metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊Q16⋊5C4, Q8.1(C4×D5), C10.65(C4×D4), C40⋊8C4.7C2, C4⋊C4.143D10, C5⋊3(Q16⋊C4), (C2×C8).172D10, Q8⋊C4.7D5, (C2×Q8).97D10, (Q8×Dic5).1C2, C22.74(D4×D5), C20.45(C22×C4), C20.Q8.1C2, C20.155(C4○D4), C4.52(D4⋊2D5), (C2×C40).190C22, (C2×C20).229C23, C2.1(Q16⋊D5), Dic10.17(C2×C4), (C2×Dic5).202D4, Dic5⋊3Q8.1C2, C20.44D4.6C2, C2.3(SD16⋊D5), C4⋊Dic5.79C22, (Q8×C10).12C22, C10.52(C8.C22), (C4×Dic5).21C22, C2.19(Dic5⋊4D4), (C2×Dic10).66C22, C4.10(C2×C4×D5), C5⋊2C8.1(C2×C4), (C5×Q8).16(C2×C4), (C2×C5⋊Q16).1C2, (C2×C10).242(C2×D4), (C5×C4⋊C4).30C22, (C5×Q8⋊C4).7C2, (C2×C5⋊2C8).26C22, (C2×C4).336(C22×D5), SmallGroup(320,416)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊Q16⋊5C4
G = < a,b,c,d | a5=b8=d4=1, c2=b4, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=b6c >
Subgroups: 342 in 108 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C5⋊2C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q16⋊C4, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C20.Q8, C40⋊8C4, C20.44D4, C5×Q8⋊C4, Dic5⋊3Q8, C2×C5⋊Q16, Q8×Dic5, C5⋊Q16⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8.C22, C4×D5, C22×D5, Q16⋊C4, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, SD16⋊D5, Q16⋊D5, C5⋊Q16⋊5C4
►Regular action on 320 points
Generators in S320
(1 159 283 29 185)(2 186 30 284 160)(3 153 285 31 187)(4 188 32 286 154)(5 155 287 25 189)(6 190 26 288 156)(7 157 281 27 191)(8 192 28 282 158)(9 320 178 152 36)(10 37 145 179 313)(11 314 180 146 38)(12 39 147 181 315)(13 316 182 148 40)(14 33 149 183 317)(15 318 184 150 34)(16 35 151 177 319)(17 312 164 139 303)(18 304 140 165 305)(19 306 166 141 297)(20 298 142 167 307)(21 308 168 143 299)(22 300 144 161 309)(23 310 162 137 301)(24 302 138 163 311)(41 67 107 215 271)(42 272 216 108 68)(43 69 109 209 265)(44 266 210 110 70)(45 71 111 211 267)(46 268 212 112 72)(47 65 105 213 269)(48 270 214 106 66)(49 99 114 225 264)(50 257 226 115 100)(51 101 116 227 258)(52 259 228 117 102)(53 103 118 229 260)(54 261 230 119 104)(55 97 120 231 262)(56 263 232 113 98)(57 254 198 204 78)(58 79 205 199 255)(59 256 200 206 80)(60 73 207 193 249)(61 250 194 208 74)(62 75 201 195 251)(63 252 196 202 76)(64 77 203 197 253)(81 222 90 237 247)(82 248 238 91 223)(83 224 92 239 241)(84 242 240 93 217)(85 218 94 233 243)(86 244 234 95 219)(87 220 96 235 245)(88 246 236 89 221)(121 295 278 130 173)(122 174 131 279 296)(123 289 280 132 175)(124 176 133 273 290)(125 291 274 134 169)(126 170 135 275 292)(127 293 276 136 171)(128 172 129 277 294)
(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)
(1 172 5 176)(2 171 6 175)(3 170 7 174)(4 169 8 173)(9 17 13 21)(10 24 14 20)(11 23 15 19)(12 22 16 18)(25 290 29 294)(26 289 30 293)(27 296 31 292)(28 295 32 291)(33 298 37 302)(34 297 38 301)(35 304 39 300)(36 303 40 299)(41 63 45 59)(42 62 46 58)(43 61 47 57)(44 60 48 64)(49 234 53 238)(50 233 54 237)(51 240 55 236)(52 239 56 235)(65 254 69 250)(66 253 70 249)(67 252 71 256)(68 251 72 255)(73 270 77 266)(74 269 78 265)(75 268 79 272)(76 267 80 271)(81 226 85 230)(82 225 86 229)(83 232 87 228)(84 231 88 227)(89 101 93 97)(90 100 94 104)(91 99 95 103)(92 98 96 102)(105 198 109 194)(106 197 110 193)(107 196 111 200)(108 195 112 199)(113 220 117 224)(114 219 118 223)(115 218 119 222)(116 217 120 221)(121 188 125 192)(122 187 126 191)(123 186 127 190)(124 185 128 189)(129 155 133 159)(130 154 134 158)(131 153 135 157)(132 160 136 156)(137 150 141 146)(138 149 142 145)(139 148 143 152)(140 147 144 151)(161 177 165 181)(162 184 166 180)(163 183 167 179)(164 182 168 178)(201 212 205 216)(202 211 206 215)(203 210 207 214)(204 209 208 213)(241 263 245 259)(242 262 246 258)(243 261 247 257)(244 260 248 264)(273 283 277 287)(274 282 278 286)(275 281 279 285)(276 288 280 284)(305 315 309 319)(306 314 310 318)(307 313 311 317)(308 320 312 316)
(1 119 22 63)(2 114 23 58)(3 117 24 61)(4 120 17 64)(5 115 18 59)(6 118 19 62)(7 113 20 57)(8 116 21 60)(9 42 173 219)(10 45 174 222)(11 48 175 217)(12 43 176 220)(13 46 169 223)(14 41 170 218)(15 44 171 221)(16 47 172 224)(25 257 165 206)(26 260 166 201)(27 263 167 204)(28 258 168 207)(29 261 161 202)(30 264 162 205)(31 259 163 208)(32 262 164 203)(33 67 135 94)(34 70 136 89)(35 65 129 92)(36 68 130 95)(37 71 131 90)(38 66 132 93)(39 69 133 96)(40 72 134 91)(49 137 199 284)(50 140 200 287)(51 143 193 282)(52 138 194 285)(53 141 195 288)(54 144 196 283)(55 139 197 286)(56 142 198 281)(73 192 227 308)(74 187 228 311)(75 190 229 306)(76 185 230 309)(77 188 231 312)(78 191 232 307)(79 186 225 310)(80 189 226 305)(81 313 267 122)(82 316 268 125)(83 319 269 128)(84 314 270 123)(85 317 271 126)(86 320 272 121)(87 315 265 124)(88 318 266 127)(97 303 253 154)(98 298 254 157)(99 301 255 160)(100 304 256 155)(101 299 249 158)(102 302 250 153)(103 297 251 156)(104 300 252 159)(105 277 239 151)(106 280 240 146)(107 275 233 149)(108 278 234 152)(109 273 235 147)(110 276 236 150)(111 279 237 145)(112 274 238 148)(177 213 294 241)(178 216 295 244)(179 211 296 247)(180 214 289 242)(181 209 290 245)(182 212 291 248)(183 215 292 243)(184 210 293 246)
G:=sub<Sym(320)| (1,159,283,29,185)(2,186,30,284,160)(3,153,285,31,187)(4,188,32,286,154)(5,155,287,25,189)(6,190,26,288,156)(7,157,281,27,191)(8,192,28,282,158)(9,320,178,152,36)(10,37,145,179,313)(11,314,180,146,38)(12,39,147,181,315)(13,316,182,148,40)(14,33,149,183,317)(15,318,184,150,34)(16,35,151,177,319)(17,312,164,139,303)(18,304,140,165,305)(19,306,166,141,297)(20,298,142,167,307)(21,308,168,143,299)(22,300,144,161,309)(23,310,162,137,301)(24,302,138,163,311)(41,67,107,215,271)(42,272,216,108,68)(43,69,109,209,265)(44,266,210,110,70)(45,71,111,211,267)(46,268,212,112,72)(47,65,105,213,269)(48,270,214,106,66)(49,99,114,225,264)(50,257,226,115,100)(51,101,116,227,258)(52,259,228,117,102)(53,103,118,229,260)(54,261,230,119,104)(55,97,120,231,262)(56,263,232,113,98)(57,254,198,204,78)(58,79,205,199,255)(59,256,200,206,80)(60,73,207,193,249)(61,250,194,208,74)(62,75,201,195,251)(63,252,196,202,76)(64,77,203,197,253)(81,222,90,237,247)(82,248,238,91,223)(83,224,92,239,241)(84,242,240,93,217)(85,218,94,233,243)(86,244,234,95,219)(87,220,96,235,245)(88,246,236,89,221)(121,295,278,130,173)(122,174,131,279,296)(123,289,280,132,175)(124,176,133,273,290)(125,291,274,134,169)(126,170,135,275,292)(127,293,276,136,171)(128,172,129,277,294), (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), (1,172,5,176)(2,171,6,175)(3,170,7,174)(4,169,8,173)(9,17,13,21)(10,24,14,20)(11,23,15,19)(12,22,16,18)(25,290,29,294)(26,289,30,293)(27,296,31,292)(28,295,32,291)(33,298,37,302)(34,297,38,301)(35,304,39,300)(36,303,40,299)(41,63,45,59)(42,62,46,58)(43,61,47,57)(44,60,48,64)(49,234,53,238)(50,233,54,237)(51,240,55,236)(52,239,56,235)(65,254,69,250)(66,253,70,249)(67,252,71,256)(68,251,72,255)(73,270,77,266)(74,269,78,265)(75,268,79,272)(76,267,80,271)(81,226,85,230)(82,225,86,229)(83,232,87,228)(84,231,88,227)(89,101,93,97)(90,100,94,104)(91,99,95,103)(92,98,96,102)(105,198,109,194)(106,197,110,193)(107,196,111,200)(108,195,112,199)(113,220,117,224)(114,219,118,223)(115,218,119,222)(116,217,120,221)(121,188,125,192)(122,187,126,191)(123,186,127,190)(124,185,128,189)(129,155,133,159)(130,154,134,158)(131,153,135,157)(132,160,136,156)(137,150,141,146)(138,149,142,145)(139,148,143,152)(140,147,144,151)(161,177,165,181)(162,184,166,180)(163,183,167,179)(164,182,168,178)(201,212,205,216)(202,211,206,215)(203,210,207,214)(204,209,208,213)(241,263,245,259)(242,262,246,258)(243,261,247,257)(244,260,248,264)(273,283,277,287)(274,282,278,286)(275,281,279,285)(276,288,280,284)(305,315,309,319)(306,314,310,318)(307,313,311,317)(308,320,312,316), (1,119,22,63)(2,114,23,58)(3,117,24,61)(4,120,17,64)(5,115,18,59)(6,118,19,62)(7,113,20,57)(8,116,21,60)(9,42,173,219)(10,45,174,222)(11,48,175,217)(12,43,176,220)(13,46,169,223)(14,41,170,218)(15,44,171,221)(16,47,172,224)(25,257,165,206)(26,260,166,201)(27,263,167,204)(28,258,168,207)(29,261,161,202)(30,264,162,205)(31,259,163,208)(32,262,164,203)(33,67,135,94)(34,70,136,89)(35,65,129,92)(36,68,130,95)(37,71,131,90)(38,66,132,93)(39,69,133,96)(40,72,134,91)(49,137,199,284)(50,140,200,287)(51,143,193,282)(52,138,194,285)(53,141,195,288)(54,144,196,283)(55,139,197,286)(56,142,198,281)(73,192,227,308)(74,187,228,311)(75,190,229,306)(76,185,230,309)(77,188,231,312)(78,191,232,307)(79,186,225,310)(80,189,226,305)(81,313,267,122)(82,316,268,125)(83,319,269,128)(84,314,270,123)(85,317,271,126)(86,320,272,121)(87,315,265,124)(88,318,266,127)(97,303,253,154)(98,298,254,157)(99,301,255,160)(100,304,256,155)(101,299,249,158)(102,302,250,153)(103,297,251,156)(104,300,252,159)(105,277,239,151)(106,280,240,146)(107,275,233,149)(108,278,234,152)(109,273,235,147)(110,276,236,150)(111,279,237,145)(112,274,238,148)(177,213,294,241)(178,216,295,244)(179,211,296,247)(180,214,289,242)(181,209,290,245)(182,212,291,248)(183,215,292,243)(184,210,293,246)>;
G:=Group( (1,159,283,29,185)(2,186,30,284,160)(3,153,285,31,187)(4,188,32,286,154)(5,155,287,25,189)(6,190,26,288,156)(7,157,281,27,191)(8,192,28,282,158)(9,320,178,152,36)(10,37,145,179,313)(11,314,180,146,38)(12,39,147,181,315)(13,316,182,148,40)(14,33,149,183,317)(15,318,184,150,34)(16,35,151,177,319)(17,312,164,139,303)(18,304,140,165,305)(19,306,166,141,297)(20,298,142,167,307)(21,308,168,143,299)(22,300,144,161,309)(23,310,162,137,301)(24,302,138,163,311)(41,67,107,215,271)(42,272,216,108,68)(43,69,109,209,265)(44,266,210,110,70)(45,71,111,211,267)(46,268,212,112,72)(47,65,105,213,269)(48,270,214,106,66)(49,99,114,225,264)(50,257,226,115,100)(51,101,116,227,258)(52,259,228,117,102)(53,103,118,229,260)(54,261,230,119,104)(55,97,120,231,262)(56,263,232,113,98)(57,254,198,204,78)(58,79,205,199,255)(59,256,200,206,80)(60,73,207,193,249)(61,250,194,208,74)(62,75,201,195,251)(63,252,196,202,76)(64,77,203,197,253)(81,222,90,237,247)(82,248,238,91,223)(83,224,92,239,241)(84,242,240,93,217)(85,218,94,233,243)(86,244,234,95,219)(87,220,96,235,245)(88,246,236,89,221)(121,295,278,130,173)(122,174,131,279,296)(123,289,280,132,175)(124,176,133,273,290)(125,291,274,134,169)(126,170,135,275,292)(127,293,276,136,171)(128,172,129,277,294), (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), (1,172,5,176)(2,171,6,175)(3,170,7,174)(4,169,8,173)(9,17,13,21)(10,24,14,20)(11,23,15,19)(12,22,16,18)(25,290,29,294)(26,289,30,293)(27,296,31,292)(28,295,32,291)(33,298,37,302)(34,297,38,301)(35,304,39,300)(36,303,40,299)(41,63,45,59)(42,62,46,58)(43,61,47,57)(44,60,48,64)(49,234,53,238)(50,233,54,237)(51,240,55,236)(52,239,56,235)(65,254,69,250)(66,253,70,249)(67,252,71,256)(68,251,72,255)(73,270,77,266)(74,269,78,265)(75,268,79,272)(76,267,80,271)(81,226,85,230)(82,225,86,229)(83,232,87,228)(84,231,88,227)(89,101,93,97)(90,100,94,104)(91,99,95,103)(92,98,96,102)(105,198,109,194)(106,197,110,193)(107,196,111,200)(108,195,112,199)(113,220,117,224)(114,219,118,223)(115,218,119,222)(116,217,120,221)(121,188,125,192)(122,187,126,191)(123,186,127,190)(124,185,128,189)(129,155,133,159)(130,154,134,158)(131,153,135,157)(132,160,136,156)(137,150,141,146)(138,149,142,145)(139,148,143,152)(140,147,144,151)(161,177,165,181)(162,184,166,180)(163,183,167,179)(164,182,168,178)(201,212,205,216)(202,211,206,215)(203,210,207,214)(204,209,208,213)(241,263,245,259)(242,262,246,258)(243,261,247,257)(244,260,248,264)(273,283,277,287)(274,282,278,286)(275,281,279,285)(276,288,280,284)(305,315,309,319)(306,314,310,318)(307,313,311,317)(308,320,312,316), (1,119,22,63)(2,114,23,58)(3,117,24,61)(4,120,17,64)(5,115,18,59)(6,118,19,62)(7,113,20,57)(8,116,21,60)(9,42,173,219)(10,45,174,222)(11,48,175,217)(12,43,176,220)(13,46,169,223)(14,41,170,218)(15,44,171,221)(16,47,172,224)(25,257,165,206)(26,260,166,201)(27,263,167,204)(28,258,168,207)(29,261,161,202)(30,264,162,205)(31,259,163,208)(32,262,164,203)(33,67,135,94)(34,70,136,89)(35,65,129,92)(36,68,130,95)(37,71,131,90)(38,66,132,93)(39,69,133,96)(40,72,134,91)(49,137,199,284)(50,140,200,287)(51,143,193,282)(52,138,194,285)(53,141,195,288)(54,144,196,283)(55,139,197,286)(56,142,198,281)(73,192,227,308)(74,187,228,311)(75,190,229,306)(76,185,230,309)(77,188,231,312)(78,191,232,307)(79,186,225,310)(80,189,226,305)(81,313,267,122)(82,316,268,125)(83,319,269,128)(84,314,270,123)(85,317,271,126)(86,320,272,121)(87,315,265,124)(88,318,266,127)(97,303,253,154)(98,298,254,157)(99,301,255,160)(100,304,256,155)(101,299,249,158)(102,302,250,153)(103,297,251,156)(104,300,252,159)(105,277,239,151)(106,280,240,146)(107,275,233,149)(108,278,234,152)(109,273,235,147)(110,276,236,150)(111,279,237,145)(112,274,238,148)(177,213,294,241)(178,216,295,244)(179,211,296,247)(180,214,289,242)(181,209,290,245)(182,212,291,248)(183,215,292,243)(184,210,293,246) );
G=PermutationGroup([[(1,159,283,29,185),(2,186,30,284,160),(3,153,285,31,187),(4,188,32,286,154),(5,155,287,25,189),(6,190,26,288,156),(7,157,281,27,191),(8,192,28,282,158),(9,320,178,152,36),(10,37,145,179,313),(11,314,180,146,38),(12,39,147,181,315),(13,316,182,148,40),(14,33,149,183,317),(15,318,184,150,34),(16,35,151,177,319),(17,312,164,139,303),(18,304,140,165,305),(19,306,166,141,297),(20,298,142,167,307),(21,308,168,143,299),(22,300,144,161,309),(23,310,162,137,301),(24,302,138,163,311),(41,67,107,215,271),(42,272,216,108,68),(43,69,109,209,265),(44,266,210,110,70),(45,71,111,211,267),(46,268,212,112,72),(47,65,105,213,269),(48,270,214,106,66),(49,99,114,225,264),(50,257,226,115,100),(51,101,116,227,258),(52,259,228,117,102),(53,103,118,229,260),(54,261,230,119,104),(55,97,120,231,262),(56,263,232,113,98),(57,254,198,204,78),(58,79,205,199,255),(59,256,200,206,80),(60,73,207,193,249),(61,250,194,208,74),(62,75,201,195,251),(63,252,196,202,76),(64,77,203,197,253),(81,222,90,237,247),(82,248,238,91,223),(83,224,92,239,241),(84,242,240,93,217),(85,218,94,233,243),(86,244,234,95,219),(87,220,96,235,245),(88,246,236,89,221),(121,295,278,130,173),(122,174,131,279,296),(123,289,280,132,175),(124,176,133,273,290),(125,291,274,134,169),(126,170,135,275,292),(127,293,276,136,171),(128,172,129,277,294)], [(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)], [(1,172,5,176),(2,171,6,175),(3,170,7,174),(4,169,8,173),(9,17,13,21),(10,24,14,20),(11,23,15,19),(12,22,16,18),(25,290,29,294),(26,289,30,293),(27,296,31,292),(28,295,32,291),(33,298,37,302),(34,297,38,301),(35,304,39,300),(36,303,40,299),(41,63,45,59),(42,62,46,58),(43,61,47,57),(44,60,48,64),(49,234,53,238),(50,233,54,237),(51,240,55,236),(52,239,56,235),(65,254,69,250),(66,253,70,249),(67,252,71,256),(68,251,72,255),(73,270,77,266),(74,269,78,265),(75,268,79,272),(76,267,80,271),(81,226,85,230),(82,225,86,229),(83,232,87,228),(84,231,88,227),(89,101,93,97),(90,100,94,104),(91,99,95,103),(92,98,96,102),(105,198,109,194),(106,197,110,193),(107,196,111,200),(108,195,112,199),(113,220,117,224),(114,219,118,223),(115,218,119,222),(116,217,120,221),(121,188,125,192),(122,187,126,191),(123,186,127,190),(124,185,128,189),(129,155,133,159),(130,154,134,158),(131,153,135,157),(132,160,136,156),(137,150,141,146),(138,149,142,145),(139,148,143,152),(140,147,144,151),(161,177,165,181),(162,184,166,180),(163,183,167,179),(164,182,168,178),(201,212,205,216),(202,211,206,215),(203,210,207,214),(204,209,208,213),(241,263,245,259),(242,262,246,258),(243,261,247,257),(244,260,248,264),(273,283,277,287),(274,282,278,286),(275,281,279,285),(276,288,280,284),(305,315,309,319),(306,314,310,318),(307,313,311,317),(308,320,312,316)], [(1,119,22,63),(2,114,23,58),(3,117,24,61),(4,120,17,64),(5,115,18,59),(6,118,19,62),(7,113,20,57),(8,116,21,60),(9,42,173,219),(10,45,174,222),(11,48,175,217),(12,43,176,220),(13,46,169,223),(14,41,170,218),(15,44,171,221),(16,47,172,224),(25,257,165,206),(26,260,166,201),(27,263,167,204),(28,258,168,207),(29,261,161,202),(30,264,162,205),(31,259,163,208),(32,262,164,203),(33,67,135,94),(34,70,136,89),(35,65,129,92),(36,68,130,95),(37,71,131,90),(38,66,132,93),(39,69,133,96),(40,72,134,91),(49,137,199,284),(50,140,200,287),(51,143,193,282),(52,138,194,285),(53,141,195,288),(54,144,196,283),(55,139,197,286),(56,142,198,281),(73,192,227,308),(74,187,228,311),(75,190,229,306),(76,185,230,309),(77,188,231,312),(78,191,232,307),(79,186,225,310),(80,189,226,305),(81,313,267,122),(82,316,268,125),(83,319,269,128),(84,314,270,123),(85,317,271,126),(86,320,272,121),(87,315,265,124),(88,318,266,127),(97,303,253,154),(98,298,254,157),(99,301,255,160),(100,304,256,155),(101,299,249,158),(102,302,250,153),(103,297,251,156),(104,300,252,159),(105,277,239,151),(106,280,240,146),(107,275,233,149),(108,278,234,152),(109,273,235,147),(110,276,236,150),(111,279,237,145),(112,274,238,148),(177,213,294,241),(178,216,295,244),(179,211,296,247),(180,214,289,242),(181,209,290,245),(182,212,291,248),(183,215,292,243),(184,210,293,246)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4×D5 | C8.C22 | D4⋊2D5 | D4×D5 | SD16⋊D5 | Q16⋊D5 |
| kernel | C5⋊Q16⋊5C4 | C20.Q8 | C40⋊8C4 | C20.44D4 | C5×Q8⋊C4 | Dic5⋊3Q8 | C2×C5⋊Q16 | Q8×Dic5 | C5⋊Q16 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C5⋊Q16⋊5C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 38 | 7 | 0 | 0 | 0 | 0 |
| 22 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 0 | 38 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 38 | 0 | 0 |
| 40 | 37 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 27 | 7 | 0 | 0 | 0 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 14 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[38,22,0,0,0,0,7,3,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,38,0,0,14,0,0,0,0,0,0,38,0,0],[40,0,0,0,0,0,37,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[27,7,0,0,0,0,7,14,0,0,0,0,0,0,0,14,0,0,0,0,3,0,0,0,0,0,0,0,0,14,0,0,0,0,3,0] >;
C5⋊Q16⋊5C4 in GAP, Magma, Sage, TeX
C_5\rtimes Q_{16}\rtimes_5C_4 % in TeX
G:=Group("C5:Q16:5C4"); // GroupNames label
G:=SmallGroup(320,416);
// by ID
G=gap.SmallGroup(320,416);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,758,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=d^4=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^6*c>;
// generators/relations