direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×C5⋊C8, C5⋊2(C8×Q8), (C5×Q8)⋊2C8, C2.3(Q8×F5), C20.9(C2×C8), (C2×Q8).9F5, C10.7(C4×Q8), (Q8×C10).5C4, C20⋊C8.5C2, C4⋊Dic5.15C4, C2.3(Q8.F5), C10.22(C8○D4), C10.22(C22×C8), (Q8×Dic5).15C2, Dic5.31(C2×Q8), Dic5.70(C4○D4), C22.52(C22×F5), (C2×Dic5).356C23, (C4×Dic5).196C22, C4.4(C2×C5⋊C8), (C4×C5⋊C8).4C2, C2.7(C22×C5⋊C8), (C2×C4).86(C2×F5), (C2×C20).64(C2×C4), (C2×C5⋊C8).41C22, (C2×C10).85(C22×C4), (C2×Dic5).76(C2×C4), SmallGroup(320,1124)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 266 in 102 conjugacy classes, 64 normal (18 characteristic)
C1, C2 [×3], C4 [×6], C4 [×5], C22, C5, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×6], C2×C10, C4×C8 [×3], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5, C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C8×Q8, C4×Dic5 [×3], C4⋊Dic5 [×3], C2×C5⋊C8, C2×C5⋊C8 [×3], Q8×C10, C4×C5⋊C8 [×3], C20⋊C8 [×3], Q8×Dic5, Q8×C5⋊C8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C22×C8, C8○D4, C5⋊C8 [×4], C2×F5 [×3], C8×Q8, C2×C5⋊C8 [×6], C22×F5, Q8.F5, Q8×F5, C22×C5⋊C8, Q8×C5⋊C8
Generators and relations
G = < a,b,c,d | a4=c5=d8=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►Regular action on 320 points
Generators in S320
(1 121 257 75)(2 122 258 76)(3 123 259 77)(4 124 260 78)(5 125 261 79)(6 126 262 80)(7 127 263 73)(8 128 264 74)(9 283 254 46)(10 284 255 47)(11 285 256 48)(12 286 249 41)(13 287 250 42)(14 288 251 43)(15 281 252 44)(16 282 253 45)(17 266 35 304)(18 267 36 297)(19 268 37 298)(20 269 38 299)(21 270 39 300)(22 271 40 301)(23 272 33 302)(24 265 34 303)(25 247 231 139)(26 248 232 140)(27 241 225 141)(28 242 226 142)(29 243 227 143)(30 244 228 144)(31 245 229 137)(32 246 230 138)(49 194 135 280)(50 195 136 273)(51 196 129 274)(52 197 130 275)(53 198 131 276)(54 199 132 277)(55 200 133 278)(56 193 134 279)(57 109 153 167)(58 110 154 168)(59 111 155 161)(60 112 156 162)(61 105 157 163)(62 106 158 164)(63 107 159 165)(64 108 160 166)(65 176 309 117)(66 169 310 118)(67 170 311 119)(68 171 312 120)(69 172 305 113)(70 173 306 114)(71 174 307 115)(72 175 308 116)(81 289 183 217)(82 290 184 218)(83 291 177 219)(84 292 178 220)(85 293 179 221)(86 294 180 222)(87 295 181 223)(88 296 182 224)(89 320 186 234)(90 313 187 235)(91 314 188 236)(92 315 189 237)(93 316 190 238)(94 317 191 239)(95 318 192 240)(96 319 185 233)(97 216 205 149)(98 209 206 150)(99 210 207 151)(100 211 208 152)(101 212 201 145)(102 213 202 146)(103 214 203 147)(104 215 204 148)
(1 297 257 267)(2 298 258 268)(3 299 259 269)(4 300 260 270)(5 301 261 271)(6 302 262 272)(7 303 263 265)(8 304 264 266)(9 294 254 222)(10 295 255 223)(11 296 256 224)(12 289 249 217)(13 290 250 218)(14 291 251 219)(15 292 252 220)(16 293 253 221)(17 128 35 74)(18 121 36 75)(19 122 37 76)(20 123 38 77)(21 124 39 78)(22 125 40 79)(23 126 33 80)(24 127 34 73)(25 151 231 210)(26 152 232 211)(27 145 225 212)(28 146 226 213)(29 147 227 214)(30 148 228 215)(31 149 229 216)(32 150 230 209)(41 183 286 81)(42 184 287 82)(43 177 288 83)(44 178 281 84)(45 179 282 85)(46 180 283 86)(47 181 284 87)(48 182 285 88)(49 186 135 89)(50 187 136 90)(51 188 129 91)(52 189 130 92)(53 190 131 93)(54 191 132 94)(55 192 133 95)(56 185 134 96)(57 172 153 113)(58 173 154 114)(59 174 155 115)(60 175 156 116)(61 176 157 117)(62 169 158 118)(63 170 159 119)(64 171 160 120)(65 163 309 105)(66 164 310 106)(67 165 311 107)(68 166 312 108)(69 167 305 109)(70 168 306 110)(71 161 307 111)(72 162 308 112)(97 245 205 137)(98 246 206 138)(99 247 207 139)(100 248 208 140)(101 241 201 141)(102 242 202 142)(103 243 203 143)(104 244 204 144)(193 319 279 233)(194 320 280 234)(195 313 273 235)(196 314 274 236)(197 315 275 237)(198 316 276 238)(199 317 277 239)(200 318 278 240)
(1 174 252 103 193)(2 104 175 194 253)(3 195 97 254 176)(4 255 196 169 98)(5 170 256 99 197)(6 100 171 198 249)(7 199 101 250 172)(8 251 200 173 102)(9 117 259 273 205)(10 274 118 206 260)(11 207 275 261 119)(12 262 208 120 276)(13 113 263 277 201)(14 278 114 202 264)(15 203 279 257 115)(16 258 204 116 280)(17 83 192 168 226)(18 161 84 227 185)(19 228 162 186 85)(20 187 229 86 163)(21 87 188 164 230)(22 165 88 231 189)(23 232 166 190 81)(24 191 225 82 167)(25 92 40 107 182)(26 108 93 183 33)(27 184 109 34 94)(28 35 177 95 110)(29 96 36 111 178)(30 112 89 179 37)(31 180 105 38 90)(32 39 181 91 106)(41 126 211 312 131)(42 305 127 132 212)(43 133 306 213 128)(44 214 134 121 307)(45 122 215 308 135)(46 309 123 136 216)(47 129 310 209 124)(48 210 130 125 311)(49 282 76 148 72)(50 149 283 65 77)(51 66 150 78 284)(52 79 67 285 151)(53 286 80 152 68)(54 145 287 69 73)(55 70 146 74 288)(56 75 71 281 147)(57 265 239 141 290)(58 142 266 291 240)(59 292 143 233 267)(60 234 293 268 144)(61 269 235 137 294)(62 138 270 295 236)(63 296 139 237 271)(64 238 289 272 140)(153 303 317 241 218)(154 242 304 219 318)(155 220 243 319 297)(156 320 221 298 244)(157 299 313 245 222)(158 246 300 223 314)(159 224 247 315 301)(160 316 217 302 248)
(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)
G:=sub<Sym(320)| (1,121,257,75)(2,122,258,76)(3,123,259,77)(4,124,260,78)(5,125,261,79)(6,126,262,80)(7,127,263,73)(8,128,264,74)(9,283,254,46)(10,284,255,47)(11,285,256,48)(12,286,249,41)(13,287,250,42)(14,288,251,43)(15,281,252,44)(16,282,253,45)(17,266,35,304)(18,267,36,297)(19,268,37,298)(20,269,38,299)(21,270,39,300)(22,271,40,301)(23,272,33,302)(24,265,34,303)(25,247,231,139)(26,248,232,140)(27,241,225,141)(28,242,226,142)(29,243,227,143)(30,244,228,144)(31,245,229,137)(32,246,230,138)(49,194,135,280)(50,195,136,273)(51,196,129,274)(52,197,130,275)(53,198,131,276)(54,199,132,277)(55,200,133,278)(56,193,134,279)(57,109,153,167)(58,110,154,168)(59,111,155,161)(60,112,156,162)(61,105,157,163)(62,106,158,164)(63,107,159,165)(64,108,160,166)(65,176,309,117)(66,169,310,118)(67,170,311,119)(68,171,312,120)(69,172,305,113)(70,173,306,114)(71,174,307,115)(72,175,308,116)(81,289,183,217)(82,290,184,218)(83,291,177,219)(84,292,178,220)(85,293,179,221)(86,294,180,222)(87,295,181,223)(88,296,182,224)(89,320,186,234)(90,313,187,235)(91,314,188,236)(92,315,189,237)(93,316,190,238)(94,317,191,239)(95,318,192,240)(96,319,185,233)(97,216,205,149)(98,209,206,150)(99,210,207,151)(100,211,208,152)(101,212,201,145)(102,213,202,146)(103,214,203,147)(104,215,204,148), (1,297,257,267)(2,298,258,268)(3,299,259,269)(4,300,260,270)(5,301,261,271)(6,302,262,272)(7,303,263,265)(8,304,264,266)(9,294,254,222)(10,295,255,223)(11,296,256,224)(12,289,249,217)(13,290,250,218)(14,291,251,219)(15,292,252,220)(16,293,253,221)(17,128,35,74)(18,121,36,75)(19,122,37,76)(20,123,38,77)(21,124,39,78)(22,125,40,79)(23,126,33,80)(24,127,34,73)(25,151,231,210)(26,152,232,211)(27,145,225,212)(28,146,226,213)(29,147,227,214)(30,148,228,215)(31,149,229,216)(32,150,230,209)(41,183,286,81)(42,184,287,82)(43,177,288,83)(44,178,281,84)(45,179,282,85)(46,180,283,86)(47,181,284,87)(48,182,285,88)(49,186,135,89)(50,187,136,90)(51,188,129,91)(52,189,130,92)(53,190,131,93)(54,191,132,94)(55,192,133,95)(56,185,134,96)(57,172,153,113)(58,173,154,114)(59,174,155,115)(60,175,156,116)(61,176,157,117)(62,169,158,118)(63,170,159,119)(64,171,160,120)(65,163,309,105)(66,164,310,106)(67,165,311,107)(68,166,312,108)(69,167,305,109)(70,168,306,110)(71,161,307,111)(72,162,308,112)(97,245,205,137)(98,246,206,138)(99,247,207,139)(100,248,208,140)(101,241,201,141)(102,242,202,142)(103,243,203,143)(104,244,204,144)(193,319,279,233)(194,320,280,234)(195,313,273,235)(196,314,274,236)(197,315,275,237)(198,316,276,238)(199,317,277,239)(200,318,278,240), (1,174,252,103,193)(2,104,175,194,253)(3,195,97,254,176)(4,255,196,169,98)(5,170,256,99,197)(6,100,171,198,249)(7,199,101,250,172)(8,251,200,173,102)(9,117,259,273,205)(10,274,118,206,260)(11,207,275,261,119)(12,262,208,120,276)(13,113,263,277,201)(14,278,114,202,264)(15,203,279,257,115)(16,258,204,116,280)(17,83,192,168,226)(18,161,84,227,185)(19,228,162,186,85)(20,187,229,86,163)(21,87,188,164,230)(22,165,88,231,189)(23,232,166,190,81)(24,191,225,82,167)(25,92,40,107,182)(26,108,93,183,33)(27,184,109,34,94)(28,35,177,95,110)(29,96,36,111,178)(30,112,89,179,37)(31,180,105,38,90)(32,39,181,91,106)(41,126,211,312,131)(42,305,127,132,212)(43,133,306,213,128)(44,214,134,121,307)(45,122,215,308,135)(46,309,123,136,216)(47,129,310,209,124)(48,210,130,125,311)(49,282,76,148,72)(50,149,283,65,77)(51,66,150,78,284)(52,79,67,285,151)(53,286,80,152,68)(54,145,287,69,73)(55,70,146,74,288)(56,75,71,281,147)(57,265,239,141,290)(58,142,266,291,240)(59,292,143,233,267)(60,234,293,268,144)(61,269,235,137,294)(62,138,270,295,236)(63,296,139,237,271)(64,238,289,272,140)(153,303,317,241,218)(154,242,304,219,318)(155,220,243,319,297)(156,320,221,298,244)(157,299,313,245,222)(158,246,300,223,314)(159,224,247,315,301)(160,316,217,302,248), (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)>;
G:=Group( (1,121,257,75)(2,122,258,76)(3,123,259,77)(4,124,260,78)(5,125,261,79)(6,126,262,80)(7,127,263,73)(8,128,264,74)(9,283,254,46)(10,284,255,47)(11,285,256,48)(12,286,249,41)(13,287,250,42)(14,288,251,43)(15,281,252,44)(16,282,253,45)(17,266,35,304)(18,267,36,297)(19,268,37,298)(20,269,38,299)(21,270,39,300)(22,271,40,301)(23,272,33,302)(24,265,34,303)(25,247,231,139)(26,248,232,140)(27,241,225,141)(28,242,226,142)(29,243,227,143)(30,244,228,144)(31,245,229,137)(32,246,230,138)(49,194,135,280)(50,195,136,273)(51,196,129,274)(52,197,130,275)(53,198,131,276)(54,199,132,277)(55,200,133,278)(56,193,134,279)(57,109,153,167)(58,110,154,168)(59,111,155,161)(60,112,156,162)(61,105,157,163)(62,106,158,164)(63,107,159,165)(64,108,160,166)(65,176,309,117)(66,169,310,118)(67,170,311,119)(68,171,312,120)(69,172,305,113)(70,173,306,114)(71,174,307,115)(72,175,308,116)(81,289,183,217)(82,290,184,218)(83,291,177,219)(84,292,178,220)(85,293,179,221)(86,294,180,222)(87,295,181,223)(88,296,182,224)(89,320,186,234)(90,313,187,235)(91,314,188,236)(92,315,189,237)(93,316,190,238)(94,317,191,239)(95,318,192,240)(96,319,185,233)(97,216,205,149)(98,209,206,150)(99,210,207,151)(100,211,208,152)(101,212,201,145)(102,213,202,146)(103,214,203,147)(104,215,204,148), (1,297,257,267)(2,298,258,268)(3,299,259,269)(4,300,260,270)(5,301,261,271)(6,302,262,272)(7,303,263,265)(8,304,264,266)(9,294,254,222)(10,295,255,223)(11,296,256,224)(12,289,249,217)(13,290,250,218)(14,291,251,219)(15,292,252,220)(16,293,253,221)(17,128,35,74)(18,121,36,75)(19,122,37,76)(20,123,38,77)(21,124,39,78)(22,125,40,79)(23,126,33,80)(24,127,34,73)(25,151,231,210)(26,152,232,211)(27,145,225,212)(28,146,226,213)(29,147,227,214)(30,148,228,215)(31,149,229,216)(32,150,230,209)(41,183,286,81)(42,184,287,82)(43,177,288,83)(44,178,281,84)(45,179,282,85)(46,180,283,86)(47,181,284,87)(48,182,285,88)(49,186,135,89)(50,187,136,90)(51,188,129,91)(52,189,130,92)(53,190,131,93)(54,191,132,94)(55,192,133,95)(56,185,134,96)(57,172,153,113)(58,173,154,114)(59,174,155,115)(60,175,156,116)(61,176,157,117)(62,169,158,118)(63,170,159,119)(64,171,160,120)(65,163,309,105)(66,164,310,106)(67,165,311,107)(68,166,312,108)(69,167,305,109)(70,168,306,110)(71,161,307,111)(72,162,308,112)(97,245,205,137)(98,246,206,138)(99,247,207,139)(100,248,208,140)(101,241,201,141)(102,242,202,142)(103,243,203,143)(104,244,204,144)(193,319,279,233)(194,320,280,234)(195,313,273,235)(196,314,274,236)(197,315,275,237)(198,316,276,238)(199,317,277,239)(200,318,278,240), (1,174,252,103,193)(2,104,175,194,253)(3,195,97,254,176)(4,255,196,169,98)(5,170,256,99,197)(6,100,171,198,249)(7,199,101,250,172)(8,251,200,173,102)(9,117,259,273,205)(10,274,118,206,260)(11,207,275,261,119)(12,262,208,120,276)(13,113,263,277,201)(14,278,114,202,264)(15,203,279,257,115)(16,258,204,116,280)(17,83,192,168,226)(18,161,84,227,185)(19,228,162,186,85)(20,187,229,86,163)(21,87,188,164,230)(22,165,88,231,189)(23,232,166,190,81)(24,191,225,82,167)(25,92,40,107,182)(26,108,93,183,33)(27,184,109,34,94)(28,35,177,95,110)(29,96,36,111,178)(30,112,89,179,37)(31,180,105,38,90)(32,39,181,91,106)(41,126,211,312,131)(42,305,127,132,212)(43,133,306,213,128)(44,214,134,121,307)(45,122,215,308,135)(46,309,123,136,216)(47,129,310,209,124)(48,210,130,125,311)(49,282,76,148,72)(50,149,283,65,77)(51,66,150,78,284)(52,79,67,285,151)(53,286,80,152,68)(54,145,287,69,73)(55,70,146,74,288)(56,75,71,281,147)(57,265,239,141,290)(58,142,266,291,240)(59,292,143,233,267)(60,234,293,268,144)(61,269,235,137,294)(62,138,270,295,236)(63,296,139,237,271)(64,238,289,272,140)(153,303,317,241,218)(154,242,304,219,318)(155,220,243,319,297)(156,320,221,298,244)(157,299,313,245,222)(158,246,300,223,314)(159,224,247,315,301)(160,316,217,302,248), (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) );
G=PermutationGroup([(1,121,257,75),(2,122,258,76),(3,123,259,77),(4,124,260,78),(5,125,261,79),(6,126,262,80),(7,127,263,73),(8,128,264,74),(9,283,254,46),(10,284,255,47),(11,285,256,48),(12,286,249,41),(13,287,250,42),(14,288,251,43),(15,281,252,44),(16,282,253,45),(17,266,35,304),(18,267,36,297),(19,268,37,298),(20,269,38,299),(21,270,39,300),(22,271,40,301),(23,272,33,302),(24,265,34,303),(25,247,231,139),(26,248,232,140),(27,241,225,141),(28,242,226,142),(29,243,227,143),(30,244,228,144),(31,245,229,137),(32,246,230,138),(49,194,135,280),(50,195,136,273),(51,196,129,274),(52,197,130,275),(53,198,131,276),(54,199,132,277),(55,200,133,278),(56,193,134,279),(57,109,153,167),(58,110,154,168),(59,111,155,161),(60,112,156,162),(61,105,157,163),(62,106,158,164),(63,107,159,165),(64,108,160,166),(65,176,309,117),(66,169,310,118),(67,170,311,119),(68,171,312,120),(69,172,305,113),(70,173,306,114),(71,174,307,115),(72,175,308,116),(81,289,183,217),(82,290,184,218),(83,291,177,219),(84,292,178,220),(85,293,179,221),(86,294,180,222),(87,295,181,223),(88,296,182,224),(89,320,186,234),(90,313,187,235),(91,314,188,236),(92,315,189,237),(93,316,190,238),(94,317,191,239),(95,318,192,240),(96,319,185,233),(97,216,205,149),(98,209,206,150),(99,210,207,151),(100,211,208,152),(101,212,201,145),(102,213,202,146),(103,214,203,147),(104,215,204,148)], [(1,297,257,267),(2,298,258,268),(3,299,259,269),(4,300,260,270),(5,301,261,271),(6,302,262,272),(7,303,263,265),(8,304,264,266),(9,294,254,222),(10,295,255,223),(11,296,256,224),(12,289,249,217),(13,290,250,218),(14,291,251,219),(15,292,252,220),(16,293,253,221),(17,128,35,74),(18,121,36,75),(19,122,37,76),(20,123,38,77),(21,124,39,78),(22,125,40,79),(23,126,33,80),(24,127,34,73),(25,151,231,210),(26,152,232,211),(27,145,225,212),(28,146,226,213),(29,147,227,214),(30,148,228,215),(31,149,229,216),(32,150,230,209),(41,183,286,81),(42,184,287,82),(43,177,288,83),(44,178,281,84),(45,179,282,85),(46,180,283,86),(47,181,284,87),(48,182,285,88),(49,186,135,89),(50,187,136,90),(51,188,129,91),(52,189,130,92),(53,190,131,93),(54,191,132,94),(55,192,133,95),(56,185,134,96),(57,172,153,113),(58,173,154,114),(59,174,155,115),(60,175,156,116),(61,176,157,117),(62,169,158,118),(63,170,159,119),(64,171,160,120),(65,163,309,105),(66,164,310,106),(67,165,311,107),(68,166,312,108),(69,167,305,109),(70,168,306,110),(71,161,307,111),(72,162,308,112),(97,245,205,137),(98,246,206,138),(99,247,207,139),(100,248,208,140),(101,241,201,141),(102,242,202,142),(103,243,203,143),(104,244,204,144),(193,319,279,233),(194,320,280,234),(195,313,273,235),(196,314,274,236),(197,315,275,237),(198,316,276,238),(199,317,277,239),(200,318,278,240)], [(1,174,252,103,193),(2,104,175,194,253),(3,195,97,254,176),(4,255,196,169,98),(5,170,256,99,197),(6,100,171,198,249),(7,199,101,250,172),(8,251,200,173,102),(9,117,259,273,205),(10,274,118,206,260),(11,207,275,261,119),(12,262,208,120,276),(13,113,263,277,201),(14,278,114,202,264),(15,203,279,257,115),(16,258,204,116,280),(17,83,192,168,226),(18,161,84,227,185),(19,228,162,186,85),(20,187,229,86,163),(21,87,188,164,230),(22,165,88,231,189),(23,232,166,190,81),(24,191,225,82,167),(25,92,40,107,182),(26,108,93,183,33),(27,184,109,34,94),(28,35,177,95,110),(29,96,36,111,178),(30,112,89,179,37),(31,180,105,38,90),(32,39,181,91,106),(41,126,211,312,131),(42,305,127,132,212),(43,133,306,213,128),(44,214,134,121,307),(45,122,215,308,135),(46,309,123,136,216),(47,129,310,209,124),(48,210,130,125,311),(49,282,76,148,72),(50,149,283,65,77),(51,66,150,78,284),(52,79,67,285,151),(53,286,80,152,68),(54,145,287,69,73),(55,70,146,74,288),(56,75,71,281,147),(57,265,239,141,290),(58,142,266,291,240),(59,292,143,233,267),(60,234,293,268,144),(61,269,235,137,294),(62,138,270,295,236),(63,296,139,237,271),(64,238,289,272,140),(153,303,317,241,218),(154,242,304,219,318),(155,220,243,319,297),(156,320,221,298,244),(157,299,313,245,222),(158,246,300,223,314),(159,224,247,315,301),(160,316,217,302,248)], [(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)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 9 | 16 | 31 |
| 0 | 0 | 0 | 0 | 7 | 22 | 32 | 10 |
| 0 | 0 | 0 | 0 | 10 | 29 | 19 | 26 |
| 0 | 0 | 0 | 0 | 31 | 38 | 12 | 22 |
G:=sub<GL(8,GF(41))| [1,40,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,26,4,0,0,0,0,0,0,5,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,23,9,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,19,7,10,31,0,0,0,0,9,22,29,38,0,0,0,0,16,32,19,12,0,0,0,0,31,10,26,22] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | Q8 | C4○D4 | C8○D4 | F5 | C2×F5 | C5⋊C8 | Q8.F5 | Q8×F5 |
| kernel | Q8×C5⋊C8 | C4×C5⋊C8 | C20⋊C8 | Q8×Dic5 | C4⋊Dic5 | Q8×C10 | C5×Q8 | C5⋊C8 | Dic5 | C10 | C2×Q8 | C2×C4 | Q8 | C2 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 16 | 2 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
Q_8\times C_5\rtimes C_8
% in TeX
G:=Group("Q8xC5:C8"); // GroupNames label
G:=SmallGroup(320,1124);
// by ID
G=gap.SmallGroup(320,1124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,219,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^5=d^8=1,b^2=a^2,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^3>;
// generators/relations
×
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ℤ
𝔽
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