direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×Q16⋊C4, Q16⋊3C20, C8.5(C2×C20), C40.87(C2×C4), (C5×Q16)⋊15C4, C2.16(D4×C20), (C4×Q8).3C10, Q8.2(C2×C20), C4.Q8.2C10, C8⋊C4.1C10, (C2×C20).457D4, C10.148(C4×D4), C42.9(C2×C10), (C2×Q16).6C10, (Q8×C20).16C2, C4.13(C22×C20), Q8⋊C4.6C10, (C10×Q16).13C2, C22.55(D4×C10), C20.260(C4○D4), C20.217(C22×C4), (C2×C20).908C23, (C4×C20).250C22, (C2×C40).330C22, (Q8×C10).257C22, C10.130(C8.C22), C4.5(C5×C4○D4), (C5×C4.Q8).7C2, (C5×C8⋊C4).5C2, C4⋊C4.49(C2×C10), (C2×C8).19(C2×C10), (C2×C4).103(C5×D4), (C5×Q8).35(C2×C4), C2.5(C5×C8.C22), (C2×C10).631(C2×D4), (C2×Q8).42(C2×C10), (C5×C4⋊C4).370C22, (C2×C4).83(C22×C10), (C5×Q8⋊C4).15C2, SmallGroup(320,942)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q16⋊C4
G = < a,b,c,d | a5=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, cd=dc >
Subgroups: 154 in 108 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C10, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C40, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C5×Q8, Q16⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×Q16, Q8×C10, C5×C8⋊C4, C5×Q8⋊C4, C5×C4.Q8, Q8×C20, C10×Q16, C5×Q16⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C8.C22, C2×C20, C5×D4, C22×C10, Q16⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8.C22, C5×Q16⋊C4
►Regular action on 320 points
Generators in S320
(1 163 123 169 134)(2 164 124 170 135)(3 165 125 171 136)(4 166 126 172 129)(5 167 127 173 130)(6 168 128 174 131)(7 161 121 175 132)(8 162 122 176 133)(9 273 297 17 289)(10 274 298 18 290)(11 275 299 19 291)(12 276 300 20 292)(13 277 301 21 293)(14 278 302 22 294)(15 279 303 23 295)(16 280 304 24 296)(25 286 313 33 305)(26 287 314 34 306)(27 288 315 35 307)(28 281 316 36 308)(29 282 317 37 309)(30 283 318 38 310)(31 284 319 39 311)(32 285 320 40 312)(41 235 251 49 243)(42 236 252 50 244)(43 237 253 51 245)(44 238 254 52 246)(45 239 255 53 247)(46 240 256 54 248)(47 233 249 55 241)(48 234 250 56 242)(57 73 267 65 259)(58 74 268 66 260)(59 75 269 67 261)(60 76 270 68 262)(61 77 271 69 263)(62 78 272 70 264)(63 79 265 71 257)(64 80 266 72 258)(81 195 210 89 202)(82 196 211 90 203)(83 197 212 91 204)(84 198 213 92 205)(85 199 214 93 206)(86 200 215 94 207)(87 193 216 95 208)(88 194 209 96 201)(97 113 226 105 218)(98 114 227 106 219)(99 115 228 107 220)(100 116 229 108 221)(101 117 230 109 222)(102 118 231 110 223)(103 119 232 111 224)(104 120 225 112 217)(137 153 188 145 180)(138 154 189 146 181)(139 155 190 147 182)(140 156 191 148 183)(141 157 192 149 184)(142 158 185 150 177)(143 159 186 151 178)(144 160 187 152 179)
(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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 220 13 224)(10 219 14 223)(11 218 15 222)(12 217 16 221)(17 228 21 232)(18 227 22 231)(19 226 23 230)(20 225 24 229)(25 86 29 82)(26 85 30 81)(27 84 31 88)(28 83 32 87)(33 94 37 90)(34 93 38 89)(35 92 39 96)(36 91 40 95)(41 178 45 182)(42 177 46 181)(43 184 47 180)(44 183 48 179)(49 186 53 190)(50 185 54 189)(51 192 55 188)(52 191 56 187)(65 169 69 173)(66 176 70 172)(67 175 71 171)(68 174 72 170)(73 163 77 167)(74 162 78 166)(75 161 79 165)(76 168 80 164)(97 279 101 275)(98 278 102 274)(99 277 103 273)(100 276 104 280)(105 295 109 291)(106 294 110 290)(107 293 111 289)(108 292 112 296)(113 303 117 299)(114 302 118 298)(115 301 119 297)(116 300 120 304)(121 265 125 269)(122 272 126 268)(123 271 127 267)(124 270 128 266)(129 260 133 264)(130 259 134 263)(131 258 135 262)(132 257 136 261)(137 237 141 233)(138 236 142 240)(139 235 143 239)(140 234 144 238)(145 245 149 241)(146 244 150 248)(147 243 151 247)(148 242 152 246)(153 253 157 249)(154 252 158 256)(155 251 159 255)(156 250 160 254)(193 281 197 285)(194 288 198 284)(195 287 199 283)(196 286 200 282)(201 307 205 311)(202 306 206 310)(203 305 207 309)(204 312 208 308)(209 315 213 319)(210 314 214 318)(211 313 215 317)(212 320 216 316)
(1 143 27 275)(2 140 28 280)(3 137 29 277)(4 142 30 274)(5 139 31 279)(6 144 32 276)(7 141 25 273)(8 138 26 278)(9 132 184 305)(10 129 177 310)(11 134 178 307)(12 131 179 312)(13 136 180 309)(14 133 181 306)(15 130 182 311)(16 135 183 308)(17 121 192 313)(18 126 185 318)(19 123 186 315)(20 128 187 320)(21 125 188 317)(22 122 189 314)(23 127 190 319)(24 124 191 316)(33 289 175 149)(34 294 176 146)(35 291 169 151)(36 296 170 148)(37 293 171 145)(38 290 172 150)(39 295 173 147)(40 292 174 152)(41 201 222 259)(42 206 223 264)(43 203 224 261)(44 208 217 258)(45 205 218 263)(46 202 219 260)(47 207 220 257)(48 204 221 262)(49 209 230 267)(50 214 231 272)(51 211 232 269)(52 216 225 266)(53 213 226 271)(54 210 227 268)(55 215 228 265)(56 212 229 270)(57 235 88 101)(58 240 81 98)(59 237 82 103)(60 234 83 100)(61 239 84 97)(62 236 85 102)(63 233 86 99)(64 238 87 104)(65 243 96 109)(66 248 89 106)(67 245 90 111)(68 242 91 108)(69 247 92 105)(70 244 93 110)(71 241 94 107)(72 246 95 112)(73 251 194 117)(74 256 195 114)(75 253 196 119)(76 250 197 116)(77 255 198 113)(78 252 199 118)(79 249 200 115)(80 254 193 120)(153 282 301 165)(154 287 302 162)(155 284 303 167)(156 281 304 164)(157 286 297 161)(158 283 298 166)(159 288 299 163)(160 285 300 168)
G:=sub<Sym(320)| (1,163,123,169,134)(2,164,124,170,135)(3,165,125,171,136)(4,166,126,172,129)(5,167,127,173,130)(6,168,128,174,131)(7,161,121,175,132)(8,162,122,176,133)(9,273,297,17,289)(10,274,298,18,290)(11,275,299,19,291)(12,276,300,20,292)(13,277,301,21,293)(14,278,302,22,294)(15,279,303,23,295)(16,280,304,24,296)(25,286,313,33,305)(26,287,314,34,306)(27,288,315,35,307)(28,281,316,36,308)(29,282,317,37,309)(30,283,318,38,310)(31,284,319,39,311)(32,285,320,40,312)(41,235,251,49,243)(42,236,252,50,244)(43,237,253,51,245)(44,238,254,52,246)(45,239,255,53,247)(46,240,256,54,248)(47,233,249,55,241)(48,234,250,56,242)(57,73,267,65,259)(58,74,268,66,260)(59,75,269,67,261)(60,76,270,68,262)(61,77,271,69,263)(62,78,272,70,264)(63,79,265,71,257)(64,80,266,72,258)(81,195,210,89,202)(82,196,211,90,203)(83,197,212,91,204)(84,198,213,92,205)(85,199,214,93,206)(86,200,215,94,207)(87,193,216,95,208)(88,194,209,96,201)(97,113,226,105,218)(98,114,227,106,219)(99,115,228,107,220)(100,116,229,108,221)(101,117,230,109,222)(102,118,231,110,223)(103,119,232,111,224)(104,120,225,112,217)(137,153,188,145,180)(138,154,189,146,181)(139,155,190,147,182)(140,156,191,148,183)(141,157,192,149,184)(142,158,185,150,177)(143,159,186,151,178)(144,160,187,152,179), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,220,13,224)(10,219,14,223)(11,218,15,222)(12,217,16,221)(17,228,21,232)(18,227,22,231)(19,226,23,230)(20,225,24,229)(25,86,29,82)(26,85,30,81)(27,84,31,88)(28,83,32,87)(33,94,37,90)(34,93,38,89)(35,92,39,96)(36,91,40,95)(41,178,45,182)(42,177,46,181)(43,184,47,180)(44,183,48,179)(49,186,53,190)(50,185,54,189)(51,192,55,188)(52,191,56,187)(65,169,69,173)(66,176,70,172)(67,175,71,171)(68,174,72,170)(73,163,77,167)(74,162,78,166)(75,161,79,165)(76,168,80,164)(97,279,101,275)(98,278,102,274)(99,277,103,273)(100,276,104,280)(105,295,109,291)(106,294,110,290)(107,293,111,289)(108,292,112,296)(113,303,117,299)(114,302,118,298)(115,301,119,297)(116,300,120,304)(121,265,125,269)(122,272,126,268)(123,271,127,267)(124,270,128,266)(129,260,133,264)(130,259,134,263)(131,258,135,262)(132,257,136,261)(137,237,141,233)(138,236,142,240)(139,235,143,239)(140,234,144,238)(145,245,149,241)(146,244,150,248)(147,243,151,247)(148,242,152,246)(153,253,157,249)(154,252,158,256)(155,251,159,255)(156,250,160,254)(193,281,197,285)(194,288,198,284)(195,287,199,283)(196,286,200,282)(201,307,205,311)(202,306,206,310)(203,305,207,309)(204,312,208,308)(209,315,213,319)(210,314,214,318)(211,313,215,317)(212,320,216,316), (1,143,27,275)(2,140,28,280)(3,137,29,277)(4,142,30,274)(5,139,31,279)(6,144,32,276)(7,141,25,273)(8,138,26,278)(9,132,184,305)(10,129,177,310)(11,134,178,307)(12,131,179,312)(13,136,180,309)(14,133,181,306)(15,130,182,311)(16,135,183,308)(17,121,192,313)(18,126,185,318)(19,123,186,315)(20,128,187,320)(21,125,188,317)(22,122,189,314)(23,127,190,319)(24,124,191,316)(33,289,175,149)(34,294,176,146)(35,291,169,151)(36,296,170,148)(37,293,171,145)(38,290,172,150)(39,295,173,147)(40,292,174,152)(41,201,222,259)(42,206,223,264)(43,203,224,261)(44,208,217,258)(45,205,218,263)(46,202,219,260)(47,207,220,257)(48,204,221,262)(49,209,230,267)(50,214,231,272)(51,211,232,269)(52,216,225,266)(53,213,226,271)(54,210,227,268)(55,215,228,265)(56,212,229,270)(57,235,88,101)(58,240,81,98)(59,237,82,103)(60,234,83,100)(61,239,84,97)(62,236,85,102)(63,233,86,99)(64,238,87,104)(65,243,96,109)(66,248,89,106)(67,245,90,111)(68,242,91,108)(69,247,92,105)(70,244,93,110)(71,241,94,107)(72,246,95,112)(73,251,194,117)(74,256,195,114)(75,253,196,119)(76,250,197,116)(77,255,198,113)(78,252,199,118)(79,249,200,115)(80,254,193,120)(153,282,301,165)(154,287,302,162)(155,284,303,167)(156,281,304,164)(157,286,297,161)(158,283,298,166)(159,288,299,163)(160,285,300,168)>;
G:=Group( (1,163,123,169,134)(2,164,124,170,135)(3,165,125,171,136)(4,166,126,172,129)(5,167,127,173,130)(6,168,128,174,131)(7,161,121,175,132)(8,162,122,176,133)(9,273,297,17,289)(10,274,298,18,290)(11,275,299,19,291)(12,276,300,20,292)(13,277,301,21,293)(14,278,302,22,294)(15,279,303,23,295)(16,280,304,24,296)(25,286,313,33,305)(26,287,314,34,306)(27,288,315,35,307)(28,281,316,36,308)(29,282,317,37,309)(30,283,318,38,310)(31,284,319,39,311)(32,285,320,40,312)(41,235,251,49,243)(42,236,252,50,244)(43,237,253,51,245)(44,238,254,52,246)(45,239,255,53,247)(46,240,256,54,248)(47,233,249,55,241)(48,234,250,56,242)(57,73,267,65,259)(58,74,268,66,260)(59,75,269,67,261)(60,76,270,68,262)(61,77,271,69,263)(62,78,272,70,264)(63,79,265,71,257)(64,80,266,72,258)(81,195,210,89,202)(82,196,211,90,203)(83,197,212,91,204)(84,198,213,92,205)(85,199,214,93,206)(86,200,215,94,207)(87,193,216,95,208)(88,194,209,96,201)(97,113,226,105,218)(98,114,227,106,219)(99,115,228,107,220)(100,116,229,108,221)(101,117,230,109,222)(102,118,231,110,223)(103,119,232,111,224)(104,120,225,112,217)(137,153,188,145,180)(138,154,189,146,181)(139,155,190,147,182)(140,156,191,148,183)(141,157,192,149,184)(142,158,185,150,177)(143,159,186,151,178)(144,160,187,152,179), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,220,13,224)(10,219,14,223)(11,218,15,222)(12,217,16,221)(17,228,21,232)(18,227,22,231)(19,226,23,230)(20,225,24,229)(25,86,29,82)(26,85,30,81)(27,84,31,88)(28,83,32,87)(33,94,37,90)(34,93,38,89)(35,92,39,96)(36,91,40,95)(41,178,45,182)(42,177,46,181)(43,184,47,180)(44,183,48,179)(49,186,53,190)(50,185,54,189)(51,192,55,188)(52,191,56,187)(65,169,69,173)(66,176,70,172)(67,175,71,171)(68,174,72,170)(73,163,77,167)(74,162,78,166)(75,161,79,165)(76,168,80,164)(97,279,101,275)(98,278,102,274)(99,277,103,273)(100,276,104,280)(105,295,109,291)(106,294,110,290)(107,293,111,289)(108,292,112,296)(113,303,117,299)(114,302,118,298)(115,301,119,297)(116,300,120,304)(121,265,125,269)(122,272,126,268)(123,271,127,267)(124,270,128,266)(129,260,133,264)(130,259,134,263)(131,258,135,262)(132,257,136,261)(137,237,141,233)(138,236,142,240)(139,235,143,239)(140,234,144,238)(145,245,149,241)(146,244,150,248)(147,243,151,247)(148,242,152,246)(153,253,157,249)(154,252,158,256)(155,251,159,255)(156,250,160,254)(193,281,197,285)(194,288,198,284)(195,287,199,283)(196,286,200,282)(201,307,205,311)(202,306,206,310)(203,305,207,309)(204,312,208,308)(209,315,213,319)(210,314,214,318)(211,313,215,317)(212,320,216,316), (1,143,27,275)(2,140,28,280)(3,137,29,277)(4,142,30,274)(5,139,31,279)(6,144,32,276)(7,141,25,273)(8,138,26,278)(9,132,184,305)(10,129,177,310)(11,134,178,307)(12,131,179,312)(13,136,180,309)(14,133,181,306)(15,130,182,311)(16,135,183,308)(17,121,192,313)(18,126,185,318)(19,123,186,315)(20,128,187,320)(21,125,188,317)(22,122,189,314)(23,127,190,319)(24,124,191,316)(33,289,175,149)(34,294,176,146)(35,291,169,151)(36,296,170,148)(37,293,171,145)(38,290,172,150)(39,295,173,147)(40,292,174,152)(41,201,222,259)(42,206,223,264)(43,203,224,261)(44,208,217,258)(45,205,218,263)(46,202,219,260)(47,207,220,257)(48,204,221,262)(49,209,230,267)(50,214,231,272)(51,211,232,269)(52,216,225,266)(53,213,226,271)(54,210,227,268)(55,215,228,265)(56,212,229,270)(57,235,88,101)(58,240,81,98)(59,237,82,103)(60,234,83,100)(61,239,84,97)(62,236,85,102)(63,233,86,99)(64,238,87,104)(65,243,96,109)(66,248,89,106)(67,245,90,111)(68,242,91,108)(69,247,92,105)(70,244,93,110)(71,241,94,107)(72,246,95,112)(73,251,194,117)(74,256,195,114)(75,253,196,119)(76,250,197,116)(77,255,198,113)(78,252,199,118)(79,249,200,115)(80,254,193,120)(153,282,301,165)(154,287,302,162)(155,284,303,167)(156,281,304,164)(157,286,297,161)(158,283,298,166)(159,288,299,163)(160,285,300,168) );
G=PermutationGroup([[(1,163,123,169,134),(2,164,124,170,135),(3,165,125,171,136),(4,166,126,172,129),(5,167,127,173,130),(6,168,128,174,131),(7,161,121,175,132),(8,162,122,176,133),(9,273,297,17,289),(10,274,298,18,290),(11,275,299,19,291),(12,276,300,20,292),(13,277,301,21,293),(14,278,302,22,294),(15,279,303,23,295),(16,280,304,24,296),(25,286,313,33,305),(26,287,314,34,306),(27,288,315,35,307),(28,281,316,36,308),(29,282,317,37,309),(30,283,318,38,310),(31,284,319,39,311),(32,285,320,40,312),(41,235,251,49,243),(42,236,252,50,244),(43,237,253,51,245),(44,238,254,52,246),(45,239,255,53,247),(46,240,256,54,248),(47,233,249,55,241),(48,234,250,56,242),(57,73,267,65,259),(58,74,268,66,260),(59,75,269,67,261),(60,76,270,68,262),(61,77,271,69,263),(62,78,272,70,264),(63,79,265,71,257),(64,80,266,72,258),(81,195,210,89,202),(82,196,211,90,203),(83,197,212,91,204),(84,198,213,92,205),(85,199,214,93,206),(86,200,215,94,207),(87,193,216,95,208),(88,194,209,96,201),(97,113,226,105,218),(98,114,227,106,219),(99,115,228,107,220),(100,116,229,108,221),(101,117,230,109,222),(102,118,231,110,223),(103,119,232,111,224),(104,120,225,112,217),(137,153,188,145,180),(138,154,189,146,181),(139,155,190,147,182),(140,156,191,148,183),(141,157,192,149,184),(142,158,185,150,177),(143,159,186,151,178),(144,160,187,152,179)], [(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)], [(1,61,5,57),(2,60,6,64),(3,59,7,63),(4,58,8,62),(9,220,13,224),(10,219,14,223),(11,218,15,222),(12,217,16,221),(17,228,21,232),(18,227,22,231),(19,226,23,230),(20,225,24,229),(25,86,29,82),(26,85,30,81),(27,84,31,88),(28,83,32,87),(33,94,37,90),(34,93,38,89),(35,92,39,96),(36,91,40,95),(41,178,45,182),(42,177,46,181),(43,184,47,180),(44,183,48,179),(49,186,53,190),(50,185,54,189),(51,192,55,188),(52,191,56,187),(65,169,69,173),(66,176,70,172),(67,175,71,171),(68,174,72,170),(73,163,77,167),(74,162,78,166),(75,161,79,165),(76,168,80,164),(97,279,101,275),(98,278,102,274),(99,277,103,273),(100,276,104,280),(105,295,109,291),(106,294,110,290),(107,293,111,289),(108,292,112,296),(113,303,117,299),(114,302,118,298),(115,301,119,297),(116,300,120,304),(121,265,125,269),(122,272,126,268),(123,271,127,267),(124,270,128,266),(129,260,133,264),(130,259,134,263),(131,258,135,262),(132,257,136,261),(137,237,141,233),(138,236,142,240),(139,235,143,239),(140,234,144,238),(145,245,149,241),(146,244,150,248),(147,243,151,247),(148,242,152,246),(153,253,157,249),(154,252,158,256),(155,251,159,255),(156,250,160,254),(193,281,197,285),(194,288,198,284),(195,287,199,283),(196,286,200,282),(201,307,205,311),(202,306,206,310),(203,305,207,309),(204,312,208,308),(209,315,213,319),(210,314,214,318),(211,313,215,317),(212,320,216,316)], [(1,143,27,275),(2,140,28,280),(3,137,29,277),(4,142,30,274),(5,139,31,279),(6,144,32,276),(7,141,25,273),(8,138,26,278),(9,132,184,305),(10,129,177,310),(11,134,178,307),(12,131,179,312),(13,136,180,309),(14,133,181,306),(15,130,182,311),(16,135,183,308),(17,121,192,313),(18,126,185,318),(19,123,186,315),(20,128,187,320),(21,125,188,317),(22,122,189,314),(23,127,190,319),(24,124,191,316),(33,289,175,149),(34,294,176,146),(35,291,169,151),(36,296,170,148),(37,293,171,145),(38,290,172,150),(39,295,173,147),(40,292,174,152),(41,201,222,259),(42,206,223,264),(43,203,224,261),(44,208,217,258),(45,205,218,263),(46,202,219,260),(47,207,220,257),(48,204,221,262),(49,209,230,267),(50,214,231,272),(51,211,232,269),(52,216,225,266),(53,213,226,271),(54,210,227,268),(55,215,228,265),(56,212,229,270),(57,235,88,101),(58,240,81,98),(59,237,82,103),(60,234,83,100),(61,239,84,97),(62,236,85,102),(63,233,86,99),(64,238,87,104),(65,243,96,109),(66,248,89,106),(67,245,90,111),(68,242,91,108),(69,247,92,105),(70,244,93,110),(71,241,94,107),(72,246,95,112),(73,251,194,117),(74,256,195,114),(75,253,196,119),(76,250,197,116),(77,255,198,113),(78,252,199,118),(79,249,200,115),(80,254,193,120),(153,282,301,165),(154,287,302,162),(155,284,303,167),(156,281,304,164),(157,286,297,161),(158,283,298,166),(159,288,299,163),(160,285,300,168)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | ··· | 4N | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 20A | ··· | 20X | 20Y | ··· | 20BD | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×Q16⋊C4 | C5×C8⋊C4 | C5×Q8⋊C4 | C5×C4.Q8 | Q8×C20 | C10×Q16 | C5×Q16 | Q16⋊C4 | C8⋊C4 | Q8⋊C4 | C4.Q8 | C4×Q8 | C2×Q16 | Q16 | C2×C20 | C20 | C2×C4 | C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 8 | 4 | 32 | 2 | 2 | 8 | 8 | 2 | 8 |
Matrix representation of C5×Q16⋊C4 ►in GL6(𝔽41)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 17 | 5 | 0 | 0 | 0 | 0 |
| 24 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 20 | 13 | 37 |
| 0 | 0 | 33 | 38 | 25 | 12 |
| 0 | 0 | 34 | 8 | 40 | 8 |
| 0 | 0 | 1 | 37 | 17 | 11 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 19 | 29 |
| 0 | 0 | 15 | 17 | 36 | 24 |
| 0 | 0 | 0 | 27 | 34 | 14 |
| 0 | 0 | 15 | 10 | 9 | 31 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 39 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 |
G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[17,24,0,0,0,0,5,24,0,0,0,0,0,0,34,33,34,1,0,0,20,38,8,37,0,0,13,25,40,17,0,0,37,12,8,11],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,0,15,0,15,0,0,1,17,27,10,0,0,19,36,34,9,0,0,29,24,14,31],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,39,40,40,40,0,0,0,1,0,0] >;
C5×Q16⋊C4 in GAP, Magma, Sage, TeX
C_5\times Q_{16}\rtimes C_4 % in TeX
G:=Group("C5xQ16:C4"); // GroupNames label
G:=SmallGroup(320,942);
// by ID
G=gap.SmallGroup(320,942);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,3446,436,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations