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## G = C5×Q16⋊C4order 320 = 26·5

### Direct product of C5 and Q16⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×Q16⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×Q8⋊C4 — C5×Q16⋊C4
 Lower central C1 — C2 — C4 — C5×Q16⋊C4
 Upper central C1 — C2×C10 — C4×C20 — C5×Q16⋊C4

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 [×2], C4 [×2], C4 [×8], C22, C5, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], Q8 [×2], C10, C10 [×2], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], Q16 [×4], C2×Q8 [×2], C20 [×2], C20 [×8], C2×C10, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8 [×2], C2×Q16, C40 [×2], C40, C2×C20, C2×C20 [×2], C2×C20 [×4], C5×Q8 [×4], C5×Q8 [×2], Q16⋊C4, C4×C20, C4×C20 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C5×Q16 [×4], Q8×C10 [×2], C5×C8⋊C4, C5×Q8⋊C4 [×2], C5×C4.Q8, Q8×C20 [×2], C10×Q16, C5×Q16⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C8.C22 [×2], C2×C20 [×6], C5×D4 [×2], C22×C10, Q16⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8.C22 [×2], C5×Q16⋊C4

Smallest permutation representation of C5×Q16⋊C4
Regular action on 320 points
Generators in S320
(1 132 128 168 173)(2 133 121 161 174)(3 134 122 162 175)(4 135 123 163 176)(5 136 124 164 169)(6 129 125 165 170)(7 130 126 166 171)(8 131 127 167 172)(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)(81 214 93 206 197)(82 215 94 207 198)(83 216 95 208 199)(84 209 96 201 200)(85 210 89 202 193)(86 211 90 203 194)(87 212 91 204 195)(88 213 92 205 196)(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)(25 207 29 203)(26 206 30 202)(27 205 31 201)(28 204 32 208)(33 215 37 211)(34 214 38 210)(35 213 39 209)(36 212 40 216)(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)(65 168 69 164)(66 167 70 163)(67 166 71 162)(68 165 72 161)(73 132 77 136)(74 131 78 135)(75 130 79 134)(76 129 80 133)(81 318 85 314)(82 317 86 313)(83 316 87 320)(84 315 88 319)(89 306 93 310)(90 305 94 309)(91 312 95 308)(92 311 96 307)(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)(121 270 125 266)(122 269 126 265)(123 268 127 272)(124 267 128 271)(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)(169 259 173 263)(170 258 174 262)(171 257 175 261)(172 264 176 260)(193 287 197 283)(194 286 198 282)(195 285 199 281)(196 284 200 288)
(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)(9 171 184 305)(10 176 177 310)(11 173 178 307)(12 170 179 312)(13 175 180 309)(14 172 181 306)(15 169 182 311)(16 174 183 308)(17 126 192 313)(18 123 185 318)(19 128 186 315)(20 125 187 320)(21 122 188 317)(22 127 189 314)(23 124 190 319)(24 121 191 316)(33 289 166 149)(34 294 167 146)(35 291 168 151)(36 296 161 148)(37 293 162 145)(38 290 163 150)(39 295 164 147)(40 292 165 152)(41 96 222 259)(42 93 223 264)(43 90 224 261)(44 95 217 258)(45 92 218 263)(46 89 219 260)(47 94 220 257)(48 91 221 262)(49 84 230 267)(50 81 231 272)(51 86 232 269)(52 83 225 266)(53 88 226 271)(54 85 227 268)(55 82 228 265)(56 87 229 270)(57 235 201 101)(58 240 202 98)(59 237 203 103)(60 234 204 100)(61 239 205 97)(62 236 206 102)(63 233 207 99)(64 238 208 104)(65 243 209 109)(66 248 210 106)(67 245 211 111)(68 242 212 108)(69 247 213 105)(70 244 214 110)(71 241 215 107)(72 246 216 112)(73 251 200 117)(74 256 193 114)(75 253 194 119)(76 250 195 116)(77 255 196 113)(78 252 197 118)(79 249 198 115)(80 254 199 120)(129 160 285 300)(130 157 286 297)(131 154 287 302)(132 159 288 299)(133 156 281 304)(134 153 282 301)(135 158 283 298)(136 155 284 303)

G:=sub<Sym(320)| (1,132,128,168,173)(2,133,121,161,174)(3,134,122,162,175)(4,135,123,163,176)(5,136,124,164,169)(6,129,125,165,170)(7,130,126,166,171)(8,131,127,167,172)(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)(81,214,93,206,197)(82,215,94,207,198)(83,216,95,208,199)(84,209,96,201,200)(85,210,89,202,193)(86,211,90,203,194)(87,212,91,204,195)(88,213,92,205,196)(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)(25,207,29,203)(26,206,30,202)(27,205,31,201)(28,204,32,208)(33,215,37,211)(34,214,38,210)(35,213,39,209)(36,212,40,216)(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)(65,168,69,164)(66,167,70,163)(67,166,71,162)(68,165,72,161)(73,132,77,136)(74,131,78,135)(75,130,79,134)(76,129,80,133)(81,318,85,314)(82,317,86,313)(83,316,87,320)(84,315,88,319)(89,306,93,310)(90,305,94,309)(91,312,95,308)(92,311,96,307)(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)(121,270,125,266)(122,269,126,265)(123,268,127,272)(124,267,128,271)(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)(169,259,173,263)(170,258,174,262)(171,257,175,261)(172,264,176,260)(193,287,197,283)(194,286,198,282)(195,285,199,281)(196,284,200,288), (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)(9,171,184,305)(10,176,177,310)(11,173,178,307)(12,170,179,312)(13,175,180,309)(14,172,181,306)(15,169,182,311)(16,174,183,308)(17,126,192,313)(18,123,185,318)(19,128,186,315)(20,125,187,320)(21,122,188,317)(22,127,189,314)(23,124,190,319)(24,121,191,316)(33,289,166,149)(34,294,167,146)(35,291,168,151)(36,296,161,148)(37,293,162,145)(38,290,163,150)(39,295,164,147)(40,292,165,152)(41,96,222,259)(42,93,223,264)(43,90,224,261)(44,95,217,258)(45,92,218,263)(46,89,219,260)(47,94,220,257)(48,91,221,262)(49,84,230,267)(50,81,231,272)(51,86,232,269)(52,83,225,266)(53,88,226,271)(54,85,227,268)(55,82,228,265)(56,87,229,270)(57,235,201,101)(58,240,202,98)(59,237,203,103)(60,234,204,100)(61,239,205,97)(62,236,206,102)(63,233,207,99)(64,238,208,104)(65,243,209,109)(66,248,210,106)(67,245,211,111)(68,242,212,108)(69,247,213,105)(70,244,214,110)(71,241,215,107)(72,246,216,112)(73,251,200,117)(74,256,193,114)(75,253,194,119)(76,250,195,116)(77,255,196,113)(78,252,197,118)(79,249,198,115)(80,254,199,120)(129,160,285,300)(130,157,286,297)(131,154,287,302)(132,159,288,299)(133,156,281,304)(134,153,282,301)(135,158,283,298)(136,155,284,303)>;

G:=Group( (1,132,128,168,173)(2,133,121,161,174)(3,134,122,162,175)(4,135,123,163,176)(5,136,124,164,169)(6,129,125,165,170)(7,130,126,166,171)(8,131,127,167,172)(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)(81,214,93,206,197)(82,215,94,207,198)(83,216,95,208,199)(84,209,96,201,200)(85,210,89,202,193)(86,211,90,203,194)(87,212,91,204,195)(88,213,92,205,196)(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)(25,207,29,203)(26,206,30,202)(27,205,31,201)(28,204,32,208)(33,215,37,211)(34,214,38,210)(35,213,39,209)(36,212,40,216)(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)(65,168,69,164)(66,167,70,163)(67,166,71,162)(68,165,72,161)(73,132,77,136)(74,131,78,135)(75,130,79,134)(76,129,80,133)(81,318,85,314)(82,317,86,313)(83,316,87,320)(84,315,88,319)(89,306,93,310)(90,305,94,309)(91,312,95,308)(92,311,96,307)(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)(121,270,125,266)(122,269,126,265)(123,268,127,272)(124,267,128,271)(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)(169,259,173,263)(170,258,174,262)(171,257,175,261)(172,264,176,260)(193,287,197,283)(194,286,198,282)(195,285,199,281)(196,284,200,288), (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)(9,171,184,305)(10,176,177,310)(11,173,178,307)(12,170,179,312)(13,175,180,309)(14,172,181,306)(15,169,182,311)(16,174,183,308)(17,126,192,313)(18,123,185,318)(19,128,186,315)(20,125,187,320)(21,122,188,317)(22,127,189,314)(23,124,190,319)(24,121,191,316)(33,289,166,149)(34,294,167,146)(35,291,168,151)(36,296,161,148)(37,293,162,145)(38,290,163,150)(39,295,164,147)(40,292,165,152)(41,96,222,259)(42,93,223,264)(43,90,224,261)(44,95,217,258)(45,92,218,263)(46,89,219,260)(47,94,220,257)(48,91,221,262)(49,84,230,267)(50,81,231,272)(51,86,232,269)(52,83,225,266)(53,88,226,271)(54,85,227,268)(55,82,228,265)(56,87,229,270)(57,235,201,101)(58,240,202,98)(59,237,203,103)(60,234,204,100)(61,239,205,97)(62,236,206,102)(63,233,207,99)(64,238,208,104)(65,243,209,109)(66,248,210,106)(67,245,211,111)(68,242,212,108)(69,247,213,105)(70,244,214,110)(71,241,215,107)(72,246,216,112)(73,251,200,117)(74,256,193,114)(75,253,194,119)(76,250,195,116)(77,255,196,113)(78,252,197,118)(79,249,198,115)(80,254,199,120)(129,160,285,300)(130,157,286,297)(131,154,287,302)(132,159,288,299)(133,156,281,304)(134,153,282,301)(135,158,283,298)(136,155,284,303) );

G=PermutationGroup([(1,132,128,168,173),(2,133,121,161,174),(3,134,122,162,175),(4,135,123,163,176),(5,136,124,164,169),(6,129,125,165,170),(7,130,126,166,171),(8,131,127,167,172),(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),(81,214,93,206,197),(82,215,94,207,198),(83,216,95,208,199),(84,209,96,201,200),(85,210,89,202,193),(86,211,90,203,194),(87,212,91,204,195),(88,213,92,205,196),(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),(25,207,29,203),(26,206,30,202),(27,205,31,201),(28,204,32,208),(33,215,37,211),(34,214,38,210),(35,213,39,209),(36,212,40,216),(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),(65,168,69,164),(66,167,70,163),(67,166,71,162),(68,165,72,161),(73,132,77,136),(74,131,78,135),(75,130,79,134),(76,129,80,133),(81,318,85,314),(82,317,86,313),(83,316,87,320),(84,315,88,319),(89,306,93,310),(90,305,94,309),(91,312,95,308),(92,311,96,307),(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),(121,270,125,266),(122,269,126,265),(123,268,127,272),(124,267,128,271),(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),(169,259,173,263),(170,258,174,262),(171,257,175,261),(172,264,176,260),(193,287,197,283),(194,286,198,282),(195,285,199,281),(196,284,200,288)], [(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),(9,171,184,305),(10,176,177,310),(11,173,178,307),(12,170,179,312),(13,175,180,309),(14,172,181,306),(15,169,182,311),(16,174,183,308),(17,126,192,313),(18,123,185,318),(19,128,186,315),(20,125,187,320),(21,122,188,317),(22,127,189,314),(23,124,190,319),(24,121,191,316),(33,289,166,149),(34,294,167,146),(35,291,168,151),(36,296,161,148),(37,293,162,145),(38,290,163,150),(39,295,164,147),(40,292,165,152),(41,96,222,259),(42,93,223,264),(43,90,224,261),(44,95,217,258),(45,92,218,263),(46,89,219,260),(47,94,220,257),(48,91,221,262),(49,84,230,267),(50,81,231,272),(51,86,232,269),(52,83,225,266),(53,88,226,271),(54,85,227,268),(55,82,228,265),(56,87,229,270),(57,235,201,101),(58,240,202,98),(59,237,203,103),(60,234,204,100),(61,239,205,97),(62,236,206,102),(63,233,207,99),(64,238,208,104),(65,243,209,109),(66,248,210,106),(67,245,211,111),(68,242,212,108),(69,247,213,105),(70,244,214,110),(71,241,215,107),(72,246,216,112),(73,251,200,117),(74,256,193,114),(75,253,194,119),(76,250,195,116),(77,255,196,113),(78,252,197,118),(79,249,198,115),(80,254,199,120),(129,160,285,300),(130,157,286,297),(131,154,287,302),(132,159,288,299),(133,156,281,304),(134,153,282,301),(135,158,283,298),(136,155,284,303)])

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

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