310,115),(7,57,311,116),(8,58,312,117),(9,59,313,118),(10,60,314,119),(11,41,315,120),(12,42,316,101),(13,43,317,102),(14,44,318,103),(15,45,319,104),(16,46,320,105),(17,47,301,106),(18,48,302,107),(19,49,303,108),(20,50,304,109),(21,233,202,148),(22,234,203,149),(23,235,204,150),(24,236,205,151),(25,237,206,152),(26,238,207,153),(27,239,208,154),(28,240,209,155),(29,221,210,156),(30,222,211,157),(31,223,212,158),(32,224,213,159),(33,225,214,160),(34,226,215,141),(35,227,216,142),(36,228,217,143),(37,229,218,144),(38,230,219,145),(39,231,220,146),(40,232,201,147),(61,125,193,252),(62,126,194,253),(63,127,195,254),(64,128,196,255),(65,129,197,256),(66,130,198,257),(67,131,199,258),(68,132,200,259),(69,133,181,260),(70,134,182,241),(71,135,183,242),(72,136,184,243),(73,137,185,244),(74,138,186,245),(75,139,187,246),(76,140,188,247),(77,121,189,248),(78,122,190,249),(79,123,191,250),(80,124,192,251),(81,172,273,298),(82,173,274,299),(83,174,275,300),(84,175,276,281),(85,176,277,282),(86,177,278,283),(87,178,279,284),(88,179,280,285),(89,180,261,286),(90,161,262,287),(91,162,263,288),(92,163,264,289),(93,164,265,290),(94,165,266,291),(95,166,267,292),(96,167,268,293),(97,168,269,294),(98,169,270,295),(99,170,271,296),(100,171,272,297)], [(1,187,305,75),(2,188,306,76),(3,189,307,77),(4,190,308,78),(5,191,309,79),(6,192,310,80),(7,193,311,61),(8,194,312,62),(9,195,313,63),(10,196,314,64),(11,197,315,65),(12,198,316,66),(13,199,317,67),(14,200,318,68),(15,181,319,69),(16,182,320,70),(17,183,301,71),(18,184,302,72),(19,185,303,73),(20,186,304,74),(21,273,202,81),(22,274,203,82),(23,275,204,83),(24,276,205,84),(25,277,206,85),(26,278,207,86),(27,279,208,87),(28,280,209,88),(29,261,210,89),(30,262,211,90),(31,263,212,91),(32,264,213,92),(33,265,214,93),(34,266,215,94),(35,267,216,95),(36,268,217,96),(37,269,218,97),(38,270,219,98),(39,271,220,99),(40,272,201,100),(41,129,120,256),(42,130,101,257),(43,131,102,258),(44,132,103,259),(45,133,104,260),(46,134,105,241),(47,135,106,242),(48,136,107,243),(49,137,108,244),(50,138,109,245),(51,139,110,246),(52,140,111,247),(53,121,112,248),(54,122,113,249),(55,123,114,250),(56,124,115,251),(57,125,116,252),(58,126,117,253),(59,127,118,254),(60,128,119,255),(141,291,226,165),(142,292,227,166),(143,293,228,167),(144,294,229,168),(145,295,230,169),(146,296,231,170),(147,297,232,171),(148,298,233,172),(149,299,234,173),(150,300,235,174),(151,281,236,175),(152,282,237,176),(153,283,238,177),(154,284,239,178),(155,285,240,179),(156,286,221,180),(157,287,222,161),(158,288,223,162),(159,289,224,163),(160,290,225,164)], [(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)], [(1,283,11,293),(2,282,12,292),(3,281,13,291),(4,300,14,290),(5,299,15,289),(6,298,16,288),(7,297,17,287),(8,296,18,286),(9,295,19,285),(10,294,20,284),(21,134,31,124),(22,133,32,123),(23,132,33,122),(24,131,34,121),(25,130,35,140),(26,129,36,139),(27,128,37,138),(28,127,38,137),(29,126,39,136),(30,125,40,135),(41,96,51,86),(42,95,52,85),(43,94,53,84),(44,93,54,83),(45,92,55,82),(46,91,56,81),(47,90,57,100),(48,89,58,99),(49,88,59,98),(50,87,60,97),(61,147,71,157),(62,146,72,156),(63,145,73,155),(64,144,74,154),(65,143,75,153),(66,142,76,152),(67,141,77,151),(68,160,78,150),(69,159,79,149),(70,158,80,148),(101,267,111,277),(102,266,112,276),(103,265,113,275),(104,264,114,274),(105,263,115,273),(106,262,116,272),(107,261,117,271),(108,280,118,270),(109,279,119,269),(110,278,120,268),(161,311,171,301),(162,310,172,320),(163,309,173,319),(164,308,174,318),(165,307,175,317),(166,306,176,316),(167,305,177,315),(168,304,178,314),(169,303,179,313),(170,302,180,312),(181,224,191,234),(182,223,192,233),(183,222,193,232),(184,221,194,231),(185,240,195,230),(186,239,196,229),(187,238,197,228),(188,237,198,227),(189,236,199,226),(190,235,200,225),(201,242,211,252),(202,241,212,251),(203,260,213,250),(204,259,214,249),(205,258,215,248),(206,257,216,247),(207,256,217,246),(208,255,218,245),(209,254,219,244),(210,253,220,243)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
0 | 1 | 0 | 0 |
40 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 11 | 0 | 0 |
11 | 40 | 0 | 0 |
0 | 0 | 40 | 0 |
0 | 0 | 0 | 40 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 16 | 39 |
0 | 0 | 2 | 28 |
40 | 0 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 31 | 37 |
0 | 0 | 15 | 10 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,11,0,0,11,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,16,2,0,0,39,28],[40,0,0,0,0,40,0,0,0,0,31,15,0,0,37,10] >;
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 | Q8 | D5 | D10 | D10 | D10 | Dic10 | 2+ (1+4) | Q8×D5 | D4⋊8D10 |
kernel | Q8×Dic10 | C4×Dic10 | C20⋊2Q8 | C20⋊Q8 | Q8×Dic5 | Q8×C20 | Dic10 | C5×Q8 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C4 | C2 |
# reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 4 | 2 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Q_8\times Dic_{10}
% in TeX
G:=Group("Q8xDic10");
// GroupNames label
G:=SmallGroup(320,1238);
// by ID
G=gap.SmallGroup(320,1238);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,675,80,12550]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations