direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4⋊2Q16, C20⋊9Q16, C4⋊2(C5×Q16), C4⋊C8.6C10, C4⋊Q8.4C10, Q8.1(C5×D4), C4.34(D4×C10), (C5×Q8).26D4, (C4×Q8).5C10, C2.5(C10×Q16), C20.395(C2×D4), (C2×C20).324D4, (C10×Q16).9C2, (C2×Q16).2C10, (Q8×C20).18C2, C10.52(C2×Q16), C42.17(C2×C10), Q8⋊C4.2C10, C22.86(D4×C10), C20.344(C4○D4), (C4×C20).259C22, (C2×C20).921C23, (C2×C40).257C22, C10.145(C4⋊D4), (Q8×C10).160C22, C10.135(C8.C22), (C5×C4⋊C8).19C2, (C2×C8).4(C2×C10), C4.43(C5×C4○D4), (C5×C4⋊Q8).19C2, C4⋊C4.54(C2×C10), (C2×C4).130(C5×D4), C2.14(C5×C4⋊D4), (C2×C10).642(C2×D4), (C2×Q8).48(C2×C10), C2.10(C5×C8.C22), (C5×C4⋊C4).375C22, (C2×C4).96(C22×C10), (C5×Q8⋊C4).11C2, SmallGroup(320,963)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊2Q16
G = < a,b,c,d | a5=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 170 in 108 conjugacy classes, 58 normal (34 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, C20, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C4⋊2Q16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×Q16, Q8×C10, Q8×C10, C5×Q8⋊C4, C5×C4⋊C8, Q8×C20, C5×C4⋊Q8, C10×Q16, C5×C4⋊2Q16
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×Q16, C8.C22, C5×D4, C22×C10, C4⋊2Q16, C5×Q16, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×Q16, C5×C8.C22, C5×C4⋊2Q16
►Regular action on 320 points
Generators in S320
(1 31 164 171 50)(2 32 165 172 51)(3 25 166 173 52)(4 26 167 174 53)(5 27 168 175 54)(6 28 161 176 55)(7 29 162 169 56)(8 30 163 170 49)(9 124 237 37 229)(10 125 238 38 230)(11 126 239 39 231)(12 127 240 40 232)(13 128 233 33 225)(14 121 234 34 226)(15 122 235 35 227)(16 123 236 36 228)(17 209 200 63 217)(18 210 193 64 218)(19 211 194 57 219)(20 212 195 58 220)(21 213 196 59 221)(22 214 197 60 222)(23 215 198 61 223)(24 216 199 62 224)(41 77 190 69 182)(42 78 191 70 183)(43 79 192 71 184)(44 80 185 72 177)(45 73 186 65 178)(46 74 187 66 179)(47 75 188 67 180)(48 76 189 68 181)(81 273 160 265 152)(82 274 153 266 145)(83 275 154 267 146)(84 276 155 268 147)(85 277 156 269 148)(86 278 157 270 149)(87 279 158 271 150)(88 280 159 272 151)(89 281 297 97 289)(90 282 298 98 290)(91 283 299 99 291)(92 284 300 100 292)(93 285 301 101 293)(94 286 302 102 294)(95 287 303 103 295)(96 288 304 104 296)(105 204 313 113 305)(106 205 314 114 306)(107 206 315 115 307)(108 207 316 116 308)(109 208 317 117 309)(110 201 318 118 310)(111 202 319 119 311)(112 203 320 120 312)(129 242 258 137 250)(130 243 259 138 251)(131 244 260 139 252)(132 245 261 140 253)(133 246 262 141 254)(134 247 263 142 255)(135 248 264 143 256)(136 241 257 144 249)
(1 111 146 215)(2 216 147 112)(3 105 148 209)(4 210 149 106)(5 107 150 211)(6 212 151 108)(7 109 152 213)(8 214 145 110)(9 44 297 261)(10 262 298 45)(11 46 299 263)(12 264 300 47)(13 48 301 257)(14 258 302 41)(15 42 303 259)(16 260 304 43)(17 52 305 269)(18 270 306 53)(19 54 307 271)(20 272 308 55)(21 56 309 265)(22 266 310 49)(23 50 311 267)(24 268 312 51)(25 204 85 200)(26 193 86 205)(27 206 87 194)(28 195 88 207)(29 208 81 196)(30 197 82 201)(31 202 83 198)(32 199 84 203)(33 68 93 136)(34 129 94 69)(35 70 95 130)(36 131 96 71)(37 72 89 132)(38 133 90 65)(39 66 91 134)(40 135 92 67)(57 168 315 279)(58 280 316 161)(59 162 317 273)(60 274 318 163)(61 164 319 275)(62 276 320 165)(63 166 313 277)(64 278 314 167)(73 125 141 98)(74 99 142 126)(75 127 143 100)(76 101 144 128)(77 121 137 102)(78 103 138 122)(79 123 139 104)(80 97 140 124)(113 156 217 173)(114 174 218 157)(115 158 219 175)(116 176 220 159)(117 160 221 169)(118 170 222 153)(119 154 223 171)(120 172 224 155)(177 281 245 229)(178 230 246 282)(179 283 247 231)(180 232 248 284)(181 285 241 225)(182 226 242 286)(183 287 243 227)(184 228 244 288)(185 289 253 237)(186 238 254 290)(187 291 255 239)(188 240 256 292)(189 293 249 233)(190 234 250 294)(191 295 251 235)(192 236 252 296)
(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)
(1 179 5 183)(2 178 6 182)(3 177 7 181)(4 184 8 180)(9 196 13 200)(10 195 14 199)(11 194 15 198)(12 193 16 197)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(25 44 29 48)(26 43 30 47)(27 42 31 46)(28 41 32 45)(49 67 53 71)(50 66 54 70)(51 65 55 69)(52 72 56 68)(57 122 61 126)(58 121 62 125)(59 128 63 124)(60 127 64 123)(73 161 77 165)(74 168 78 164)(75 167 79 163)(76 166 80 162)(81 257 85 261)(82 264 86 260)(83 263 87 259)(84 262 88 258)(89 309 93 305)(90 308 94 312)(91 307 95 311)(92 306 96 310)(97 317 101 313)(98 316 102 320)(99 315 103 319)(100 314 104 318)(105 281 109 285)(106 288 110 284)(107 287 111 283)(108 286 112 282)(113 289 117 293)(114 296 118 292)(115 295 119 291)(116 294 120 290)(129 268 133 272)(130 267 134 271)(131 266 135 270)(132 265 136 269)(137 276 141 280)(138 275 142 279)(139 274 143 278)(140 273 144 277)(145 248 149 244)(146 247 150 243)(147 246 151 242)(148 245 152 241)(153 256 157 252)(154 255 158 251)(155 254 159 250)(156 253 160 249)(169 189 173 185)(170 188 174 192)(171 187 175 191)(172 186 176 190)(201 300 205 304)(202 299 206 303)(203 298 207 302)(204 297 208 301)(209 229 213 225)(210 228 214 232)(211 227 215 231)(212 226 216 230)(217 237 221 233)(218 236 222 240)(219 235 223 239)(220 234 224 238)
G:=sub<Sym(320)| (1,31,164,171,50)(2,32,165,172,51)(3,25,166,173,52)(4,26,167,174,53)(5,27,168,175,54)(6,28,161,176,55)(7,29,162,169,56)(8,30,163,170,49)(9,124,237,37,229)(10,125,238,38,230)(11,126,239,39,231)(12,127,240,40,232)(13,128,233,33,225)(14,121,234,34,226)(15,122,235,35,227)(16,123,236,36,228)(17,209,200,63,217)(18,210,193,64,218)(19,211,194,57,219)(20,212,195,58,220)(21,213,196,59,221)(22,214,197,60,222)(23,215,198,61,223)(24,216,199,62,224)(41,77,190,69,182)(42,78,191,70,183)(43,79,192,71,184)(44,80,185,72,177)(45,73,186,65,178)(46,74,187,66,179)(47,75,188,67,180)(48,76,189,68,181)(81,273,160,265,152)(82,274,153,266,145)(83,275,154,267,146)(84,276,155,268,147)(85,277,156,269,148)(86,278,157,270,149)(87,279,158,271,150)(88,280,159,272,151)(89,281,297,97,289)(90,282,298,98,290)(91,283,299,99,291)(92,284,300,100,292)(93,285,301,101,293)(94,286,302,102,294)(95,287,303,103,295)(96,288,304,104,296)(105,204,313,113,305)(106,205,314,114,306)(107,206,315,115,307)(108,207,316,116,308)(109,208,317,117,309)(110,201,318,118,310)(111,202,319,119,311)(112,203,320,120,312)(129,242,258,137,250)(130,243,259,138,251)(131,244,260,139,252)(132,245,261,140,253)(133,246,262,141,254)(134,247,263,142,255)(135,248,264,143,256)(136,241,257,144,249), (1,111,146,215)(2,216,147,112)(3,105,148,209)(4,210,149,106)(5,107,150,211)(6,212,151,108)(7,109,152,213)(8,214,145,110)(9,44,297,261)(10,262,298,45)(11,46,299,263)(12,264,300,47)(13,48,301,257)(14,258,302,41)(15,42,303,259)(16,260,304,43)(17,52,305,269)(18,270,306,53)(19,54,307,271)(20,272,308,55)(21,56,309,265)(22,266,310,49)(23,50,311,267)(24,268,312,51)(25,204,85,200)(26,193,86,205)(27,206,87,194)(28,195,88,207)(29,208,81,196)(30,197,82,201)(31,202,83,198)(32,199,84,203)(33,68,93,136)(34,129,94,69)(35,70,95,130)(36,131,96,71)(37,72,89,132)(38,133,90,65)(39,66,91,134)(40,135,92,67)(57,168,315,279)(58,280,316,161)(59,162,317,273)(60,274,318,163)(61,164,319,275)(62,276,320,165)(63,166,313,277)(64,278,314,167)(73,125,141,98)(74,99,142,126)(75,127,143,100)(76,101,144,128)(77,121,137,102)(78,103,138,122)(79,123,139,104)(80,97,140,124)(113,156,217,173)(114,174,218,157)(115,158,219,175)(116,176,220,159)(117,160,221,169)(118,170,222,153)(119,154,223,171)(120,172,224,155)(177,281,245,229)(178,230,246,282)(179,283,247,231)(180,232,248,284)(181,285,241,225)(182,226,242,286)(183,287,243,227)(184,228,244,288)(185,289,253,237)(186,238,254,290)(187,291,255,239)(188,240,256,292)(189,293,249,233)(190,234,250,294)(191,295,251,235)(192,236,252,296), (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), (1,179,5,183)(2,178,6,182)(3,177,7,181)(4,184,8,180)(9,196,13,200)(10,195,14,199)(11,194,15,198)(12,193,16,197)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(25,44,29,48)(26,43,30,47)(27,42,31,46)(28,41,32,45)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)(57,122,61,126)(58,121,62,125)(59,128,63,124)(60,127,64,123)(73,161,77,165)(74,168,78,164)(75,167,79,163)(76,166,80,162)(81,257,85,261)(82,264,86,260)(83,263,87,259)(84,262,88,258)(89,309,93,305)(90,308,94,312)(91,307,95,311)(92,306,96,310)(97,317,101,313)(98,316,102,320)(99,315,103,319)(100,314,104,318)(105,281,109,285)(106,288,110,284)(107,287,111,283)(108,286,112,282)(113,289,117,293)(114,296,118,292)(115,295,119,291)(116,294,120,290)(129,268,133,272)(130,267,134,271)(131,266,135,270)(132,265,136,269)(137,276,141,280)(138,275,142,279)(139,274,143,278)(140,273,144,277)(145,248,149,244)(146,247,150,243)(147,246,151,242)(148,245,152,241)(153,256,157,252)(154,255,158,251)(155,254,159,250)(156,253,160,249)(169,189,173,185)(170,188,174,192)(171,187,175,191)(172,186,176,190)(201,300,205,304)(202,299,206,303)(203,298,207,302)(204,297,208,301)(209,229,213,225)(210,228,214,232)(211,227,215,231)(212,226,216,230)(217,237,221,233)(218,236,222,240)(219,235,223,239)(220,234,224,238)>;
G:=Group( (1,31,164,171,50)(2,32,165,172,51)(3,25,166,173,52)(4,26,167,174,53)(5,27,168,175,54)(6,28,161,176,55)(7,29,162,169,56)(8,30,163,170,49)(9,124,237,37,229)(10,125,238,38,230)(11,126,239,39,231)(12,127,240,40,232)(13,128,233,33,225)(14,121,234,34,226)(15,122,235,35,227)(16,123,236,36,228)(17,209,200,63,217)(18,210,193,64,218)(19,211,194,57,219)(20,212,195,58,220)(21,213,196,59,221)(22,214,197,60,222)(23,215,198,61,223)(24,216,199,62,224)(41,77,190,69,182)(42,78,191,70,183)(43,79,192,71,184)(44,80,185,72,177)(45,73,186,65,178)(46,74,187,66,179)(47,75,188,67,180)(48,76,189,68,181)(81,273,160,265,152)(82,274,153,266,145)(83,275,154,267,146)(84,276,155,268,147)(85,277,156,269,148)(86,278,157,270,149)(87,279,158,271,150)(88,280,159,272,151)(89,281,297,97,289)(90,282,298,98,290)(91,283,299,99,291)(92,284,300,100,292)(93,285,301,101,293)(94,286,302,102,294)(95,287,303,103,295)(96,288,304,104,296)(105,204,313,113,305)(106,205,314,114,306)(107,206,315,115,307)(108,207,316,116,308)(109,208,317,117,309)(110,201,318,118,310)(111,202,319,119,311)(112,203,320,120,312)(129,242,258,137,250)(130,243,259,138,251)(131,244,260,139,252)(132,245,261,140,253)(133,246,262,141,254)(134,247,263,142,255)(135,248,264,143,256)(136,241,257,144,249), (1,111,146,215)(2,216,147,112)(3,105,148,209)(4,210,149,106)(5,107,150,211)(6,212,151,108)(7,109,152,213)(8,214,145,110)(9,44,297,261)(10,262,298,45)(11,46,299,263)(12,264,300,47)(13,48,301,257)(14,258,302,41)(15,42,303,259)(16,260,304,43)(17,52,305,269)(18,270,306,53)(19,54,307,271)(20,272,308,55)(21,56,309,265)(22,266,310,49)(23,50,311,267)(24,268,312,51)(25,204,85,200)(26,193,86,205)(27,206,87,194)(28,195,88,207)(29,208,81,196)(30,197,82,201)(31,202,83,198)(32,199,84,203)(33,68,93,136)(34,129,94,69)(35,70,95,130)(36,131,96,71)(37,72,89,132)(38,133,90,65)(39,66,91,134)(40,135,92,67)(57,168,315,279)(58,280,316,161)(59,162,317,273)(60,274,318,163)(61,164,319,275)(62,276,320,165)(63,166,313,277)(64,278,314,167)(73,125,141,98)(74,99,142,126)(75,127,143,100)(76,101,144,128)(77,121,137,102)(78,103,138,122)(79,123,139,104)(80,97,140,124)(113,156,217,173)(114,174,218,157)(115,158,219,175)(116,176,220,159)(117,160,221,169)(118,170,222,153)(119,154,223,171)(120,172,224,155)(177,281,245,229)(178,230,246,282)(179,283,247,231)(180,232,248,284)(181,285,241,225)(182,226,242,286)(183,287,243,227)(184,228,244,288)(185,289,253,237)(186,238,254,290)(187,291,255,239)(188,240,256,292)(189,293,249,233)(190,234,250,294)(191,295,251,235)(192,236,252,296), (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), (1,179,5,183)(2,178,6,182)(3,177,7,181)(4,184,8,180)(9,196,13,200)(10,195,14,199)(11,194,15,198)(12,193,16,197)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(25,44,29,48)(26,43,30,47)(27,42,31,46)(28,41,32,45)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)(57,122,61,126)(58,121,62,125)(59,128,63,124)(60,127,64,123)(73,161,77,165)(74,168,78,164)(75,167,79,163)(76,166,80,162)(81,257,85,261)(82,264,86,260)(83,263,87,259)(84,262,88,258)(89,309,93,305)(90,308,94,312)(91,307,95,311)(92,306,96,310)(97,317,101,313)(98,316,102,320)(99,315,103,319)(100,314,104,318)(105,281,109,285)(106,288,110,284)(107,287,111,283)(108,286,112,282)(113,289,117,293)(114,296,118,292)(115,295,119,291)(116,294,120,290)(129,268,133,272)(130,267,134,271)(131,266,135,270)(132,265,136,269)(137,276,141,280)(138,275,142,279)(139,274,143,278)(140,273,144,277)(145,248,149,244)(146,247,150,243)(147,246,151,242)(148,245,152,241)(153,256,157,252)(154,255,158,251)(155,254,159,250)(156,253,160,249)(169,189,173,185)(170,188,174,192)(171,187,175,191)(172,186,176,190)(201,300,205,304)(202,299,206,303)(203,298,207,302)(204,297,208,301)(209,229,213,225)(210,228,214,232)(211,227,215,231)(212,226,216,230)(217,237,221,233)(218,236,222,240)(219,235,223,239)(220,234,224,238) );
G=PermutationGroup([[(1,31,164,171,50),(2,32,165,172,51),(3,25,166,173,52),(4,26,167,174,53),(5,27,168,175,54),(6,28,161,176,55),(7,29,162,169,56),(8,30,163,170,49),(9,124,237,37,229),(10,125,238,38,230),(11,126,239,39,231),(12,127,240,40,232),(13,128,233,33,225),(14,121,234,34,226),(15,122,235,35,227),(16,123,236,36,228),(17,209,200,63,217),(18,210,193,64,218),(19,211,194,57,219),(20,212,195,58,220),(21,213,196,59,221),(22,214,197,60,222),(23,215,198,61,223),(24,216,199,62,224),(41,77,190,69,182),(42,78,191,70,183),(43,79,192,71,184),(44,80,185,72,177),(45,73,186,65,178),(46,74,187,66,179),(47,75,188,67,180),(48,76,189,68,181),(81,273,160,265,152),(82,274,153,266,145),(83,275,154,267,146),(84,276,155,268,147),(85,277,156,269,148),(86,278,157,270,149),(87,279,158,271,150),(88,280,159,272,151),(89,281,297,97,289),(90,282,298,98,290),(91,283,299,99,291),(92,284,300,100,292),(93,285,301,101,293),(94,286,302,102,294),(95,287,303,103,295),(96,288,304,104,296),(105,204,313,113,305),(106,205,314,114,306),(107,206,315,115,307),(108,207,316,116,308),(109,208,317,117,309),(110,201,318,118,310),(111,202,319,119,311),(112,203,320,120,312),(129,242,258,137,250),(130,243,259,138,251),(131,244,260,139,252),(132,245,261,140,253),(133,246,262,141,254),(134,247,263,142,255),(135,248,264,143,256),(136,241,257,144,249)], [(1,111,146,215),(2,216,147,112),(3,105,148,209),(4,210,149,106),(5,107,150,211),(6,212,151,108),(7,109,152,213),(8,214,145,110),(9,44,297,261),(10,262,298,45),(11,46,299,263),(12,264,300,47),(13,48,301,257),(14,258,302,41),(15,42,303,259),(16,260,304,43),(17,52,305,269),(18,270,306,53),(19,54,307,271),(20,272,308,55),(21,56,309,265),(22,266,310,49),(23,50,311,267),(24,268,312,51),(25,204,85,200),(26,193,86,205),(27,206,87,194),(28,195,88,207),(29,208,81,196),(30,197,82,201),(31,202,83,198),(32,199,84,203),(33,68,93,136),(34,129,94,69),(35,70,95,130),(36,131,96,71),(37,72,89,132),(38,133,90,65),(39,66,91,134),(40,135,92,67),(57,168,315,279),(58,280,316,161),(59,162,317,273),(60,274,318,163),(61,164,319,275),(62,276,320,165),(63,166,313,277),(64,278,314,167),(73,125,141,98),(74,99,142,126),(75,127,143,100),(76,101,144,128),(77,121,137,102),(78,103,138,122),(79,123,139,104),(80,97,140,124),(113,156,217,173),(114,174,218,157),(115,158,219,175),(116,176,220,159),(117,160,221,169),(118,170,222,153),(119,154,223,171),(120,172,224,155),(177,281,245,229),(178,230,246,282),(179,283,247,231),(180,232,248,284),(181,285,241,225),(182,226,242,286),(183,287,243,227),(184,228,244,288),(185,289,253,237),(186,238,254,290),(187,291,255,239),(188,240,256,292),(189,293,249,233),(190,234,250,294),(191,295,251,235),(192,236,252,296)], [(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)], [(1,179,5,183),(2,178,6,182),(3,177,7,181),(4,184,8,180),(9,196,13,200),(10,195,14,199),(11,194,15,198),(12,193,16,197),(17,37,21,33),(18,36,22,40),(19,35,23,39),(20,34,24,38),(25,44,29,48),(26,43,30,47),(27,42,31,46),(28,41,32,45),(49,67,53,71),(50,66,54,70),(51,65,55,69),(52,72,56,68),(57,122,61,126),(58,121,62,125),(59,128,63,124),(60,127,64,123),(73,161,77,165),(74,168,78,164),(75,167,79,163),(76,166,80,162),(81,257,85,261),(82,264,86,260),(83,263,87,259),(84,262,88,258),(89,309,93,305),(90,308,94,312),(91,307,95,311),(92,306,96,310),(97,317,101,313),(98,316,102,320),(99,315,103,319),(100,314,104,318),(105,281,109,285),(106,288,110,284),(107,287,111,283),(108,286,112,282),(113,289,117,293),(114,296,118,292),(115,295,119,291),(116,294,120,290),(129,268,133,272),(130,267,134,271),(131,266,135,270),(132,265,136,269),(137,276,141,280),(138,275,142,279),(139,274,143,278),(140,273,144,277),(145,248,149,244),(146,247,150,243),(147,246,151,242),(148,245,152,241),(153,256,157,252),(154,255,158,251),(155,254,159,250),(156,253,160,249),(169,189,173,185),(170,188,174,192),(171,187,175,191),(172,186,176,190),(201,300,205,304),(202,299,206,303),(203,298,207,302),(204,297,208,301),(209,229,213,225),(210,228,214,232),(211,227,215,231),(212,226,216,230),(217,237,221,233),(218,236,222,240),(219,235,223,239),(220,234,224,238)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20AJ | 20AK | ··· | 20AR | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | Q16 | C4○D4 | C5×D4 | C5×D4 | C5×Q16 | C5×C4○D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×C4⋊2Q16 | C5×Q8⋊C4 | C5×C4⋊C8 | Q8×C20 | C5×C4⋊Q8 | C10×Q16 | C4⋊2Q16 | Q8⋊C4 | C4⋊C8 | C4×Q8 | C4⋊Q8 | C2×Q16 | C2×C20 | C5×Q8 | C20 | C20 | C2×C4 | Q8 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 1 | 4 |
Matrix representation of C5×C4⋊2Q16 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 9 | 2 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 13 | 13 | 0 | 0 |
| 6 | 28 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 11 |
| 0 | 0 | 11 | 1 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[9,0,0,0,2,32,0,0,0,0,40,0,0,0,0,40],[13,6,0,0,13,28,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,40,11,0,0,11,1] >;
C5×C4⋊2Q16 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes_2Q_{16} % in TeX
G:=Group("C5xC4:2Q16"); // GroupNames label
G:=SmallGroup(320,963);
// by ID
G=gap.SmallGroup(320,963);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766,856,10085,2539,124]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations