direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.5Q8, C40.21Q8, C8.4(C5×Q8), (C4×C8).9C10, C4.6(Q8×C10), (C4×C40).27C2, C20.95(C2×Q8), C2.D8.6C10, C4.Q8.7C10, (C2×C20).368D4, C10.41(C4⋊Q8), C42.84(C2×C10), C42.C2.4C10, C10.132(C4○D8), (C2×C20).955C23, (C4×C20).368C22, (C2×C40).441C22, C22.120(D4×C10), C2.7(C5×C4⋊Q8), C2.19(C5×C4○D8), (C2×C4).58(C5×D4), C4⋊C4.24(C2×C10), (C2×C8).82(C2×C10), (C5×C4.Q8).14C2, (C5×C2.D8).15C2, (C2×C10).676(C2×D4), (C5×C4⋊C4).244C22, (C5×C42.C2).11C2, (C2×C4).130(C22×C10), SmallGroup(320,1000)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.5Q8
G = < a,b,c,d | a5=b8=c4=1, d2=b4c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >
Subgroups: 130 in 86 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C20, C20, C2×C10, C4×C8, C4.Q8, C2.D8, C42.C2, C40, C2×C20, C2×C20, C2×C20, C8.5Q8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C4×C40, C5×C4.Q8, C5×C2.D8, C5×C42.C2, C5×C8.5Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C2×C10, C4⋊Q8, C4○D8, C5×D4, C5×Q8, C22×C10, C8.5Q8, D4×C10, Q8×C10, C5×C4⋊Q8, C5×C4○D8, C5×C8.5Q8
►Regular action on 320 points
Generators in S320
(1 57 22 54 9)(2 58 23 55 10)(3 59 24 56 11)(4 60 17 49 12)(5 61 18 50 13)(6 62 19 51 14)(7 63 20 52 15)(8 64 21 53 16)(25 81 73 33 65)(26 82 74 34 66)(27 83 75 35 67)(28 84 76 36 68)(29 85 77 37 69)(30 86 78 38 70)(31 87 79 39 71)(32 88 80 40 72)(41 317 275 309 267)(42 318 276 310 268)(43 319 277 311 269)(44 320 278 312 270)(45 313 279 305 271)(46 314 280 306 272)(47 315 273 307 265)(48 316 274 308 266)(89 138 97 134 128)(90 139 98 135 121)(91 140 99 136 122)(92 141 100 129 123)(93 142 101 130 124)(94 143 102 131 125)(95 144 103 132 126)(96 137 104 133 127)(105 167 153 113 145)(106 168 154 114 146)(107 161 155 115 147)(108 162 156 116 148)(109 163 157 117 149)(110 164 158 118 150)(111 165 159 119 151)(112 166 160 120 152)(169 208 219 177 211)(170 201 220 178 212)(171 202 221 179 213)(172 203 222 180 214)(173 204 223 181 215)(174 205 224 182 216)(175 206 217 183 209)(176 207 218 184 210)(185 241 235 193 227)(186 242 236 194 228)(187 243 237 195 229)(188 244 238 196 230)(189 245 239 197 231)(190 246 240 198 232)(191 247 233 199 225)(192 248 234 200 226)(249 288 299 257 291)(250 281 300 258 292)(251 282 301 259 293)(252 283 302 260 294)(253 284 303 261 295)(254 285 304 262 296)(255 286 297 263 289)(256 287 298 264 290)
(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)
(1 107 27 141)(2 108 28 142)(3 109 29 143)(4 110 30 144)(5 111 31 137)(6 112 32 138)(7 105 25 139)(8 106 26 140)(9 147 67 92)(10 148 68 93)(11 149 69 94)(12 150 70 95)(13 151 71 96)(14 152 72 89)(15 145 65 90)(16 146 66 91)(17 158 78 132)(18 159 79 133)(19 160 80 134)(20 153 73 135)(21 154 74 136)(22 155 75 129)(23 156 76 130)(24 157 77 131)(33 121 52 113)(34 122 53 114)(35 123 54 115)(36 124 55 116)(37 125 56 117)(38 126 49 118)(39 127 50 119)(40 128 51 120)(41 206 282 247)(42 207 283 248)(43 208 284 241)(44 201 285 242)(45 202 286 243)(46 203 287 244)(47 204 288 245)(48 205 281 246)(57 161 83 100)(58 162 84 101)(59 163 85 102)(60 164 86 103)(61 165 87 104)(62 166 88 97)(63 167 81 98)(64 168 82 99)(169 253 185 269)(170 254 186 270)(171 255 187 271)(172 256 188 272)(173 249 189 265)(174 250 190 266)(175 251 191 267)(176 252 192 268)(177 261 193 277)(178 262 194 278)(179 263 195 279)(180 264 196 280)(181 257 197 273)(182 258 198 274)(183 259 199 275)(184 260 200 276)(209 293 225 309)(210 294 226 310)(211 295 227 311)(212 296 228 312)(213 289 229 305)(214 290 230 306)(215 291 231 307)(216 292 232 308)(217 301 233 317)(218 302 234 318)(219 303 235 319)(220 304 236 320)(221 297 237 313)(222 298 238 314)(223 299 239 315)(224 300 240 316)
(1 187 31 175)(2 190 32 170)(3 185 25 173)(4 188 26 176)(5 191 27 171)(6 186 28 174)(7 189 29 169)(8 192 30 172)(9 229 71 209)(10 232 72 212)(11 227 65 215)(12 230 66 210)(13 225 67 213)(14 228 68 216)(15 231 69 211)(16 226 70 214)(17 238 74 218)(18 233 75 221)(19 236 76 224)(20 239 77 219)(21 234 78 222)(22 237 79 217)(23 240 80 220)(24 235 73 223)(33 181 56 193)(34 184 49 196)(35 179 50 199)(36 182 51 194)(37 177 52 197)(38 180 53 200)(39 183 54 195)(40 178 55 198)(41 165 286 100)(42 168 287 103)(43 163 288 98)(44 166 281 101)(45 161 282 104)(46 164 283 99)(47 167 284 102)(48 162 285 97)(57 243 87 206)(58 246 88 201)(59 241 81 204)(60 244 82 207)(61 247 83 202)(62 242 84 205)(63 245 85 208)(64 248 86 203)(89 308 148 296)(90 311 149 291)(91 306 150 294)(92 309 151 289)(93 312 152 292)(94 307 145 295)(95 310 146 290)(96 305 147 293)(105 253 143 265)(106 256 144 268)(107 251 137 271)(108 254 138 266)(109 249 139 269)(110 252 140 272)(111 255 141 267)(112 250 142 270)(113 261 125 273)(114 264 126 276)(115 259 127 279)(116 262 128 274)(117 257 121 277)(118 260 122 280)(119 263 123 275)(120 258 124 278)(129 317 159 297)(130 320 160 300)(131 315 153 303)(132 318 154 298)(133 313 155 301)(134 316 156 304)(135 319 157 299)(136 314 158 302)
G:=sub<Sym(320)| (1,57,22,54,9)(2,58,23,55,10)(3,59,24,56,11)(4,60,17,49,12)(5,61,18,50,13)(6,62,19,51,14)(7,63,20,52,15)(8,64,21,53,16)(25,81,73,33,65)(26,82,74,34,66)(27,83,75,35,67)(28,84,76,36,68)(29,85,77,37,69)(30,86,78,38,70)(31,87,79,39,71)(32,88,80,40,72)(41,317,275,309,267)(42,318,276,310,268)(43,319,277,311,269)(44,320,278,312,270)(45,313,279,305,271)(46,314,280,306,272)(47,315,273,307,265)(48,316,274,308,266)(89,138,97,134,128)(90,139,98,135,121)(91,140,99,136,122)(92,141,100,129,123)(93,142,101,130,124)(94,143,102,131,125)(95,144,103,132,126)(96,137,104,133,127)(105,167,153,113,145)(106,168,154,114,146)(107,161,155,115,147)(108,162,156,116,148)(109,163,157,117,149)(110,164,158,118,150)(111,165,159,119,151)(112,166,160,120,152)(169,208,219,177,211)(170,201,220,178,212)(171,202,221,179,213)(172,203,222,180,214)(173,204,223,181,215)(174,205,224,182,216)(175,206,217,183,209)(176,207,218,184,210)(185,241,235,193,227)(186,242,236,194,228)(187,243,237,195,229)(188,244,238,196,230)(189,245,239,197,231)(190,246,240,198,232)(191,247,233,199,225)(192,248,234,200,226)(249,288,299,257,291)(250,281,300,258,292)(251,282,301,259,293)(252,283,302,260,294)(253,284,303,261,295)(254,285,304,262,296)(255,286,297,263,289)(256,287,298,264,290), (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), (1,107,27,141)(2,108,28,142)(3,109,29,143)(4,110,30,144)(5,111,31,137)(6,112,32,138)(7,105,25,139)(8,106,26,140)(9,147,67,92)(10,148,68,93)(11,149,69,94)(12,150,70,95)(13,151,71,96)(14,152,72,89)(15,145,65,90)(16,146,66,91)(17,158,78,132)(18,159,79,133)(19,160,80,134)(20,153,73,135)(21,154,74,136)(22,155,75,129)(23,156,76,130)(24,157,77,131)(33,121,52,113)(34,122,53,114)(35,123,54,115)(36,124,55,116)(37,125,56,117)(38,126,49,118)(39,127,50,119)(40,128,51,120)(41,206,282,247)(42,207,283,248)(43,208,284,241)(44,201,285,242)(45,202,286,243)(46,203,287,244)(47,204,288,245)(48,205,281,246)(57,161,83,100)(58,162,84,101)(59,163,85,102)(60,164,86,103)(61,165,87,104)(62,166,88,97)(63,167,81,98)(64,168,82,99)(169,253,185,269)(170,254,186,270)(171,255,187,271)(172,256,188,272)(173,249,189,265)(174,250,190,266)(175,251,191,267)(176,252,192,268)(177,261,193,277)(178,262,194,278)(179,263,195,279)(180,264,196,280)(181,257,197,273)(182,258,198,274)(183,259,199,275)(184,260,200,276)(209,293,225,309)(210,294,226,310)(211,295,227,311)(212,296,228,312)(213,289,229,305)(214,290,230,306)(215,291,231,307)(216,292,232,308)(217,301,233,317)(218,302,234,318)(219,303,235,319)(220,304,236,320)(221,297,237,313)(222,298,238,314)(223,299,239,315)(224,300,240,316), (1,187,31,175)(2,190,32,170)(3,185,25,173)(4,188,26,176)(5,191,27,171)(6,186,28,174)(7,189,29,169)(8,192,30,172)(9,229,71,209)(10,232,72,212)(11,227,65,215)(12,230,66,210)(13,225,67,213)(14,228,68,216)(15,231,69,211)(16,226,70,214)(17,238,74,218)(18,233,75,221)(19,236,76,224)(20,239,77,219)(21,234,78,222)(22,237,79,217)(23,240,80,220)(24,235,73,223)(33,181,56,193)(34,184,49,196)(35,179,50,199)(36,182,51,194)(37,177,52,197)(38,180,53,200)(39,183,54,195)(40,178,55,198)(41,165,286,100)(42,168,287,103)(43,163,288,98)(44,166,281,101)(45,161,282,104)(46,164,283,99)(47,167,284,102)(48,162,285,97)(57,243,87,206)(58,246,88,201)(59,241,81,204)(60,244,82,207)(61,247,83,202)(62,242,84,205)(63,245,85,208)(64,248,86,203)(89,308,148,296)(90,311,149,291)(91,306,150,294)(92,309,151,289)(93,312,152,292)(94,307,145,295)(95,310,146,290)(96,305,147,293)(105,253,143,265)(106,256,144,268)(107,251,137,271)(108,254,138,266)(109,249,139,269)(110,252,140,272)(111,255,141,267)(112,250,142,270)(113,261,125,273)(114,264,126,276)(115,259,127,279)(116,262,128,274)(117,257,121,277)(118,260,122,280)(119,263,123,275)(120,258,124,278)(129,317,159,297)(130,320,160,300)(131,315,153,303)(132,318,154,298)(133,313,155,301)(134,316,156,304)(135,319,157,299)(136,314,158,302)>;
G:=Group( (1,57,22,54,9)(2,58,23,55,10)(3,59,24,56,11)(4,60,17,49,12)(5,61,18,50,13)(6,62,19,51,14)(7,63,20,52,15)(8,64,21,53,16)(25,81,73,33,65)(26,82,74,34,66)(27,83,75,35,67)(28,84,76,36,68)(29,85,77,37,69)(30,86,78,38,70)(31,87,79,39,71)(32,88,80,40,72)(41,317,275,309,267)(42,318,276,310,268)(43,319,277,311,269)(44,320,278,312,270)(45,313,279,305,271)(46,314,280,306,272)(47,315,273,307,265)(48,316,274,308,266)(89,138,97,134,128)(90,139,98,135,121)(91,140,99,136,122)(92,141,100,129,123)(93,142,101,130,124)(94,143,102,131,125)(95,144,103,132,126)(96,137,104,133,127)(105,167,153,113,145)(106,168,154,114,146)(107,161,155,115,147)(108,162,156,116,148)(109,163,157,117,149)(110,164,158,118,150)(111,165,159,119,151)(112,166,160,120,152)(169,208,219,177,211)(170,201,220,178,212)(171,202,221,179,213)(172,203,222,180,214)(173,204,223,181,215)(174,205,224,182,216)(175,206,217,183,209)(176,207,218,184,210)(185,241,235,193,227)(186,242,236,194,228)(187,243,237,195,229)(188,244,238,196,230)(189,245,239,197,231)(190,246,240,198,232)(191,247,233,199,225)(192,248,234,200,226)(249,288,299,257,291)(250,281,300,258,292)(251,282,301,259,293)(252,283,302,260,294)(253,284,303,261,295)(254,285,304,262,296)(255,286,297,263,289)(256,287,298,264,290), (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), (1,107,27,141)(2,108,28,142)(3,109,29,143)(4,110,30,144)(5,111,31,137)(6,112,32,138)(7,105,25,139)(8,106,26,140)(9,147,67,92)(10,148,68,93)(11,149,69,94)(12,150,70,95)(13,151,71,96)(14,152,72,89)(15,145,65,90)(16,146,66,91)(17,158,78,132)(18,159,79,133)(19,160,80,134)(20,153,73,135)(21,154,74,136)(22,155,75,129)(23,156,76,130)(24,157,77,131)(33,121,52,113)(34,122,53,114)(35,123,54,115)(36,124,55,116)(37,125,56,117)(38,126,49,118)(39,127,50,119)(40,128,51,120)(41,206,282,247)(42,207,283,248)(43,208,284,241)(44,201,285,242)(45,202,286,243)(46,203,287,244)(47,204,288,245)(48,205,281,246)(57,161,83,100)(58,162,84,101)(59,163,85,102)(60,164,86,103)(61,165,87,104)(62,166,88,97)(63,167,81,98)(64,168,82,99)(169,253,185,269)(170,254,186,270)(171,255,187,271)(172,256,188,272)(173,249,189,265)(174,250,190,266)(175,251,191,267)(176,252,192,268)(177,261,193,277)(178,262,194,278)(179,263,195,279)(180,264,196,280)(181,257,197,273)(182,258,198,274)(183,259,199,275)(184,260,200,276)(209,293,225,309)(210,294,226,310)(211,295,227,311)(212,296,228,312)(213,289,229,305)(214,290,230,306)(215,291,231,307)(216,292,232,308)(217,301,233,317)(218,302,234,318)(219,303,235,319)(220,304,236,320)(221,297,237,313)(222,298,238,314)(223,299,239,315)(224,300,240,316), (1,187,31,175)(2,190,32,170)(3,185,25,173)(4,188,26,176)(5,191,27,171)(6,186,28,174)(7,189,29,169)(8,192,30,172)(9,229,71,209)(10,232,72,212)(11,227,65,215)(12,230,66,210)(13,225,67,213)(14,228,68,216)(15,231,69,211)(16,226,70,214)(17,238,74,218)(18,233,75,221)(19,236,76,224)(20,239,77,219)(21,234,78,222)(22,237,79,217)(23,240,80,220)(24,235,73,223)(33,181,56,193)(34,184,49,196)(35,179,50,199)(36,182,51,194)(37,177,52,197)(38,180,53,200)(39,183,54,195)(40,178,55,198)(41,165,286,100)(42,168,287,103)(43,163,288,98)(44,166,281,101)(45,161,282,104)(46,164,283,99)(47,167,284,102)(48,162,285,97)(57,243,87,206)(58,246,88,201)(59,241,81,204)(60,244,82,207)(61,247,83,202)(62,242,84,205)(63,245,85,208)(64,248,86,203)(89,308,148,296)(90,311,149,291)(91,306,150,294)(92,309,151,289)(93,312,152,292)(94,307,145,295)(95,310,146,290)(96,305,147,293)(105,253,143,265)(106,256,144,268)(107,251,137,271)(108,254,138,266)(109,249,139,269)(110,252,140,272)(111,255,141,267)(112,250,142,270)(113,261,125,273)(114,264,126,276)(115,259,127,279)(116,262,128,274)(117,257,121,277)(118,260,122,280)(119,263,123,275)(120,258,124,278)(129,317,159,297)(130,320,160,300)(131,315,153,303)(132,318,154,298)(133,313,155,301)(134,316,156,304)(135,319,157,299)(136,314,158,302) );
G=PermutationGroup([[(1,57,22,54,9),(2,58,23,55,10),(3,59,24,56,11),(4,60,17,49,12),(5,61,18,50,13),(6,62,19,51,14),(7,63,20,52,15),(8,64,21,53,16),(25,81,73,33,65),(26,82,74,34,66),(27,83,75,35,67),(28,84,76,36,68),(29,85,77,37,69),(30,86,78,38,70),(31,87,79,39,71),(32,88,80,40,72),(41,317,275,309,267),(42,318,276,310,268),(43,319,277,311,269),(44,320,278,312,270),(45,313,279,305,271),(46,314,280,306,272),(47,315,273,307,265),(48,316,274,308,266),(89,138,97,134,128),(90,139,98,135,121),(91,140,99,136,122),(92,141,100,129,123),(93,142,101,130,124),(94,143,102,131,125),(95,144,103,132,126),(96,137,104,133,127),(105,167,153,113,145),(106,168,154,114,146),(107,161,155,115,147),(108,162,156,116,148),(109,163,157,117,149),(110,164,158,118,150),(111,165,159,119,151),(112,166,160,120,152),(169,208,219,177,211),(170,201,220,178,212),(171,202,221,179,213),(172,203,222,180,214),(173,204,223,181,215),(174,205,224,182,216),(175,206,217,183,209),(176,207,218,184,210),(185,241,235,193,227),(186,242,236,194,228),(187,243,237,195,229),(188,244,238,196,230),(189,245,239,197,231),(190,246,240,198,232),(191,247,233,199,225),(192,248,234,200,226),(249,288,299,257,291),(250,281,300,258,292),(251,282,301,259,293),(252,283,302,260,294),(253,284,303,261,295),(254,285,304,262,296),(255,286,297,263,289),(256,287,298,264,290)], [(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)], [(1,107,27,141),(2,108,28,142),(3,109,29,143),(4,110,30,144),(5,111,31,137),(6,112,32,138),(7,105,25,139),(8,106,26,140),(9,147,67,92),(10,148,68,93),(11,149,69,94),(12,150,70,95),(13,151,71,96),(14,152,72,89),(15,145,65,90),(16,146,66,91),(17,158,78,132),(18,159,79,133),(19,160,80,134),(20,153,73,135),(21,154,74,136),(22,155,75,129),(23,156,76,130),(24,157,77,131),(33,121,52,113),(34,122,53,114),(35,123,54,115),(36,124,55,116),(37,125,56,117),(38,126,49,118),(39,127,50,119),(40,128,51,120),(41,206,282,247),(42,207,283,248),(43,208,284,241),(44,201,285,242),(45,202,286,243),(46,203,287,244),(47,204,288,245),(48,205,281,246),(57,161,83,100),(58,162,84,101),(59,163,85,102),(60,164,86,103),(61,165,87,104),(62,166,88,97),(63,167,81,98),(64,168,82,99),(169,253,185,269),(170,254,186,270),(171,255,187,271),(172,256,188,272),(173,249,189,265),(174,250,190,266),(175,251,191,267),(176,252,192,268),(177,261,193,277),(178,262,194,278),(179,263,195,279),(180,264,196,280),(181,257,197,273),(182,258,198,274),(183,259,199,275),(184,260,200,276),(209,293,225,309),(210,294,226,310),(211,295,227,311),(212,296,228,312),(213,289,229,305),(214,290,230,306),(215,291,231,307),(216,292,232,308),(217,301,233,317),(218,302,234,318),(219,303,235,319),(220,304,236,320),(221,297,237,313),(222,298,238,314),(223,299,239,315),(224,300,240,316)], [(1,187,31,175),(2,190,32,170),(3,185,25,173),(4,188,26,176),(5,191,27,171),(6,186,28,174),(7,189,29,169),(8,192,30,172),(9,229,71,209),(10,232,72,212),(11,227,65,215),(12,230,66,210),(13,225,67,213),(14,228,68,216),(15,231,69,211),(16,226,70,214),(17,238,74,218),(18,233,75,221),(19,236,76,224),(20,239,77,219),(21,234,78,222),(22,237,79,217),(23,240,80,220),(24,235,73,223),(33,181,56,193),(34,184,49,196),(35,179,50,199),(36,182,51,194),(37,177,52,197),(38,180,53,200),(39,183,54,195),(40,178,55,198),(41,165,286,100),(42,168,287,103),(43,163,288,98),(44,166,281,101),(45,161,282,104),(46,164,283,99),(47,167,284,102),(48,162,285,97),(57,243,87,206),(58,246,88,201),(59,241,81,204),(60,244,82,207),(61,247,83,202),(62,242,84,205),(63,245,85,208),(64,248,86,203),(89,308,148,296),(90,311,149,291),(91,306,150,294),(92,309,151,289),(93,312,152,292),(94,307,145,295),(95,310,146,290),(96,305,147,293),(105,253,143,265),(106,256,144,268),(107,251,137,271),(108,254,138,266),(109,249,139,269),(110,252,140,272),(111,255,141,267),(112,250,142,270),(113,261,125,273),(114,264,126,276),(115,259,127,279),(116,262,128,274),(117,257,121,277),(118,260,122,280),(119,263,123,275),(120,258,124,278),(129,317,159,297),(130,320,160,300),(131,315,153,303),(132,318,154,298),(133,313,155,301),(134,316,156,304),(135,319,157,299),(136,314,158,302)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | Q8 | D4 | C4○D8 | C5×Q8 | C5×D4 | C5×C4○D8 |
| kernel | C5×C8.5Q8 | C4×C40 | C5×C4.Q8 | C5×C2.D8 | C5×C42.C2 | C8.5Q8 | C4×C8 | C4.Q8 | C2.D8 | C42.C2 | C40 | C2×C20 | C10 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 2 | 8 | 16 | 8 | 32 |
Matrix representation of C5×C8.5Q8 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 26 | 26 |
| 40 | 39 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 32 | 0 |
| 23 | 40 | 0 | 0 |
| 38 | 18 | 0 | 0 |
| 0 | 0 | 38 | 21 |
| 0 | 0 | 21 | 3 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,26,26,0,0,15,26],[40,1,0,0,39,1,0,0,0,0,0,32,0,0,9,0],[23,38,0,0,40,18,0,0,0,0,38,21,0,0,21,3] >;
C5×C8.5Q8 in GAP, Magma, Sage, TeX
C_5\times C_8._5Q_8
% in TeX
G:=Group("C5xC8.5Q8"); // GroupNames label
G:=SmallGroup(320,1000);
// by ID
G=gap.SmallGroup(320,1000);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,1408,1766,436,10085,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations