direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8⋊4Q8, C40⋊14Q8, C20.44M4(2), C8⋊4(C5×Q8), C4⋊C4.9C20, C4⋊C8.10C10, C2.6(Q8×C20), (C4×C40).33C2, (C4×C8).15C10, (C2×Q8).7C20, (C4×Q8).4C10, C10.43(C4×Q8), C4.25(Q8×C10), C8⋊C4.5C10, (Q8×C20).17C2, (Q8×C10).27C4, C20.131(C2×Q8), C4.3(C5×M4(2)), C10.76(C8○D4), C42.75(C2×C10), C20.357(C4○D4), (C4×C20).253C22, (C2×C40).447C22, (C2×C20).994C23, C2.11(C10×M4(2)), C10.89(C2×M4(2)), C22.49(C22×C20), C2.9(C5×C8○D4), (C5×C4⋊C8).23C2, (C5×C4⋊C4).34C4, C4.55(C5×C4○D4), (C2×C8).55(C2×C10), (C2×C4).30(C2×C20), (C5×C8⋊C4).11C2, (C2×C20).375(C2×C4), (C2×C10).345(C22×C4), (C2×C4).162(C22×C10), SmallGroup(320,947)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊4Q8
G = < a,b,c,d | a5=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c-1 >
Subgroups: 114 in 94 conjugacy classes, 74 normal (38 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C20, C20, C20, C2×C10, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C40, C40, C2×C20, C2×C20, C5×Q8, C8⋊4Q8, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, Q8×C10, C4×C40, C5×C8⋊C4, C5×C4⋊C8, C5×C4⋊C8, Q8×C20, C5×C8⋊4Q8
Quotients: C1, C2, C4, C22, C5, C2×C4, Q8, C23, C10, M4(2), C22×C4, C2×Q8, C4○D4, C20, C2×C10, C4×Q8, C2×M4(2), C8○D4, C2×C20, C5×Q8, C22×C10, C8⋊4Q8, C5×M4(2), C22×C20, Q8×C10, C5×C4○D4, Q8×C20, C10×M4(2), C5×C8○D4, C5×C8⋊4Q8
►Regular action on 320 points
Generators in S320
(1 87 79 39 71)(2 88 80 40 72)(3 81 73 33 65)(4 82 74 34 66)(5 83 75 35 67)(6 84 76 36 68)(7 85 77 37 69)(8 86 78 38 70)(9 59 19 51 31)(10 60 20 52 32)(11 61 21 53 25)(12 62 22 54 26)(13 63 23 55 27)(14 64 24 56 28)(15 57 17 49 29)(16 58 18 50 30)(41 313 280 305 272)(42 314 273 306 265)(43 315 274 307 266)(44 316 275 308 267)(45 317 276 309 268)(46 318 277 310 269)(47 319 278 311 270)(48 320 279 312 271)(89 124 138 98 130)(90 125 139 99 131)(91 126 140 100 132)(92 127 141 101 133)(93 128 142 102 134)(94 121 143 103 135)(95 122 144 104 136)(96 123 137 97 129)(105 168 153 113 145)(106 161 154 114 146)(107 162 155 115 147)(108 163 156 116 148)(109 164 157 117 149)(110 165 158 118 150)(111 166 159 119 151)(112 167 160 120 152)(169 205 218 177 210)(170 206 219 178 211)(171 207 220 179 212)(172 208 221 180 213)(173 201 222 181 214)(174 202 223 182 215)(175 203 224 183 216)(176 204 217 184 209)(185 241 234 193 226)(186 242 235 194 227)(187 243 236 195 228)(188 244 237 196 229)(189 245 238 197 230)(190 246 239 198 231)(191 247 240 199 232)(192 248 233 200 225)(249 285 298 257 290)(250 286 299 258 291)(251 287 300 259 292)(252 288 301 260 293)(253 281 302 261 294)(254 282 303 262 295)(255 283 304 263 296)(256 284 297 264 289)
(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)
(1 133 53 111)(2 134 54 112)(3 135 55 105)(4 136 56 106)(5 129 49 107)(6 130 50 108)(7 131 51 109)(8 132 52 110)(9 157 77 125)(10 158 78 126)(11 159 79 127)(12 160 80 128)(13 153 73 121)(14 154 74 122)(15 155 75 123)(16 156 76 124)(17 147 67 97)(18 148 68 98)(19 149 69 99)(20 150 70 100)(21 151 71 101)(22 152 72 102)(23 145 65 103)(24 146 66 104)(25 166 87 92)(26 167 88 93)(27 168 81 94)(28 161 82 95)(29 162 83 96)(30 163 84 89)(31 164 85 90)(32 165 86 91)(33 143 63 113)(34 144 64 114)(35 137 57 115)(36 138 58 116)(37 139 59 117)(38 140 60 118)(39 141 61 119)(40 142 62 120)(41 204 284 248)(42 205 285 241)(43 206 286 242)(44 207 287 243)(45 208 288 244)(46 201 281 245)(47 202 282 246)(48 203 283 247)(169 249 185 265)(170 250 186 266)(171 251 187 267)(172 252 188 268)(173 253 189 269)(174 254 190 270)(175 255 191 271)(176 256 192 272)(177 257 193 273)(178 258 194 274)(179 259 195 275)(180 260 196 276)(181 261 197 277)(182 262 198 278)(183 263 199 279)(184 264 200 280)(209 289 225 305)(210 290 226 306)(211 291 227 307)(212 292 228 308)(213 293 229 309)(214 294 230 310)(215 295 231 311)(216 296 232 312)(217 297 233 313)(218 298 234 314)(219 299 235 315)(220 300 236 316)(221 301 237 317)(222 302 238 318)(223 303 239 319)(224 304 240 320)
(1 173 53 189)(2 170 54 186)(3 175 55 191)(4 172 56 188)(5 169 49 185)(6 174 50 190)(7 171 51 187)(8 176 52 192)(9 236 77 220)(10 233 78 217)(11 238 79 222)(12 235 80 219)(13 240 73 224)(14 237 74 221)(15 234 75 218)(16 239 76 223)(17 226 67 210)(18 231 68 215)(19 228 69 212)(20 225 70 209)(21 230 71 214)(22 227 72 211)(23 232 65 216)(24 229 66 213)(25 245 87 201)(26 242 88 206)(27 247 81 203)(28 244 82 208)(29 241 83 205)(30 246 84 202)(31 243 85 207)(32 248 86 204)(33 183 63 199)(34 180 64 196)(35 177 57 193)(36 182 58 198)(37 179 59 195)(38 184 60 200)(39 181 61 197)(40 178 62 194)(41 165 284 91)(42 162 285 96)(43 167 286 93)(44 164 287 90)(45 161 288 95)(46 166 281 92)(47 163 282 89)(48 168 283 94)(97 306 147 290)(98 311 148 295)(99 308 149 292)(100 305 150 289)(101 310 151 294)(102 307 152 291)(103 312 145 296)(104 309 146 293)(105 255 135 271)(106 252 136 268)(107 249 129 265)(108 254 130 270)(109 251 131 267)(110 256 132 272)(111 253 133 269)(112 250 134 266)(113 263 143 279)(114 260 144 276)(115 257 137 273)(116 262 138 278)(117 259 139 275)(118 264 140 280)(119 261 141 277)(120 258 142 274)(121 320 153 304)(122 317 154 301)(123 314 155 298)(124 319 156 303)(125 316 157 300)(126 313 158 297)(127 318 159 302)(128 315 160 299)
G:=sub<Sym(320)| (1,87,79,39,71)(2,88,80,40,72)(3,81,73,33,65)(4,82,74,34,66)(5,83,75,35,67)(6,84,76,36,68)(7,85,77,37,69)(8,86,78,38,70)(9,59,19,51,31)(10,60,20,52,32)(11,61,21,53,25)(12,62,22,54,26)(13,63,23,55,27)(14,64,24,56,28)(15,57,17,49,29)(16,58,18,50,30)(41,313,280,305,272)(42,314,273,306,265)(43,315,274,307,266)(44,316,275,308,267)(45,317,276,309,268)(46,318,277,310,269)(47,319,278,311,270)(48,320,279,312,271)(89,124,138,98,130)(90,125,139,99,131)(91,126,140,100,132)(92,127,141,101,133)(93,128,142,102,134)(94,121,143,103,135)(95,122,144,104,136)(96,123,137,97,129)(105,168,153,113,145)(106,161,154,114,146)(107,162,155,115,147)(108,163,156,116,148)(109,164,157,117,149)(110,165,158,118,150)(111,166,159,119,151)(112,167,160,120,152)(169,205,218,177,210)(170,206,219,178,211)(171,207,220,179,212)(172,208,221,180,213)(173,201,222,181,214)(174,202,223,182,215)(175,203,224,183,216)(176,204,217,184,209)(185,241,234,193,226)(186,242,235,194,227)(187,243,236,195,228)(188,244,237,196,229)(189,245,238,197,230)(190,246,239,198,231)(191,247,240,199,232)(192,248,233,200,225)(249,285,298,257,290)(250,286,299,258,291)(251,287,300,259,292)(252,288,301,260,293)(253,281,302,261,294)(254,282,303,262,295)(255,283,304,263,296)(256,284,297,264,289), (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), (1,133,53,111)(2,134,54,112)(3,135,55,105)(4,136,56,106)(5,129,49,107)(6,130,50,108)(7,131,51,109)(8,132,52,110)(9,157,77,125)(10,158,78,126)(11,159,79,127)(12,160,80,128)(13,153,73,121)(14,154,74,122)(15,155,75,123)(16,156,76,124)(17,147,67,97)(18,148,68,98)(19,149,69,99)(20,150,70,100)(21,151,71,101)(22,152,72,102)(23,145,65,103)(24,146,66,104)(25,166,87,92)(26,167,88,93)(27,168,81,94)(28,161,82,95)(29,162,83,96)(30,163,84,89)(31,164,85,90)(32,165,86,91)(33,143,63,113)(34,144,64,114)(35,137,57,115)(36,138,58,116)(37,139,59,117)(38,140,60,118)(39,141,61,119)(40,142,62,120)(41,204,284,248)(42,205,285,241)(43,206,286,242)(44,207,287,243)(45,208,288,244)(46,201,281,245)(47,202,282,246)(48,203,283,247)(169,249,185,265)(170,250,186,266)(171,251,187,267)(172,252,188,268)(173,253,189,269)(174,254,190,270)(175,255,191,271)(176,256,192,272)(177,257,193,273)(178,258,194,274)(179,259,195,275)(180,260,196,276)(181,261,197,277)(182,262,198,278)(183,263,199,279)(184,264,200,280)(209,289,225,305)(210,290,226,306)(211,291,227,307)(212,292,228,308)(213,293,229,309)(214,294,230,310)(215,295,231,311)(216,296,232,312)(217,297,233,313)(218,298,234,314)(219,299,235,315)(220,300,236,316)(221,301,237,317)(222,302,238,318)(223,303,239,319)(224,304,240,320), (1,173,53,189)(2,170,54,186)(3,175,55,191)(4,172,56,188)(5,169,49,185)(6,174,50,190)(7,171,51,187)(8,176,52,192)(9,236,77,220)(10,233,78,217)(11,238,79,222)(12,235,80,219)(13,240,73,224)(14,237,74,221)(15,234,75,218)(16,239,76,223)(17,226,67,210)(18,231,68,215)(19,228,69,212)(20,225,70,209)(21,230,71,214)(22,227,72,211)(23,232,65,216)(24,229,66,213)(25,245,87,201)(26,242,88,206)(27,247,81,203)(28,244,82,208)(29,241,83,205)(30,246,84,202)(31,243,85,207)(32,248,86,204)(33,183,63,199)(34,180,64,196)(35,177,57,193)(36,182,58,198)(37,179,59,195)(38,184,60,200)(39,181,61,197)(40,178,62,194)(41,165,284,91)(42,162,285,96)(43,167,286,93)(44,164,287,90)(45,161,288,95)(46,166,281,92)(47,163,282,89)(48,168,283,94)(97,306,147,290)(98,311,148,295)(99,308,149,292)(100,305,150,289)(101,310,151,294)(102,307,152,291)(103,312,145,296)(104,309,146,293)(105,255,135,271)(106,252,136,268)(107,249,129,265)(108,254,130,270)(109,251,131,267)(110,256,132,272)(111,253,133,269)(112,250,134,266)(113,263,143,279)(114,260,144,276)(115,257,137,273)(116,262,138,278)(117,259,139,275)(118,264,140,280)(119,261,141,277)(120,258,142,274)(121,320,153,304)(122,317,154,301)(123,314,155,298)(124,319,156,303)(125,316,157,300)(126,313,158,297)(127,318,159,302)(128,315,160,299)>;
G:=Group( (1,87,79,39,71)(2,88,80,40,72)(3,81,73,33,65)(4,82,74,34,66)(5,83,75,35,67)(6,84,76,36,68)(7,85,77,37,69)(8,86,78,38,70)(9,59,19,51,31)(10,60,20,52,32)(11,61,21,53,25)(12,62,22,54,26)(13,63,23,55,27)(14,64,24,56,28)(15,57,17,49,29)(16,58,18,50,30)(41,313,280,305,272)(42,314,273,306,265)(43,315,274,307,266)(44,316,275,308,267)(45,317,276,309,268)(46,318,277,310,269)(47,319,278,311,270)(48,320,279,312,271)(89,124,138,98,130)(90,125,139,99,131)(91,126,140,100,132)(92,127,141,101,133)(93,128,142,102,134)(94,121,143,103,135)(95,122,144,104,136)(96,123,137,97,129)(105,168,153,113,145)(106,161,154,114,146)(107,162,155,115,147)(108,163,156,116,148)(109,164,157,117,149)(110,165,158,118,150)(111,166,159,119,151)(112,167,160,120,152)(169,205,218,177,210)(170,206,219,178,211)(171,207,220,179,212)(172,208,221,180,213)(173,201,222,181,214)(174,202,223,182,215)(175,203,224,183,216)(176,204,217,184,209)(185,241,234,193,226)(186,242,235,194,227)(187,243,236,195,228)(188,244,237,196,229)(189,245,238,197,230)(190,246,239,198,231)(191,247,240,199,232)(192,248,233,200,225)(249,285,298,257,290)(250,286,299,258,291)(251,287,300,259,292)(252,288,301,260,293)(253,281,302,261,294)(254,282,303,262,295)(255,283,304,263,296)(256,284,297,264,289), (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), (1,133,53,111)(2,134,54,112)(3,135,55,105)(4,136,56,106)(5,129,49,107)(6,130,50,108)(7,131,51,109)(8,132,52,110)(9,157,77,125)(10,158,78,126)(11,159,79,127)(12,160,80,128)(13,153,73,121)(14,154,74,122)(15,155,75,123)(16,156,76,124)(17,147,67,97)(18,148,68,98)(19,149,69,99)(20,150,70,100)(21,151,71,101)(22,152,72,102)(23,145,65,103)(24,146,66,104)(25,166,87,92)(26,167,88,93)(27,168,81,94)(28,161,82,95)(29,162,83,96)(30,163,84,89)(31,164,85,90)(32,165,86,91)(33,143,63,113)(34,144,64,114)(35,137,57,115)(36,138,58,116)(37,139,59,117)(38,140,60,118)(39,141,61,119)(40,142,62,120)(41,204,284,248)(42,205,285,241)(43,206,286,242)(44,207,287,243)(45,208,288,244)(46,201,281,245)(47,202,282,246)(48,203,283,247)(169,249,185,265)(170,250,186,266)(171,251,187,267)(172,252,188,268)(173,253,189,269)(174,254,190,270)(175,255,191,271)(176,256,192,272)(177,257,193,273)(178,258,194,274)(179,259,195,275)(180,260,196,276)(181,261,197,277)(182,262,198,278)(183,263,199,279)(184,264,200,280)(209,289,225,305)(210,290,226,306)(211,291,227,307)(212,292,228,308)(213,293,229,309)(214,294,230,310)(215,295,231,311)(216,296,232,312)(217,297,233,313)(218,298,234,314)(219,299,235,315)(220,300,236,316)(221,301,237,317)(222,302,238,318)(223,303,239,319)(224,304,240,320), (1,173,53,189)(2,170,54,186)(3,175,55,191)(4,172,56,188)(5,169,49,185)(6,174,50,190)(7,171,51,187)(8,176,52,192)(9,236,77,220)(10,233,78,217)(11,238,79,222)(12,235,80,219)(13,240,73,224)(14,237,74,221)(15,234,75,218)(16,239,76,223)(17,226,67,210)(18,231,68,215)(19,228,69,212)(20,225,70,209)(21,230,71,214)(22,227,72,211)(23,232,65,216)(24,229,66,213)(25,245,87,201)(26,242,88,206)(27,247,81,203)(28,244,82,208)(29,241,83,205)(30,246,84,202)(31,243,85,207)(32,248,86,204)(33,183,63,199)(34,180,64,196)(35,177,57,193)(36,182,58,198)(37,179,59,195)(38,184,60,200)(39,181,61,197)(40,178,62,194)(41,165,284,91)(42,162,285,96)(43,167,286,93)(44,164,287,90)(45,161,288,95)(46,166,281,92)(47,163,282,89)(48,168,283,94)(97,306,147,290)(98,311,148,295)(99,308,149,292)(100,305,150,289)(101,310,151,294)(102,307,152,291)(103,312,145,296)(104,309,146,293)(105,255,135,271)(106,252,136,268)(107,249,129,265)(108,254,130,270)(109,251,131,267)(110,256,132,272)(111,253,133,269)(112,250,134,266)(113,263,143,279)(114,260,144,276)(115,257,137,273)(116,262,138,278)(117,259,139,275)(118,264,140,280)(119,261,141,277)(120,258,142,274)(121,320,153,304)(122,317,154,301)(123,314,155,298)(124,319,156,303)(125,316,157,300)(126,313,158,297)(127,318,159,302)(128,315,160,299) );
G=PermutationGroup([[(1,87,79,39,71),(2,88,80,40,72),(3,81,73,33,65),(4,82,74,34,66),(5,83,75,35,67),(6,84,76,36,68),(7,85,77,37,69),(8,86,78,38,70),(9,59,19,51,31),(10,60,20,52,32),(11,61,21,53,25),(12,62,22,54,26),(13,63,23,55,27),(14,64,24,56,28),(15,57,17,49,29),(16,58,18,50,30),(41,313,280,305,272),(42,314,273,306,265),(43,315,274,307,266),(44,316,275,308,267),(45,317,276,309,268),(46,318,277,310,269),(47,319,278,311,270),(48,320,279,312,271),(89,124,138,98,130),(90,125,139,99,131),(91,126,140,100,132),(92,127,141,101,133),(93,128,142,102,134),(94,121,143,103,135),(95,122,144,104,136),(96,123,137,97,129),(105,168,153,113,145),(106,161,154,114,146),(107,162,155,115,147),(108,163,156,116,148),(109,164,157,117,149),(110,165,158,118,150),(111,166,159,119,151),(112,167,160,120,152),(169,205,218,177,210),(170,206,219,178,211),(171,207,220,179,212),(172,208,221,180,213),(173,201,222,181,214),(174,202,223,182,215),(175,203,224,183,216),(176,204,217,184,209),(185,241,234,193,226),(186,242,235,194,227),(187,243,236,195,228),(188,244,237,196,229),(189,245,238,197,230),(190,246,239,198,231),(191,247,240,199,232),(192,248,233,200,225),(249,285,298,257,290),(250,286,299,258,291),(251,287,300,259,292),(252,288,301,260,293),(253,281,302,261,294),(254,282,303,262,295),(255,283,304,263,296),(256,284,297,264,289)], [(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)], [(1,133,53,111),(2,134,54,112),(3,135,55,105),(4,136,56,106),(5,129,49,107),(6,130,50,108),(7,131,51,109),(8,132,52,110),(9,157,77,125),(10,158,78,126),(11,159,79,127),(12,160,80,128),(13,153,73,121),(14,154,74,122),(15,155,75,123),(16,156,76,124),(17,147,67,97),(18,148,68,98),(19,149,69,99),(20,150,70,100),(21,151,71,101),(22,152,72,102),(23,145,65,103),(24,146,66,104),(25,166,87,92),(26,167,88,93),(27,168,81,94),(28,161,82,95),(29,162,83,96),(30,163,84,89),(31,164,85,90),(32,165,86,91),(33,143,63,113),(34,144,64,114),(35,137,57,115),(36,138,58,116),(37,139,59,117),(38,140,60,118),(39,141,61,119),(40,142,62,120),(41,204,284,248),(42,205,285,241),(43,206,286,242),(44,207,287,243),(45,208,288,244),(46,201,281,245),(47,202,282,246),(48,203,283,247),(169,249,185,265),(170,250,186,266),(171,251,187,267),(172,252,188,268),(173,253,189,269),(174,254,190,270),(175,255,191,271),(176,256,192,272),(177,257,193,273),(178,258,194,274),(179,259,195,275),(180,260,196,276),(181,261,197,277),(182,262,198,278),(183,263,199,279),(184,264,200,280),(209,289,225,305),(210,290,226,306),(211,291,227,307),(212,292,228,308),(213,293,229,309),(214,294,230,310),(215,295,231,311),(216,296,232,312),(217,297,233,313),(218,298,234,314),(219,299,235,315),(220,300,236,316),(221,301,237,317),(222,302,238,318),(223,303,239,319),(224,304,240,320)], [(1,173,53,189),(2,170,54,186),(3,175,55,191),(4,172,56,188),(5,169,49,185),(6,174,50,190),(7,171,51,187),(8,176,52,192),(9,236,77,220),(10,233,78,217),(11,238,79,222),(12,235,80,219),(13,240,73,224),(14,237,74,221),(15,234,75,218),(16,239,76,223),(17,226,67,210),(18,231,68,215),(19,228,69,212),(20,225,70,209),(21,230,71,214),(22,227,72,211),(23,232,65,216),(24,229,66,213),(25,245,87,201),(26,242,88,206),(27,247,81,203),(28,244,82,208),(29,241,83,205),(30,246,84,202),(31,243,85,207),(32,248,86,204),(33,183,63,199),(34,180,64,196),(35,177,57,193),(36,182,58,198),(37,179,59,195),(38,184,60,200),(39,181,61,197),(40,178,62,194),(41,165,284,91),(42,162,285,96),(43,167,286,93),(44,164,287,90),(45,161,288,95),(46,166,281,92),(47,163,282,89),(48,168,283,94),(97,306,147,290),(98,311,148,295),(99,308,149,292),(100,305,150,289),(101,310,151,294),(102,307,152,291),(103,312,145,296),(104,309,146,293),(105,255,135,271),(106,252,136,268),(107,249,129,265),(108,254,130,270),(109,251,131,267),(110,256,132,272),(111,253,133,269),(112,250,134,266),(113,263,143,279),(114,260,144,276),(115,257,137,273),(116,262,138,278),(117,259,139,275),(118,264,140,280),(119,261,141,277),(120,258,142,274),(121,320,153,304),(122,317,154,301),(123,314,155,298),(124,319,156,303),(125,316,157,300),(126,313,158,297),(127,318,159,302),(128,315,160,299)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AV | 40A | ··· | 40AF | 40AG | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | Q8 | M4(2) | C4○D4 | C8○D4 | C5×Q8 | C5×M4(2) | C5×C4○D4 | C5×C8○D4 |
| kernel | C5×C8⋊4Q8 | C4×C40 | C5×C8⋊C4 | C5×C4⋊C8 | Q8×C20 | C5×C4⋊C4 | Q8×C10 | C8⋊4Q8 | C4×C8 | C8⋊C4 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | C40 | C20 | C20 | C10 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 6 | 2 | 4 | 4 | 8 | 12 | 4 | 24 | 8 | 2 | 4 | 2 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of C5×C8⋊4Q8 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 1 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 32 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 3 | 0 |
| 16 | 9 | 0 | 0 |
| 40 | 25 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 8 | 40 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,32,0,0,1,0,0,0,0,0,0,32,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,27,0],[16,40,0,0,9,25,0,0,0,0,1,8,0,0,10,40] >;
C5×C8⋊4Q8 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_4Q_8
% in TeX
G:=Group("C5xC8:4Q8"); // GroupNames label
G:=SmallGroup(320,947);
// by ID
G=gap.SmallGroup(320,947);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,3446,646,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^-1>;
// generators/relations