direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4⋊Q16, C20⋊6Q16, C40.71D4, C4⋊1(C5×Q16), C8.8(C5×D4), (C4×C8).7C10, C4.3(D4×C10), C4⋊Q8.9C10, (C4×C40).25C2, (C2×C20).423D4, C20.310(C2×D4), (C2×Q16).3C10, C10.57(C2×Q16), C2.10(C10×Q16), C42.81(C2×C10), (C10×Q16).10C2, C10.44(C4⋊1D4), (C2×C20).950C23, (C4×C20).365C22, (C2×C40).424C22, C22.115(D4×C10), (Q8×C10).177C22, (C2×C4).79(C5×D4), C2.7(C5×C4⋊1D4), (C5×C4⋊Q8).24C2, (C2×C8).80(C2×C10), (C2×C10).671(C2×D4), (C2×Q8).21(C2×C10), (C2×C4).125(C22×C10), SmallGroup(320,995)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊Q16
G = < a,b,c,d | a5=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 194 in 122 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C20, C20, C2×C10, C4×C8, C4⋊Q8, C2×Q16, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C4⋊Q16, C4×C20, C5×C4⋊C4, C2×C40, C5×Q16, Q8×C10, C4×C40, C5×C4⋊Q8, C10×Q16, C5×C4⋊Q16
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C2×C10, C4⋊1D4, C2×Q16, C5×D4, C22×C10, C4⋊Q16, C5×Q16, D4×C10, C5×C4⋊1D4, C10×Q16, C5×C4⋊Q16
►Regular action on 320 points
Generators in S320
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 90 42 114 34)(10 91 43 115 35)(11 92 44 116 36)(12 93 45 117 37)(13 94 46 118 38)(14 95 47 119 39)(15 96 48 120 40)(16 89 41 113 33)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 165 153 73 145)(66 166 154 74 146)(67 167 155 75 147)(68 168 156 76 148)(69 161 157 77 149)(70 162 158 78 150)(71 163 159 79 151)(72 164 160 80 152)(81 316 240 308 232)(82 317 233 309 225)(83 318 234 310 226)(84 319 235 311 227)(85 320 236 312 228)(86 313 237 305 229)(87 314 238 306 230)(88 315 239 307 231)(169 185 261 177 253)(170 186 262 178 254)(171 187 263 179 255)(172 188 264 180 256)(173 189 257 181 249)(174 190 258 182 250)(175 191 259 183 251)(176 192 260 184 252)(193 269 246 201 277)(194 270 247 202 278)(195 271 248 203 279)(196 272 241 204 280)(197 265 242 205 273)(198 266 243 206 274)(199 267 244 207 275)(200 268 245 208 276)(209 285 301 217 293)(210 286 302 218 294)(211 287 303 219 295)(212 288 304 220 296)(213 281 297 221 289)(214 282 298 222 290)(215 283 299 223 291)(216 284 300 224 292)
(1 69 127 39)(2 70 128 40)(3 71 121 33)(4 72 122 34)(5 65 123 35)(6 66 124 36)(7 67 125 37)(8 68 126 38)(9 108 164 138)(10 109 165 139)(11 110 166 140)(12 111 167 141)(13 112 168 142)(14 105 161 143)(15 106 162 144)(16 107 163 137)(17 149 55 119)(18 150 56 120)(19 151 49 113)(20 152 50 114)(21 145 51 115)(22 146 52 116)(23 147 53 117)(24 148 54 118)(25 157 63 95)(26 158 64 96)(27 159 57 89)(28 160 58 90)(29 153 59 91)(30 154 60 92)(31 155 61 93)(32 156 62 94)(41 99 79 129)(42 100 80 130)(43 101 73 131)(44 102 74 132)(45 103 75 133)(46 104 76 134)(47 97 77 135)(48 98 78 136)(81 186 241 298)(82 187 242 299)(83 188 243 300)(84 189 244 301)(85 190 245 302)(86 191 246 303)(87 192 247 304)(88 185 248 297)(169 271 281 231)(170 272 282 232)(171 265 283 225)(172 266 284 226)(173 267 285 227)(174 268 286 228)(175 269 287 229)(176 270 288 230)(177 279 289 239)(178 280 290 240)(179 273 291 233)(180 274 292 234)(181 275 293 235)(182 276 294 236)(183 277 295 237)(184 278 296 238)(193 211 305 251)(194 212 306 252)(195 213 307 253)(196 214 308 254)(197 215 309 255)(198 216 310 256)(199 209 311 249)(200 210 312 250)(201 219 313 259)(202 220 314 260)(203 221 315 261)(204 222 316 262)(205 223 317 263)(206 224 318 264)(207 217 319 257)(208 218 320 258)
(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)
(1 173 5 169)(2 172 6 176)(3 171 7 175)(4 170 8 174)(9 241 13 245)(10 248 14 244)(11 247 15 243)(12 246 16 242)(17 249 21 253)(18 256 22 252)(19 255 23 251)(20 254 24 250)(25 257 29 261)(26 264 30 260)(27 263 31 259)(28 262 32 258)(33 265 37 269)(34 272 38 268)(35 271 39 267)(36 270 40 266)(41 273 45 277)(42 280 46 276)(43 279 47 275)(44 278 48 274)(49 215 53 211)(50 214 54 210)(51 213 55 209)(52 212 56 216)(57 223 61 219)(58 222 62 218)(59 221 63 217)(60 220 64 224)(65 231 69 227)(66 230 70 226)(67 229 71 225)(68 228 72 232)(73 239 77 235)(74 238 78 234)(75 237 79 233)(76 236 80 240)(81 168 85 164)(82 167 86 163)(83 166 87 162)(84 165 88 161)(89 205 93 201)(90 204 94 208)(91 203 95 207)(92 202 96 206)(97 181 101 177)(98 180 102 184)(99 179 103 183)(100 178 104 182)(105 189 109 185)(106 188 110 192)(107 187 111 191)(108 186 112 190)(113 197 117 193)(114 196 118 200)(115 195 119 199)(116 194 120 198)(121 283 125 287)(122 282 126 286)(123 281 127 285)(124 288 128 284)(129 291 133 295)(130 290 134 294)(131 289 135 293)(132 296 136 292)(137 299 141 303)(138 298 142 302)(139 297 143 301)(140 304 144 300)(145 307 149 311)(146 306 150 310)(147 305 151 309)(148 312 152 308)(153 315 157 319)(154 314 158 318)(155 313 159 317)(156 320 160 316)
G:=sub<Sym(320)| (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,90,42,114,34)(10,91,43,115,35)(11,92,44,116,36)(12,93,45,117,37)(13,94,46,118,38)(14,95,47,119,39)(15,96,48,120,40)(16,89,41,113,33)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,165,153,73,145)(66,166,154,74,146)(67,167,155,75,147)(68,168,156,76,148)(69,161,157,77,149)(70,162,158,78,150)(71,163,159,79,151)(72,164,160,80,152)(81,316,240,308,232)(82,317,233,309,225)(83,318,234,310,226)(84,319,235,311,227)(85,320,236,312,228)(86,313,237,305,229)(87,314,238,306,230)(88,315,239,307,231)(169,185,261,177,253)(170,186,262,178,254)(171,187,263,179,255)(172,188,264,180,256)(173,189,257,181,249)(174,190,258,182,250)(175,191,259,183,251)(176,192,260,184,252)(193,269,246,201,277)(194,270,247,202,278)(195,271,248,203,279)(196,272,241,204,280)(197,265,242,205,273)(198,266,243,206,274)(199,267,244,207,275)(200,268,245,208,276)(209,285,301,217,293)(210,286,302,218,294)(211,287,303,219,295)(212,288,304,220,296)(213,281,297,221,289)(214,282,298,222,290)(215,283,299,223,291)(216,284,300,224,292), (1,69,127,39)(2,70,128,40)(3,71,121,33)(4,72,122,34)(5,65,123,35)(6,66,124,36)(7,67,125,37)(8,68,126,38)(9,108,164,138)(10,109,165,139)(11,110,166,140)(12,111,167,141)(13,112,168,142)(14,105,161,143)(15,106,162,144)(16,107,163,137)(17,149,55,119)(18,150,56,120)(19,151,49,113)(20,152,50,114)(21,145,51,115)(22,146,52,116)(23,147,53,117)(24,148,54,118)(25,157,63,95)(26,158,64,96)(27,159,57,89)(28,160,58,90)(29,153,59,91)(30,154,60,92)(31,155,61,93)(32,156,62,94)(41,99,79,129)(42,100,80,130)(43,101,73,131)(44,102,74,132)(45,103,75,133)(46,104,76,134)(47,97,77,135)(48,98,78,136)(81,186,241,298)(82,187,242,299)(83,188,243,300)(84,189,244,301)(85,190,245,302)(86,191,246,303)(87,192,247,304)(88,185,248,297)(169,271,281,231)(170,272,282,232)(171,265,283,225)(172,266,284,226)(173,267,285,227)(174,268,286,228)(175,269,287,229)(176,270,288,230)(177,279,289,239)(178,280,290,240)(179,273,291,233)(180,274,292,234)(181,275,293,235)(182,276,294,236)(183,277,295,237)(184,278,296,238)(193,211,305,251)(194,212,306,252)(195,213,307,253)(196,214,308,254)(197,215,309,255)(198,216,310,256)(199,209,311,249)(200,210,312,250)(201,219,313,259)(202,220,314,260)(203,221,315,261)(204,222,316,262)(205,223,317,263)(206,224,318,264)(207,217,319,257)(208,218,320,258), (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), (1,173,5,169)(2,172,6,176)(3,171,7,175)(4,170,8,174)(9,241,13,245)(10,248,14,244)(11,247,15,243)(12,246,16,242)(17,249,21,253)(18,256,22,252)(19,255,23,251)(20,254,24,250)(25,257,29,261)(26,264,30,260)(27,263,31,259)(28,262,32,258)(33,265,37,269)(34,272,38,268)(35,271,39,267)(36,270,40,266)(41,273,45,277)(42,280,46,276)(43,279,47,275)(44,278,48,274)(49,215,53,211)(50,214,54,210)(51,213,55,209)(52,212,56,216)(57,223,61,219)(58,222,62,218)(59,221,63,217)(60,220,64,224)(65,231,69,227)(66,230,70,226)(67,229,71,225)(68,228,72,232)(73,239,77,235)(74,238,78,234)(75,237,79,233)(76,236,80,240)(81,168,85,164)(82,167,86,163)(83,166,87,162)(84,165,88,161)(89,205,93,201)(90,204,94,208)(91,203,95,207)(92,202,96,206)(97,181,101,177)(98,180,102,184)(99,179,103,183)(100,178,104,182)(105,189,109,185)(106,188,110,192)(107,187,111,191)(108,186,112,190)(113,197,117,193)(114,196,118,200)(115,195,119,199)(116,194,120,198)(121,283,125,287)(122,282,126,286)(123,281,127,285)(124,288,128,284)(129,291,133,295)(130,290,134,294)(131,289,135,293)(132,296,136,292)(137,299,141,303)(138,298,142,302)(139,297,143,301)(140,304,144,300)(145,307,149,311)(146,306,150,310)(147,305,151,309)(148,312,152,308)(153,315,157,319)(154,314,158,318)(155,313,159,317)(156,320,160,316)>;
G:=Group( (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,90,42,114,34)(10,91,43,115,35)(11,92,44,116,36)(12,93,45,117,37)(13,94,46,118,38)(14,95,47,119,39)(15,96,48,120,40)(16,89,41,113,33)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,165,153,73,145)(66,166,154,74,146)(67,167,155,75,147)(68,168,156,76,148)(69,161,157,77,149)(70,162,158,78,150)(71,163,159,79,151)(72,164,160,80,152)(81,316,240,308,232)(82,317,233,309,225)(83,318,234,310,226)(84,319,235,311,227)(85,320,236,312,228)(86,313,237,305,229)(87,314,238,306,230)(88,315,239,307,231)(169,185,261,177,253)(170,186,262,178,254)(171,187,263,179,255)(172,188,264,180,256)(173,189,257,181,249)(174,190,258,182,250)(175,191,259,183,251)(176,192,260,184,252)(193,269,246,201,277)(194,270,247,202,278)(195,271,248,203,279)(196,272,241,204,280)(197,265,242,205,273)(198,266,243,206,274)(199,267,244,207,275)(200,268,245,208,276)(209,285,301,217,293)(210,286,302,218,294)(211,287,303,219,295)(212,288,304,220,296)(213,281,297,221,289)(214,282,298,222,290)(215,283,299,223,291)(216,284,300,224,292), (1,69,127,39)(2,70,128,40)(3,71,121,33)(4,72,122,34)(5,65,123,35)(6,66,124,36)(7,67,125,37)(8,68,126,38)(9,108,164,138)(10,109,165,139)(11,110,166,140)(12,111,167,141)(13,112,168,142)(14,105,161,143)(15,106,162,144)(16,107,163,137)(17,149,55,119)(18,150,56,120)(19,151,49,113)(20,152,50,114)(21,145,51,115)(22,146,52,116)(23,147,53,117)(24,148,54,118)(25,157,63,95)(26,158,64,96)(27,159,57,89)(28,160,58,90)(29,153,59,91)(30,154,60,92)(31,155,61,93)(32,156,62,94)(41,99,79,129)(42,100,80,130)(43,101,73,131)(44,102,74,132)(45,103,75,133)(46,104,76,134)(47,97,77,135)(48,98,78,136)(81,186,241,298)(82,187,242,299)(83,188,243,300)(84,189,244,301)(85,190,245,302)(86,191,246,303)(87,192,247,304)(88,185,248,297)(169,271,281,231)(170,272,282,232)(171,265,283,225)(172,266,284,226)(173,267,285,227)(174,268,286,228)(175,269,287,229)(176,270,288,230)(177,279,289,239)(178,280,290,240)(179,273,291,233)(180,274,292,234)(181,275,293,235)(182,276,294,236)(183,277,295,237)(184,278,296,238)(193,211,305,251)(194,212,306,252)(195,213,307,253)(196,214,308,254)(197,215,309,255)(198,216,310,256)(199,209,311,249)(200,210,312,250)(201,219,313,259)(202,220,314,260)(203,221,315,261)(204,222,316,262)(205,223,317,263)(206,224,318,264)(207,217,319,257)(208,218,320,258), (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), (1,173,5,169)(2,172,6,176)(3,171,7,175)(4,170,8,174)(9,241,13,245)(10,248,14,244)(11,247,15,243)(12,246,16,242)(17,249,21,253)(18,256,22,252)(19,255,23,251)(20,254,24,250)(25,257,29,261)(26,264,30,260)(27,263,31,259)(28,262,32,258)(33,265,37,269)(34,272,38,268)(35,271,39,267)(36,270,40,266)(41,273,45,277)(42,280,46,276)(43,279,47,275)(44,278,48,274)(49,215,53,211)(50,214,54,210)(51,213,55,209)(52,212,56,216)(57,223,61,219)(58,222,62,218)(59,221,63,217)(60,220,64,224)(65,231,69,227)(66,230,70,226)(67,229,71,225)(68,228,72,232)(73,239,77,235)(74,238,78,234)(75,237,79,233)(76,236,80,240)(81,168,85,164)(82,167,86,163)(83,166,87,162)(84,165,88,161)(89,205,93,201)(90,204,94,208)(91,203,95,207)(92,202,96,206)(97,181,101,177)(98,180,102,184)(99,179,103,183)(100,178,104,182)(105,189,109,185)(106,188,110,192)(107,187,111,191)(108,186,112,190)(113,197,117,193)(114,196,118,200)(115,195,119,199)(116,194,120,198)(121,283,125,287)(122,282,126,286)(123,281,127,285)(124,288,128,284)(129,291,133,295)(130,290,134,294)(131,289,135,293)(132,296,136,292)(137,299,141,303)(138,298,142,302)(139,297,143,301)(140,304,144,300)(145,307,149,311)(146,306,150,310)(147,305,151,309)(148,312,152,308)(153,315,157,319)(154,314,158,318)(155,313,159,317)(156,320,160,316) );
G=PermutationGroup([[(1,105,25,97,17),(2,106,26,98,18),(3,107,27,99,19),(4,108,28,100,20),(5,109,29,101,21),(6,110,30,102,22),(7,111,31,103,23),(8,112,32,104,24),(9,90,42,114,34),(10,91,43,115,35),(11,92,44,116,36),(12,93,45,117,37),(13,94,46,118,38),(14,95,47,119,39),(15,96,48,120,40),(16,89,41,113,33),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,165,153,73,145),(66,166,154,74,146),(67,167,155,75,147),(68,168,156,76,148),(69,161,157,77,149),(70,162,158,78,150),(71,163,159,79,151),(72,164,160,80,152),(81,316,240,308,232),(82,317,233,309,225),(83,318,234,310,226),(84,319,235,311,227),(85,320,236,312,228),(86,313,237,305,229),(87,314,238,306,230),(88,315,239,307,231),(169,185,261,177,253),(170,186,262,178,254),(171,187,263,179,255),(172,188,264,180,256),(173,189,257,181,249),(174,190,258,182,250),(175,191,259,183,251),(176,192,260,184,252),(193,269,246,201,277),(194,270,247,202,278),(195,271,248,203,279),(196,272,241,204,280),(197,265,242,205,273),(198,266,243,206,274),(199,267,244,207,275),(200,268,245,208,276),(209,285,301,217,293),(210,286,302,218,294),(211,287,303,219,295),(212,288,304,220,296),(213,281,297,221,289),(214,282,298,222,290),(215,283,299,223,291),(216,284,300,224,292)], [(1,69,127,39),(2,70,128,40),(3,71,121,33),(4,72,122,34),(5,65,123,35),(6,66,124,36),(7,67,125,37),(8,68,126,38),(9,108,164,138),(10,109,165,139),(11,110,166,140),(12,111,167,141),(13,112,168,142),(14,105,161,143),(15,106,162,144),(16,107,163,137),(17,149,55,119),(18,150,56,120),(19,151,49,113),(20,152,50,114),(21,145,51,115),(22,146,52,116),(23,147,53,117),(24,148,54,118),(25,157,63,95),(26,158,64,96),(27,159,57,89),(28,160,58,90),(29,153,59,91),(30,154,60,92),(31,155,61,93),(32,156,62,94),(41,99,79,129),(42,100,80,130),(43,101,73,131),(44,102,74,132),(45,103,75,133),(46,104,76,134),(47,97,77,135),(48,98,78,136),(81,186,241,298),(82,187,242,299),(83,188,243,300),(84,189,244,301),(85,190,245,302),(86,191,246,303),(87,192,247,304),(88,185,248,297),(169,271,281,231),(170,272,282,232),(171,265,283,225),(172,266,284,226),(173,267,285,227),(174,268,286,228),(175,269,287,229),(176,270,288,230),(177,279,289,239),(178,280,290,240),(179,273,291,233),(180,274,292,234),(181,275,293,235),(182,276,294,236),(183,277,295,237),(184,278,296,238),(193,211,305,251),(194,212,306,252),(195,213,307,253),(196,214,308,254),(197,215,309,255),(198,216,310,256),(199,209,311,249),(200,210,312,250),(201,219,313,259),(202,220,314,260),(203,221,315,261),(204,222,316,262),(205,223,317,263),(206,224,318,264),(207,217,319,257),(208,218,320,258)], [(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)], [(1,173,5,169),(2,172,6,176),(3,171,7,175),(4,170,8,174),(9,241,13,245),(10,248,14,244),(11,247,15,243),(12,246,16,242),(17,249,21,253),(18,256,22,252),(19,255,23,251),(20,254,24,250),(25,257,29,261),(26,264,30,260),(27,263,31,259),(28,262,32,258),(33,265,37,269),(34,272,38,268),(35,271,39,267),(36,270,40,266),(41,273,45,277),(42,280,46,276),(43,279,47,275),(44,278,48,274),(49,215,53,211),(50,214,54,210),(51,213,55,209),(52,212,56,216),(57,223,61,219),(58,222,62,218),(59,221,63,217),(60,220,64,224),(65,231,69,227),(66,230,70,226),(67,229,71,225),(68,228,72,232),(73,239,77,235),(74,238,78,234),(75,237,79,233),(76,236,80,240),(81,168,85,164),(82,167,86,163),(83,166,87,162),(84,165,88,161),(89,205,93,201),(90,204,94,208),(91,203,95,207),(92,202,96,206),(97,181,101,177),(98,180,102,184),(99,179,103,183),(100,178,104,182),(105,189,109,185),(106,188,110,192),(107,187,111,191),(108,186,112,190),(113,197,117,193),(114,196,118,200),(115,195,119,199),(116,194,120,198),(121,283,125,287),(122,282,126,286),(123,281,127,285),(124,288,128,284),(129,291,133,295),(130,290,134,294),(131,289,135,293),(132,296,136,292),(137,299,141,303),(138,298,142,302),(139,297,143,301),(140,304,144,300),(145,307,149,311),(146,306,150,310),(147,305,151,309),(148,312,152,308),(153,315,157,319),(154,314,158,318),(155,313,159,317),(156,320,160,316)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | Q16 | C5×D4 | C5×D4 | C5×Q16 |
| kernel | C5×C4⋊Q16 | C4×C40 | C5×C4⋊Q8 | C10×Q16 | C4⋊Q16 | C4×C8 | C4⋊Q8 | C2×Q16 | C40 | C2×C20 | C20 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 4 | 2 | 8 | 16 | 8 | 32 |
Matrix representation of C5×C4⋊Q16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 24 | 24 |
| 0 | 0 | 29 | 0 |
| 39 | 28 | 0 | 0 |
| 16 | 2 | 0 | 0 |
| 0 | 0 | 23 | 40 |
| 0 | 0 | 38 | 18 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,40,0,0,2,40,0,0,0,0,40,1,0,0,39,1],[1,40,0,0,2,40,0,0,0,0,24,29,0,0,24,0],[39,16,0,0,28,2,0,0,0,0,23,38,0,0,40,18] >;
C5×C4⋊Q16 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes Q_{16} % in TeX
G:=Group("C5xC4:Q16"); // GroupNames label
G:=SmallGroup(320,995);
// by ID
G=gap.SmallGroup(320,995);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,436,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations