direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C5⋊Q16, C20⋊8Q16, C42.212D10, C5⋊5(C4×Q16), (C4×Q8).6D5, Q8.5(C4×D5), (Q8×C20).7C2, C4⋊C4.255D10, C10.105(C4×D4), (C2×C20).259D4, C10.35(C2×Q16), C4.42(C4○D20), C20.62(C4○D4), C10.94(C4○D8), C20.62(C22×C4), (C2×Q8).162D10, (C4×C20).100C22, (C2×C20).349C23, (C4×Dic10).14C2, Dic10.31(C2×C4), C10.D8.18C2, Q8⋊Dic5.17C2, C10.Q16.17C2, C2.6(D4.8D10), C4⋊Dic5.332C22, (Q8×C10).197C22, (C2×Dic10).274C22, C4.27(C2×C4×D5), (C4×C5⋊2C8).9C2, C2.21(C4×C5⋊D4), C2.3(C2×C5⋊Q16), C5⋊2C8.25(C2×C4), (C5×Q8).27(C2×C4), (C2×C10).480(C2×D4), (C2×C5⋊Q16).10C2, C22.81(C2×C5⋊D4), (C2×C4).104(C5⋊D4), (C5×C4⋊C4).286C22, (C2×C4).449(C22×D5), (C2×C5⋊2C8).255C22, SmallGroup(320,656)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C5⋊Q16
G = < a,b,c,d | a4=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 310 in 110 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C20, C2×C10, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C4×Q8, C2×Q16, C5⋊2C8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C4×Q16, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C5⋊Q16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×C5⋊2C8, C10.D8, C10.Q16, Q8⋊Dic5, C4×Dic10, C2×C5⋊Q16, Q8×C20, C4×C5⋊Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, Q16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×Q16, C4○D8, C4×D5, C5⋊D4, C22×D5, C4×Q16, C5⋊Q16, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C2×C5⋊Q16, D4.8D10, C4×C5⋊Q16
►Regular action on 320 points
Generators in S320
(1 264 50 102)(2 257 51 103)(3 258 52 104)(4 259 53 97)(5 260 54 98)(6 261 55 99)(7 262 56 100)(8 263 49 101)(9 43 82 108)(10 44 83 109)(11 45 84 110)(12 46 85 111)(13 47 86 112)(14 48 87 105)(15 41 88 106)(16 42 81 107)(17 237 222 200)(18 238 223 193)(19 239 224 194)(20 240 217 195)(21 233 218 196)(22 234 219 197)(23 235 220 198)(24 236 221 199)(25 137 153 65)(26 138 154 66)(27 139 155 67)(28 140 156 68)(29 141 157 69)(30 142 158 70)(31 143 159 71)(32 144 160 72)(33 125 151 73)(34 126 152 74)(35 127 145 75)(36 128 146 76)(37 121 147 77)(38 122 148 78)(39 123 149 79)(40 124 150 80)(57 91 172 268)(58 92 173 269)(59 93 174 270)(60 94 175 271)(61 95 176 272)(62 96 169 265)(63 89 170 266)(64 90 171 267)(113 315 297 184)(114 316 298 177)(115 317 299 178)(116 318 300 179)(117 319 301 180)(118 320 302 181)(119 313 303 182)(120 314 304 183)(129 210 295 255)(130 211 296 256)(131 212 289 249)(132 213 290 250)(133 214 291 251)(134 215 292 252)(135 216 293 253)(136 209 294 254)(161 280 309 185)(162 273 310 186)(163 274 311 187)(164 275 312 188)(165 276 305 189)(166 277 306 190)(167 278 307 191)(168 279 308 192)(201 281 241 225)(202 282 242 226)(203 283 243 227)(204 284 244 228)(205 285 245 229)(206 286 246 230)(207 287 247 231)(208 288 248 232)
(1 44 115 306 63)(2 64 307 116 45)(3 46 117 308 57)(4 58 309 118 47)(5 48 119 310 59)(6 60 311 120 41)(7 42 113 312 61)(8 62 305 114 43)(9 101 265 276 177)(10 178 277 266 102)(11 103 267 278 179)(12 180 279 268 104)(13 97 269 280 181)(14 182 273 270 98)(15 99 271 274 183)(16 184 275 272 100)(17 226 129 70 123)(18 124 71 130 227)(19 228 131 72 125)(20 126 65 132 229)(21 230 133 66 127)(22 128 67 134 231)(23 232 135 68 121)(24 122 69 136 225)(25 213 205 240 152)(26 145 233 206 214)(27 215 207 234 146)(28 147 235 208 216)(29 209 201 236 148)(30 149 237 202 210)(31 211 203 238 150)(32 151 239 204 212)(33 194 244 249 160)(34 153 250 245 195)(35 196 246 251 154)(36 155 252 247 197)(37 198 248 253 156)(38 157 254 241 199)(39 200 242 255 158)(40 159 256 243 193)(49 169 165 298 108)(50 109 299 166 170)(51 171 167 300 110)(52 111 301 168 172)(53 173 161 302 112)(54 105 303 162 174)(55 175 163 304 106)(56 107 297 164 176)(73 224 284 289 144)(74 137 290 285 217)(75 218 286 291 138)(76 139 292 287 219)(77 220 288 293 140)(78 141 294 281 221)(79 222 282 295 142)(80 143 296 283 223)(81 315 188 95 262)(82 263 96 189 316)(83 317 190 89 264)(84 257 90 191 318)(85 319 192 91 258)(86 259 92 185 320)(87 313 186 93 260)(88 261 94 187 314)
(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 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 124 13 128)(10 123 14 127)(11 122 15 126)(12 121 16 125)(17 182 21 178)(18 181 22 177)(19 180 23 184)(20 179 24 183)(33 111 37 107)(34 110 38 106)(35 109 39 105)(36 108 40 112)(41 152 45 148)(42 151 46 147)(43 150 47 146)(44 149 48 145)(49 159 53 155)(50 158 54 154)(51 157 55 153)(52 156 56 160)(57 216 61 212)(58 215 62 211)(59 214 63 210)(60 213 64 209)(65 103 69 99)(66 102 70 98)(67 101 71 97)(68 100 72 104)(73 85 77 81)(74 84 78 88)(75 83 79 87)(76 82 80 86)(89 295 93 291)(90 294 94 290)(91 293 95 289)(92 292 96 296)(113 239 117 235)(114 238 118 234)(115 237 119 233)(116 236 120 240)(129 270 133 266)(130 269 134 265)(131 268 135 272)(132 267 136 271)(137 257 141 261)(138 264 142 260)(139 263 143 259)(140 262 144 258)(161 247 165 243)(162 246 166 242)(163 245 167 241)(164 244 168 248)(169 256 173 252)(170 255 174 251)(171 254 175 250)(172 253 176 249)(185 287 189 283)(186 286 190 282)(187 285 191 281)(188 284 192 288)(193 302 197 298)(194 301 198 297)(195 300 199 304)(196 299 200 303)(201 311 205 307)(202 310 206 306)(203 309 207 305)(204 308 208 312)(217 318 221 314)(218 317 222 313)(219 316 223 320)(220 315 224 319)(225 274 229 278)(226 273 230 277)(227 280 231 276)(228 279 232 275)
G:=sub<Sym(320)| (1,264,50,102)(2,257,51,103)(3,258,52,104)(4,259,53,97)(5,260,54,98)(6,261,55,99)(7,262,56,100)(8,263,49,101)(9,43,82,108)(10,44,83,109)(11,45,84,110)(12,46,85,111)(13,47,86,112)(14,48,87,105)(15,41,88,106)(16,42,81,107)(17,237,222,200)(18,238,223,193)(19,239,224,194)(20,240,217,195)(21,233,218,196)(22,234,219,197)(23,235,220,198)(24,236,221,199)(25,137,153,65)(26,138,154,66)(27,139,155,67)(28,140,156,68)(29,141,157,69)(30,142,158,70)(31,143,159,71)(32,144,160,72)(33,125,151,73)(34,126,152,74)(35,127,145,75)(36,128,146,76)(37,121,147,77)(38,122,148,78)(39,123,149,79)(40,124,150,80)(57,91,172,268)(58,92,173,269)(59,93,174,270)(60,94,175,271)(61,95,176,272)(62,96,169,265)(63,89,170,266)(64,90,171,267)(113,315,297,184)(114,316,298,177)(115,317,299,178)(116,318,300,179)(117,319,301,180)(118,320,302,181)(119,313,303,182)(120,314,304,183)(129,210,295,255)(130,211,296,256)(131,212,289,249)(132,213,290,250)(133,214,291,251)(134,215,292,252)(135,216,293,253)(136,209,294,254)(161,280,309,185)(162,273,310,186)(163,274,311,187)(164,275,312,188)(165,276,305,189)(166,277,306,190)(167,278,307,191)(168,279,308,192)(201,281,241,225)(202,282,242,226)(203,283,243,227)(204,284,244,228)(205,285,245,229)(206,286,246,230)(207,287,247,231)(208,288,248,232), (1,44,115,306,63)(2,64,307,116,45)(3,46,117,308,57)(4,58,309,118,47)(5,48,119,310,59)(6,60,311,120,41)(7,42,113,312,61)(8,62,305,114,43)(9,101,265,276,177)(10,178,277,266,102)(11,103,267,278,179)(12,180,279,268,104)(13,97,269,280,181)(14,182,273,270,98)(15,99,271,274,183)(16,184,275,272,100)(17,226,129,70,123)(18,124,71,130,227)(19,228,131,72,125)(20,126,65,132,229)(21,230,133,66,127)(22,128,67,134,231)(23,232,135,68,121)(24,122,69,136,225)(25,213,205,240,152)(26,145,233,206,214)(27,215,207,234,146)(28,147,235,208,216)(29,209,201,236,148)(30,149,237,202,210)(31,211,203,238,150)(32,151,239,204,212)(33,194,244,249,160)(34,153,250,245,195)(35,196,246,251,154)(36,155,252,247,197)(37,198,248,253,156)(38,157,254,241,199)(39,200,242,255,158)(40,159,256,243,193)(49,169,165,298,108)(50,109,299,166,170)(51,171,167,300,110)(52,111,301,168,172)(53,173,161,302,112)(54,105,303,162,174)(55,175,163,304,106)(56,107,297,164,176)(73,224,284,289,144)(74,137,290,285,217)(75,218,286,291,138)(76,139,292,287,219)(77,220,288,293,140)(78,141,294,281,221)(79,222,282,295,142)(80,143,296,283,223)(81,315,188,95,262)(82,263,96,189,316)(83,317,190,89,264)(84,257,90,191,318)(85,319,192,91,258)(86,259,92,185,320)(87,313,186,93,260)(88,261,94,187,314), (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), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,124,13,128)(10,123,14,127)(11,122,15,126)(12,121,16,125)(17,182,21,178)(18,181,22,177)(19,180,23,184)(20,179,24,183)(33,111,37,107)(34,110,38,106)(35,109,39,105)(36,108,40,112)(41,152,45,148)(42,151,46,147)(43,150,47,146)(44,149,48,145)(49,159,53,155)(50,158,54,154)(51,157,55,153)(52,156,56,160)(57,216,61,212)(58,215,62,211)(59,214,63,210)(60,213,64,209)(65,103,69,99)(66,102,70,98)(67,101,71,97)(68,100,72,104)(73,85,77,81)(74,84,78,88)(75,83,79,87)(76,82,80,86)(89,295,93,291)(90,294,94,290)(91,293,95,289)(92,292,96,296)(113,239,117,235)(114,238,118,234)(115,237,119,233)(116,236,120,240)(129,270,133,266)(130,269,134,265)(131,268,135,272)(132,267,136,271)(137,257,141,261)(138,264,142,260)(139,263,143,259)(140,262,144,258)(161,247,165,243)(162,246,166,242)(163,245,167,241)(164,244,168,248)(169,256,173,252)(170,255,174,251)(171,254,175,250)(172,253,176,249)(185,287,189,283)(186,286,190,282)(187,285,191,281)(188,284,192,288)(193,302,197,298)(194,301,198,297)(195,300,199,304)(196,299,200,303)(201,311,205,307)(202,310,206,306)(203,309,207,305)(204,308,208,312)(217,318,221,314)(218,317,222,313)(219,316,223,320)(220,315,224,319)(225,274,229,278)(226,273,230,277)(227,280,231,276)(228,279,232,275)>;
G:=Group( (1,264,50,102)(2,257,51,103)(3,258,52,104)(4,259,53,97)(5,260,54,98)(6,261,55,99)(7,262,56,100)(8,263,49,101)(9,43,82,108)(10,44,83,109)(11,45,84,110)(12,46,85,111)(13,47,86,112)(14,48,87,105)(15,41,88,106)(16,42,81,107)(17,237,222,200)(18,238,223,193)(19,239,224,194)(20,240,217,195)(21,233,218,196)(22,234,219,197)(23,235,220,198)(24,236,221,199)(25,137,153,65)(26,138,154,66)(27,139,155,67)(28,140,156,68)(29,141,157,69)(30,142,158,70)(31,143,159,71)(32,144,160,72)(33,125,151,73)(34,126,152,74)(35,127,145,75)(36,128,146,76)(37,121,147,77)(38,122,148,78)(39,123,149,79)(40,124,150,80)(57,91,172,268)(58,92,173,269)(59,93,174,270)(60,94,175,271)(61,95,176,272)(62,96,169,265)(63,89,170,266)(64,90,171,267)(113,315,297,184)(114,316,298,177)(115,317,299,178)(116,318,300,179)(117,319,301,180)(118,320,302,181)(119,313,303,182)(120,314,304,183)(129,210,295,255)(130,211,296,256)(131,212,289,249)(132,213,290,250)(133,214,291,251)(134,215,292,252)(135,216,293,253)(136,209,294,254)(161,280,309,185)(162,273,310,186)(163,274,311,187)(164,275,312,188)(165,276,305,189)(166,277,306,190)(167,278,307,191)(168,279,308,192)(201,281,241,225)(202,282,242,226)(203,283,243,227)(204,284,244,228)(205,285,245,229)(206,286,246,230)(207,287,247,231)(208,288,248,232), (1,44,115,306,63)(2,64,307,116,45)(3,46,117,308,57)(4,58,309,118,47)(5,48,119,310,59)(6,60,311,120,41)(7,42,113,312,61)(8,62,305,114,43)(9,101,265,276,177)(10,178,277,266,102)(11,103,267,278,179)(12,180,279,268,104)(13,97,269,280,181)(14,182,273,270,98)(15,99,271,274,183)(16,184,275,272,100)(17,226,129,70,123)(18,124,71,130,227)(19,228,131,72,125)(20,126,65,132,229)(21,230,133,66,127)(22,128,67,134,231)(23,232,135,68,121)(24,122,69,136,225)(25,213,205,240,152)(26,145,233,206,214)(27,215,207,234,146)(28,147,235,208,216)(29,209,201,236,148)(30,149,237,202,210)(31,211,203,238,150)(32,151,239,204,212)(33,194,244,249,160)(34,153,250,245,195)(35,196,246,251,154)(36,155,252,247,197)(37,198,248,253,156)(38,157,254,241,199)(39,200,242,255,158)(40,159,256,243,193)(49,169,165,298,108)(50,109,299,166,170)(51,171,167,300,110)(52,111,301,168,172)(53,173,161,302,112)(54,105,303,162,174)(55,175,163,304,106)(56,107,297,164,176)(73,224,284,289,144)(74,137,290,285,217)(75,218,286,291,138)(76,139,292,287,219)(77,220,288,293,140)(78,141,294,281,221)(79,222,282,295,142)(80,143,296,283,223)(81,315,188,95,262)(82,263,96,189,316)(83,317,190,89,264)(84,257,90,191,318)(85,319,192,91,258)(86,259,92,185,320)(87,313,186,93,260)(88,261,94,187,314), (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), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,124,13,128)(10,123,14,127)(11,122,15,126)(12,121,16,125)(17,182,21,178)(18,181,22,177)(19,180,23,184)(20,179,24,183)(33,111,37,107)(34,110,38,106)(35,109,39,105)(36,108,40,112)(41,152,45,148)(42,151,46,147)(43,150,47,146)(44,149,48,145)(49,159,53,155)(50,158,54,154)(51,157,55,153)(52,156,56,160)(57,216,61,212)(58,215,62,211)(59,214,63,210)(60,213,64,209)(65,103,69,99)(66,102,70,98)(67,101,71,97)(68,100,72,104)(73,85,77,81)(74,84,78,88)(75,83,79,87)(76,82,80,86)(89,295,93,291)(90,294,94,290)(91,293,95,289)(92,292,96,296)(113,239,117,235)(114,238,118,234)(115,237,119,233)(116,236,120,240)(129,270,133,266)(130,269,134,265)(131,268,135,272)(132,267,136,271)(137,257,141,261)(138,264,142,260)(139,263,143,259)(140,262,144,258)(161,247,165,243)(162,246,166,242)(163,245,167,241)(164,244,168,248)(169,256,173,252)(170,255,174,251)(171,254,175,250)(172,253,176,249)(185,287,189,283)(186,286,190,282)(187,285,191,281)(188,284,192,288)(193,302,197,298)(194,301,198,297)(195,300,199,304)(196,299,200,303)(201,311,205,307)(202,310,206,306)(203,309,207,305)(204,308,208,312)(217,318,221,314)(218,317,222,313)(219,316,223,320)(220,315,224,319)(225,274,229,278)(226,273,230,277)(227,280,231,276)(228,279,232,275) );
G=PermutationGroup([[(1,264,50,102),(2,257,51,103),(3,258,52,104),(4,259,53,97),(5,260,54,98),(6,261,55,99),(7,262,56,100),(8,263,49,101),(9,43,82,108),(10,44,83,109),(11,45,84,110),(12,46,85,111),(13,47,86,112),(14,48,87,105),(15,41,88,106),(16,42,81,107),(17,237,222,200),(18,238,223,193),(19,239,224,194),(20,240,217,195),(21,233,218,196),(22,234,219,197),(23,235,220,198),(24,236,221,199),(25,137,153,65),(26,138,154,66),(27,139,155,67),(28,140,156,68),(29,141,157,69),(30,142,158,70),(31,143,159,71),(32,144,160,72),(33,125,151,73),(34,126,152,74),(35,127,145,75),(36,128,146,76),(37,121,147,77),(38,122,148,78),(39,123,149,79),(40,124,150,80),(57,91,172,268),(58,92,173,269),(59,93,174,270),(60,94,175,271),(61,95,176,272),(62,96,169,265),(63,89,170,266),(64,90,171,267),(113,315,297,184),(114,316,298,177),(115,317,299,178),(116,318,300,179),(117,319,301,180),(118,320,302,181),(119,313,303,182),(120,314,304,183),(129,210,295,255),(130,211,296,256),(131,212,289,249),(132,213,290,250),(133,214,291,251),(134,215,292,252),(135,216,293,253),(136,209,294,254),(161,280,309,185),(162,273,310,186),(163,274,311,187),(164,275,312,188),(165,276,305,189),(166,277,306,190),(167,278,307,191),(168,279,308,192),(201,281,241,225),(202,282,242,226),(203,283,243,227),(204,284,244,228),(205,285,245,229),(206,286,246,230),(207,287,247,231),(208,288,248,232)], [(1,44,115,306,63),(2,64,307,116,45),(3,46,117,308,57),(4,58,309,118,47),(5,48,119,310,59),(6,60,311,120,41),(7,42,113,312,61),(8,62,305,114,43),(9,101,265,276,177),(10,178,277,266,102),(11,103,267,278,179),(12,180,279,268,104),(13,97,269,280,181),(14,182,273,270,98),(15,99,271,274,183),(16,184,275,272,100),(17,226,129,70,123),(18,124,71,130,227),(19,228,131,72,125),(20,126,65,132,229),(21,230,133,66,127),(22,128,67,134,231),(23,232,135,68,121),(24,122,69,136,225),(25,213,205,240,152),(26,145,233,206,214),(27,215,207,234,146),(28,147,235,208,216),(29,209,201,236,148),(30,149,237,202,210),(31,211,203,238,150),(32,151,239,204,212),(33,194,244,249,160),(34,153,250,245,195),(35,196,246,251,154),(36,155,252,247,197),(37,198,248,253,156),(38,157,254,241,199),(39,200,242,255,158),(40,159,256,243,193),(49,169,165,298,108),(50,109,299,166,170),(51,171,167,300,110),(52,111,301,168,172),(53,173,161,302,112),(54,105,303,162,174),(55,175,163,304,106),(56,107,297,164,176),(73,224,284,289,144),(74,137,290,285,217),(75,218,286,291,138),(76,139,292,287,219),(77,220,288,293,140),(78,141,294,281,221),(79,222,282,295,142),(80,143,296,283,223),(81,315,188,95,262),(82,263,96,189,316),(83,317,190,89,264),(84,257,90,191,318),(85,319,192,91,258),(86,259,92,185,320),(87,313,186,93,260),(88,261,94,187,314)], [(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)], [(1,30,5,26),(2,29,6,25),(3,28,7,32),(4,27,8,31),(9,124,13,128),(10,123,14,127),(11,122,15,126),(12,121,16,125),(17,182,21,178),(18,181,22,177),(19,180,23,184),(20,179,24,183),(33,111,37,107),(34,110,38,106),(35,109,39,105),(36,108,40,112),(41,152,45,148),(42,151,46,147),(43,150,47,146),(44,149,48,145),(49,159,53,155),(50,158,54,154),(51,157,55,153),(52,156,56,160),(57,216,61,212),(58,215,62,211),(59,214,63,210),(60,213,64,209),(65,103,69,99),(66,102,70,98),(67,101,71,97),(68,100,72,104),(73,85,77,81),(74,84,78,88),(75,83,79,87),(76,82,80,86),(89,295,93,291),(90,294,94,290),(91,293,95,289),(92,292,96,296),(113,239,117,235),(114,238,118,234),(115,237,119,233),(116,236,120,240),(129,270,133,266),(130,269,134,265),(131,268,135,272),(132,267,136,271),(137,257,141,261),(138,264,142,260),(139,263,143,259),(140,262,144,258),(161,247,165,243),(162,246,166,242),(163,245,167,241),(164,244,168,248),(169,256,173,252),(170,255,174,251),(171,254,175,250),(172,253,176,249),(185,287,189,283),(186,286,190,282),(187,285,191,281),(188,284,192,288),(193,302,197,298),(194,301,198,297),(195,300,199,304),(196,299,200,303),(201,311,205,307),(202,310,206,306),(203,309,207,305),(204,308,208,312),(217,318,221,314),(218,317,222,313),(219,316,223,320),(220,315,224,319),(225,274,229,278),(226,273,230,277),(227,280,231,276),(228,279,232,275)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | Q16 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C4×D5 | C4○D20 | C5⋊Q16 | D4.8D10 |
| kernel | C4×C5⋊Q16 | C4×C5⋊2C8 | C10.D8 | C10.Q16 | Q8⋊Dic5 | C4×Dic10 | C2×C5⋊Q16 | Q8×C20 | C5⋊Q16 | C2×C20 | C4×Q8 | C20 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C2×C4 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C4×C5⋊Q16 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 11 | 0 | 0 |
| 30 | 32 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 15 | 26 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[9,30,0,0,11,32,0,0,0,0,12,12,0,0,29,12],[40,0,0,0,0,40,0,0,0,0,15,15,0,0,15,26] >;
C4×C5⋊Q16 in GAP, Magma, Sage, TeX
C_4\times C_5\rtimes Q_{16} % in TeX
G:=Group("C4xC5:Q16"); // GroupNames label
G:=SmallGroup(320,656);
// by ID
G=gap.SmallGroup(320,656);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,58,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations