direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×Q8⋊C8, Q8⋊C40, C20.30Q16, C20.55SD16, C20.43M4(2), (C5×Q8)⋊5C8, C4⋊C4.4C20, C4⋊C8.1C10, (C4×C8).1C10, (C4×C40).3C2, C4.2(C2×C40), C10.33C4≀C2, C4.8(C5×Q16), C20.65(C2×C8), (C4×Q8).1C10, (C2×Q8).4C20, (C2×C20).529D4, (Q8×C10).24C4, (Q8×C20).14C2, C4.14(C5×SD16), C4.2(C5×M4(2)), C42.63(C2×C10), C10.38(C22⋊C8), (C4×C20).347C22, C10.21(Q8⋊C4), C2.2(C5×C4≀C2), (C5×C4⋊C8).7C2, (C5×C4⋊C4).29C4, (C2×C4).94(C5×D4), C2.6(C5×C22⋊C8), (C2×C4).39(C2×C20), C2.1(C5×Q8⋊C4), (C2×C20).432(C2×C4), C22.26(C5×C22⋊C4), (C2×C10).185(C22⋊C4), SmallGroup(320,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊C8
G = < a,b,c,d | a5=b4=d8=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 106 in 70 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C4×C8, C4⋊C8, C4×Q8, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8⋊C8, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, C4×C40, C5×C4⋊C8, Q8×C20, C5×Q8⋊C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C10, C22⋊C4, C2×C8, M4(2), SD16, Q16, C20, C2×C10, C22⋊C8, Q8⋊C4, C4≀C2, C40, C2×C20, C5×D4, Q8⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×SD16, C5×Q16, C5×C22⋊C8, C5×Q8⋊C4, C5×C4≀C2, C5×Q8⋊C8
►Regular action on 320 points
Generators in S320
(1 51 43 35 27)(2 52 44 36 28)(3 53 45 37 29)(4 54 46 38 30)(5 55 47 39 31)(6 56 48 40 32)(7 49 41 33 25)(8 50 42 34 26)(9 235 227 219 21)(10 236 228 220 22)(11 237 229 221 23)(12 238 230 222 24)(13 239 231 223 17)(14 240 232 224 18)(15 233 225 217 19)(16 234 226 218 20)(57 174 81 73 65)(58 175 82 74 66)(59 176 83 75 67)(60 169 84 76 68)(61 170 85 77 69)(62 171 86 78 70)(63 172 87 79 71)(64 173 88 80 72)(89 121 113 105 97)(90 122 114 106 98)(91 123 115 107 99)(92 124 116 108 100)(93 125 117 109 101)(94 126 118 110 102)(95 127 119 111 103)(96 128 120 112 104)(129 167 153 145 137)(130 168 154 146 138)(131 161 155 147 139)(132 162 156 148 140)(133 163 157 149 141)(134 164 158 150 142)(135 165 159 151 143)(136 166 160 152 144)(177 209 201 193 185)(178 210 202 194 186)(179 211 203 195 187)(180 212 204 196 188)(181 213 205 197 189)(182 214 206 198 190)(183 215 207 199 191)(184 216 208 200 192)(241 273 265 257 249)(242 274 266 258 250)(243 275 267 259 251)(244 276 268 260 252)(245 277 269 261 253)(246 278 270 262 254)(247 279 271 263 255)(248 280 272 264 256)(281 313 305 297 289)(282 314 306 298 290)(283 315 307 299 291)(284 316 308 300 292)(285 317 309 301 293)(286 318 310 302 294)(287 319 311 303 295)(288 320 312 304 296)
(1 95 247 179)(2 180 248 96)(3 89 241 181)(4 182 242 90)(5 91 243 183)(6 184 244 92)(7 93 245 177)(8 178 246 94)(9 311 159 83)(10 84 160 312)(11 305 153 85)(12 86 154 306)(13 307 155 87)(14 88 156 308)(15 309 157 81)(16 82 158 310)(17 315 161 172)(18 173 162 316)(19 317 163 174)(20 175 164 318)(21 319 165 176)(22 169 166 320)(23 313 167 170)(24 171 168 314)(25 101 253 185)(26 186 254 102)(27 103 255 187)(28 188 256 104)(29 97 249 189)(30 190 250 98)(31 99 251 191)(32 192 252 100)(33 109 261 193)(34 194 262 110)(35 111 263 195)(36 196 264 112)(37 105 257 197)(38 198 258 106)(39 107 259 199)(40 200 260 108)(41 117 269 201)(42 202 270 118)(43 119 271 203)(44 204 272 120)(45 113 265 205)(46 206 266 114)(47 115 267 207)(48 208 268 116)(49 125 277 209)(50 210 278 126)(51 127 279 211)(52 212 280 128)(53 121 273 213)(54 214 274 122)(55 123 275 215)(56 216 276 124)(57 217 285 133)(58 134 286 218)(59 219 287 135)(60 136 288 220)(61 221 281 129)(62 130 282 222)(63 223 283 131)(64 132 284 224)(65 225 293 141)(66 142 294 226)(67 227 295 143)(68 144 296 228)(69 229 289 137)(70 138 290 230)(71 231 291 139)(72 140 292 232)(73 233 301 149)(74 150 302 234)(75 235 303 151)(76 152 304 236)(77 237 297 145)(78 146 298 238)(79 239 299 147)(80 148 300 240)
(1 283 247 63)(2 132 248 224)(3 285 241 57)(4 134 242 218)(5 287 243 59)(6 136 244 220)(7 281 245 61)(8 130 246 222)(9 207 159 115)(10 48 160 268)(11 201 153 117)(12 42 154 270)(13 203 155 119)(14 44 156 272)(15 205 157 113)(16 46 158 266)(17 211 161 127)(18 52 162 280)(19 213 163 121)(20 54 164 274)(21 215 165 123)(22 56 166 276)(23 209 167 125)(24 50 168 278)(25 289 253 69)(26 138 254 230)(27 291 255 71)(28 140 256 232)(29 293 249 65)(30 142 250 226)(31 295 251 67)(32 144 252 228)(33 297 261 77)(34 146 262 238)(35 299 263 79)(36 148 264 240)(37 301 257 73)(38 150 258 234)(39 303 259 75)(40 152 260 236)(41 305 269 85)(43 307 271 87)(45 309 265 81)(47 311 267 83)(49 313 277 170)(51 315 279 172)(53 317 273 174)(55 319 275 176)(58 90 286 182)(60 92 288 184)(62 94 282 178)(64 96 284 180)(66 98 294 190)(68 100 296 192)(70 102 290 186)(72 104 292 188)(74 106 302 198)(76 108 304 200)(78 110 298 194)(80 112 300 196)(82 114 310 206)(84 116 312 208)(86 118 306 202)(88 120 308 204)(89 217 181 133)(91 219 183 135)(93 221 177 129)(95 223 179 131)(97 225 189 141)(99 227 191 143)(101 229 185 137)(103 231 187 139)(105 233 197 149)(107 235 199 151)(109 237 193 145)(111 239 195 147)(122 318 214 175)(124 320 216 169)(126 314 210 171)(128 316 212 173)
(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)| (1,51,43,35,27)(2,52,44,36,28)(3,53,45,37,29)(4,54,46,38,30)(5,55,47,39,31)(6,56,48,40,32)(7,49,41,33,25)(8,50,42,34,26)(9,235,227,219,21)(10,236,228,220,22)(11,237,229,221,23)(12,238,230,222,24)(13,239,231,223,17)(14,240,232,224,18)(15,233,225,217,19)(16,234,226,218,20)(57,174,81,73,65)(58,175,82,74,66)(59,176,83,75,67)(60,169,84,76,68)(61,170,85,77,69)(62,171,86,78,70)(63,172,87,79,71)(64,173,88,80,72)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,167,153,145,137)(130,168,154,146,138)(131,161,155,147,139)(132,162,156,148,140)(133,163,157,149,141)(134,164,158,150,142)(135,165,159,151,143)(136,166,160,152,144)(177,209,201,193,185)(178,210,202,194,186)(179,211,203,195,187)(180,212,204,196,188)(181,213,205,197,189)(182,214,206,198,190)(183,215,207,199,191)(184,216,208,200,192)(241,273,265,257,249)(242,274,266,258,250)(243,275,267,259,251)(244,276,268,260,252)(245,277,269,261,253)(246,278,270,262,254)(247,279,271,263,255)(248,280,272,264,256)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,95,247,179)(2,180,248,96)(3,89,241,181)(4,182,242,90)(5,91,243,183)(6,184,244,92)(7,93,245,177)(8,178,246,94)(9,311,159,83)(10,84,160,312)(11,305,153,85)(12,86,154,306)(13,307,155,87)(14,88,156,308)(15,309,157,81)(16,82,158,310)(17,315,161,172)(18,173,162,316)(19,317,163,174)(20,175,164,318)(21,319,165,176)(22,169,166,320)(23,313,167,170)(24,171,168,314)(25,101,253,185)(26,186,254,102)(27,103,255,187)(28,188,256,104)(29,97,249,189)(30,190,250,98)(31,99,251,191)(32,192,252,100)(33,109,261,193)(34,194,262,110)(35,111,263,195)(36,196,264,112)(37,105,257,197)(38,198,258,106)(39,107,259,199)(40,200,260,108)(41,117,269,201)(42,202,270,118)(43,119,271,203)(44,204,272,120)(45,113,265,205)(46,206,266,114)(47,115,267,207)(48,208,268,116)(49,125,277,209)(50,210,278,126)(51,127,279,211)(52,212,280,128)(53,121,273,213)(54,214,274,122)(55,123,275,215)(56,216,276,124)(57,217,285,133)(58,134,286,218)(59,219,287,135)(60,136,288,220)(61,221,281,129)(62,130,282,222)(63,223,283,131)(64,132,284,224)(65,225,293,141)(66,142,294,226)(67,227,295,143)(68,144,296,228)(69,229,289,137)(70,138,290,230)(71,231,291,139)(72,140,292,232)(73,233,301,149)(74,150,302,234)(75,235,303,151)(76,152,304,236)(77,237,297,145)(78,146,298,238)(79,239,299,147)(80,148,300,240), (1,283,247,63)(2,132,248,224)(3,285,241,57)(4,134,242,218)(5,287,243,59)(6,136,244,220)(7,281,245,61)(8,130,246,222)(9,207,159,115)(10,48,160,268)(11,201,153,117)(12,42,154,270)(13,203,155,119)(14,44,156,272)(15,205,157,113)(16,46,158,266)(17,211,161,127)(18,52,162,280)(19,213,163,121)(20,54,164,274)(21,215,165,123)(22,56,166,276)(23,209,167,125)(24,50,168,278)(25,289,253,69)(26,138,254,230)(27,291,255,71)(28,140,256,232)(29,293,249,65)(30,142,250,226)(31,295,251,67)(32,144,252,228)(33,297,261,77)(34,146,262,238)(35,299,263,79)(36,148,264,240)(37,301,257,73)(38,150,258,234)(39,303,259,75)(40,152,260,236)(41,305,269,85)(43,307,271,87)(45,309,265,81)(47,311,267,83)(49,313,277,170)(51,315,279,172)(53,317,273,174)(55,319,275,176)(58,90,286,182)(60,92,288,184)(62,94,282,178)(64,96,284,180)(66,98,294,190)(68,100,296,192)(70,102,290,186)(72,104,292,188)(74,106,302,198)(76,108,304,200)(78,110,298,194)(80,112,300,196)(82,114,310,206)(84,116,312,208)(86,118,306,202)(88,120,308,204)(89,217,181,133)(91,219,183,135)(93,221,177,129)(95,223,179,131)(97,225,189,141)(99,227,191,143)(101,229,185,137)(103,231,187,139)(105,233,197,149)(107,235,199,151)(109,237,193,145)(111,239,195,147)(122,318,214,175)(124,320,216,169)(126,314,210,171)(128,316,212,173), (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( (1,51,43,35,27)(2,52,44,36,28)(3,53,45,37,29)(4,54,46,38,30)(5,55,47,39,31)(6,56,48,40,32)(7,49,41,33,25)(8,50,42,34,26)(9,235,227,219,21)(10,236,228,220,22)(11,237,229,221,23)(12,238,230,222,24)(13,239,231,223,17)(14,240,232,224,18)(15,233,225,217,19)(16,234,226,218,20)(57,174,81,73,65)(58,175,82,74,66)(59,176,83,75,67)(60,169,84,76,68)(61,170,85,77,69)(62,171,86,78,70)(63,172,87,79,71)(64,173,88,80,72)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,167,153,145,137)(130,168,154,146,138)(131,161,155,147,139)(132,162,156,148,140)(133,163,157,149,141)(134,164,158,150,142)(135,165,159,151,143)(136,166,160,152,144)(177,209,201,193,185)(178,210,202,194,186)(179,211,203,195,187)(180,212,204,196,188)(181,213,205,197,189)(182,214,206,198,190)(183,215,207,199,191)(184,216,208,200,192)(241,273,265,257,249)(242,274,266,258,250)(243,275,267,259,251)(244,276,268,260,252)(245,277,269,261,253)(246,278,270,262,254)(247,279,271,263,255)(248,280,272,264,256)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,95,247,179)(2,180,248,96)(3,89,241,181)(4,182,242,90)(5,91,243,183)(6,184,244,92)(7,93,245,177)(8,178,246,94)(9,311,159,83)(10,84,160,312)(11,305,153,85)(12,86,154,306)(13,307,155,87)(14,88,156,308)(15,309,157,81)(16,82,158,310)(17,315,161,172)(18,173,162,316)(19,317,163,174)(20,175,164,318)(21,319,165,176)(22,169,166,320)(23,313,167,170)(24,171,168,314)(25,101,253,185)(26,186,254,102)(27,103,255,187)(28,188,256,104)(29,97,249,189)(30,190,250,98)(31,99,251,191)(32,192,252,100)(33,109,261,193)(34,194,262,110)(35,111,263,195)(36,196,264,112)(37,105,257,197)(38,198,258,106)(39,107,259,199)(40,200,260,108)(41,117,269,201)(42,202,270,118)(43,119,271,203)(44,204,272,120)(45,113,265,205)(46,206,266,114)(47,115,267,207)(48,208,268,116)(49,125,277,209)(50,210,278,126)(51,127,279,211)(52,212,280,128)(53,121,273,213)(54,214,274,122)(55,123,275,215)(56,216,276,124)(57,217,285,133)(58,134,286,218)(59,219,287,135)(60,136,288,220)(61,221,281,129)(62,130,282,222)(63,223,283,131)(64,132,284,224)(65,225,293,141)(66,142,294,226)(67,227,295,143)(68,144,296,228)(69,229,289,137)(70,138,290,230)(71,231,291,139)(72,140,292,232)(73,233,301,149)(74,150,302,234)(75,235,303,151)(76,152,304,236)(77,237,297,145)(78,146,298,238)(79,239,299,147)(80,148,300,240), (1,283,247,63)(2,132,248,224)(3,285,241,57)(4,134,242,218)(5,287,243,59)(6,136,244,220)(7,281,245,61)(8,130,246,222)(9,207,159,115)(10,48,160,268)(11,201,153,117)(12,42,154,270)(13,203,155,119)(14,44,156,272)(15,205,157,113)(16,46,158,266)(17,211,161,127)(18,52,162,280)(19,213,163,121)(20,54,164,274)(21,215,165,123)(22,56,166,276)(23,209,167,125)(24,50,168,278)(25,289,253,69)(26,138,254,230)(27,291,255,71)(28,140,256,232)(29,293,249,65)(30,142,250,226)(31,295,251,67)(32,144,252,228)(33,297,261,77)(34,146,262,238)(35,299,263,79)(36,148,264,240)(37,301,257,73)(38,150,258,234)(39,303,259,75)(40,152,260,236)(41,305,269,85)(43,307,271,87)(45,309,265,81)(47,311,267,83)(49,313,277,170)(51,315,279,172)(53,317,273,174)(55,319,275,176)(58,90,286,182)(60,92,288,184)(62,94,282,178)(64,96,284,180)(66,98,294,190)(68,100,296,192)(70,102,290,186)(72,104,292,188)(74,106,302,198)(76,108,304,200)(78,110,298,194)(80,112,300,196)(82,114,310,206)(84,116,312,208)(86,118,306,202)(88,120,308,204)(89,217,181,133)(91,219,183,135)(93,221,177,129)(95,223,179,131)(97,225,189,141)(99,227,191,143)(101,229,185,137)(103,231,187,139)(105,233,197,149)(107,235,199,151)(109,237,193,145)(111,239,195,147)(122,318,214,175)(124,320,216,169)(126,314,210,171)(128,316,212,173), (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([[(1,51,43,35,27),(2,52,44,36,28),(3,53,45,37,29),(4,54,46,38,30),(5,55,47,39,31),(6,56,48,40,32),(7,49,41,33,25),(8,50,42,34,26),(9,235,227,219,21),(10,236,228,220,22),(11,237,229,221,23),(12,238,230,222,24),(13,239,231,223,17),(14,240,232,224,18),(15,233,225,217,19),(16,234,226,218,20),(57,174,81,73,65),(58,175,82,74,66),(59,176,83,75,67),(60,169,84,76,68),(61,170,85,77,69),(62,171,86,78,70),(63,172,87,79,71),(64,173,88,80,72),(89,121,113,105,97),(90,122,114,106,98),(91,123,115,107,99),(92,124,116,108,100),(93,125,117,109,101),(94,126,118,110,102),(95,127,119,111,103),(96,128,120,112,104),(129,167,153,145,137),(130,168,154,146,138),(131,161,155,147,139),(132,162,156,148,140),(133,163,157,149,141),(134,164,158,150,142),(135,165,159,151,143),(136,166,160,152,144),(177,209,201,193,185),(178,210,202,194,186),(179,211,203,195,187),(180,212,204,196,188),(181,213,205,197,189),(182,214,206,198,190),(183,215,207,199,191),(184,216,208,200,192),(241,273,265,257,249),(242,274,266,258,250),(243,275,267,259,251),(244,276,268,260,252),(245,277,269,261,253),(246,278,270,262,254),(247,279,271,263,255),(248,280,272,264,256),(281,313,305,297,289),(282,314,306,298,290),(283,315,307,299,291),(284,316,308,300,292),(285,317,309,301,293),(286,318,310,302,294),(287,319,311,303,295),(288,320,312,304,296)], [(1,95,247,179),(2,180,248,96),(3,89,241,181),(4,182,242,90),(5,91,243,183),(6,184,244,92),(7,93,245,177),(8,178,246,94),(9,311,159,83),(10,84,160,312),(11,305,153,85),(12,86,154,306),(13,307,155,87),(14,88,156,308),(15,309,157,81),(16,82,158,310),(17,315,161,172),(18,173,162,316),(19,317,163,174),(20,175,164,318),(21,319,165,176),(22,169,166,320),(23,313,167,170),(24,171,168,314),(25,101,253,185),(26,186,254,102),(27,103,255,187),(28,188,256,104),(29,97,249,189),(30,190,250,98),(31,99,251,191),(32,192,252,100),(33,109,261,193),(34,194,262,110),(35,111,263,195),(36,196,264,112),(37,105,257,197),(38,198,258,106),(39,107,259,199),(40,200,260,108),(41,117,269,201),(42,202,270,118),(43,119,271,203),(44,204,272,120),(45,113,265,205),(46,206,266,114),(47,115,267,207),(48,208,268,116),(49,125,277,209),(50,210,278,126),(51,127,279,211),(52,212,280,128),(53,121,273,213),(54,214,274,122),(55,123,275,215),(56,216,276,124),(57,217,285,133),(58,134,286,218),(59,219,287,135),(60,136,288,220),(61,221,281,129),(62,130,282,222),(63,223,283,131),(64,132,284,224),(65,225,293,141),(66,142,294,226),(67,227,295,143),(68,144,296,228),(69,229,289,137),(70,138,290,230),(71,231,291,139),(72,140,292,232),(73,233,301,149),(74,150,302,234),(75,235,303,151),(76,152,304,236),(77,237,297,145),(78,146,298,238),(79,239,299,147),(80,148,300,240)], [(1,283,247,63),(2,132,248,224),(3,285,241,57),(4,134,242,218),(5,287,243,59),(6,136,244,220),(7,281,245,61),(8,130,246,222),(9,207,159,115),(10,48,160,268),(11,201,153,117),(12,42,154,270),(13,203,155,119),(14,44,156,272),(15,205,157,113),(16,46,158,266),(17,211,161,127),(18,52,162,280),(19,213,163,121),(20,54,164,274),(21,215,165,123),(22,56,166,276),(23,209,167,125),(24,50,168,278),(25,289,253,69),(26,138,254,230),(27,291,255,71),(28,140,256,232),(29,293,249,65),(30,142,250,226),(31,295,251,67),(32,144,252,228),(33,297,261,77),(34,146,262,238),(35,299,263,79),(36,148,264,240),(37,301,257,73),(38,150,258,234),(39,303,259,75),(40,152,260,236),(41,305,269,85),(43,307,271,87),(45,309,265,81),(47,311,267,83),(49,313,277,170),(51,315,279,172),(53,317,273,174),(55,319,275,176),(58,90,286,182),(60,92,288,184),(62,94,282,178),(64,96,284,180),(66,98,294,190),(68,100,296,192),(70,102,290,186),(72,104,292,188),(74,106,302,198),(76,108,304,200),(78,110,298,194),(80,112,300,196),(82,114,310,206),(84,116,312,208),(86,118,306,202),(88,120,308,204),(89,217,181,133),(91,219,183,135),(93,221,177,129),(95,223,179,131),(97,225,189,141),(99,227,191,143),(101,229,185,137),(103,231,187,139),(105,233,197,149),(107,235,199,151),(109,237,193,145),(111,239,195,147),(122,318,214,175),(124,320,216,169),(126,314,210,171),(128,316,212,173)], [(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)]])
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 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | D4 | M4(2) | SD16 | Q16 | C4≀C2 | C5×D4 | C5×M4(2) | C5×SD16 | C5×Q16 | C5×C4≀C2 |
| kernel | C5×Q8⋊C8 | C4×C40 | C5×C4⋊C8 | Q8×C20 | C5×C4⋊C4 | Q8×C10 | Q8⋊C8 | C5×Q8 | C4×C8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C2×C20 | C20 | C20 | C20 | C10 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 32 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C5×Q8⋊C8 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 27 | 7 |
| 0 | 0 | 7 | 14 |
| 21 | 9 | 0 | 0 |
| 32 | 20 | 0 | 0 |
| 0 | 0 | 13 | 2 |
| 0 | 0 | 2 | 28 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,27,7,0,0,7,14],[21,32,0,0,9,20,0,0,0,0,13,2,0,0,2,28] >;
C5×Q8⋊C8 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes C_8
% in TeX
G:=Group("C5xQ8:C8"); // GroupNames label
G:=SmallGroup(320,131);
// by ID
G=gap.SmallGroup(320,131);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,1410,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^8=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations