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G = C5⋊Q165C4order 320 = 26·5

1st semidirect product of C5⋊Q16 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊Q165C4, Q8.1(C4×D5), C10.65(C4×D4), C408C4.7C2, C4⋊C4.143D10, C53(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(D42D5), (C2×C40).190C22, (C2×C20).229C23, C2.1(Q16⋊D5), Dic10.17(C2×C4), (C2×Dic5).202D4, Dic53Q8.1C2, C20.44D4.6C2, C2.3(SD16⋊D5), C4⋊Dic5.79C22, (Q8×C10).12C22, C10.52(C8.C22), (C4×Dic5).21C22, C2.19(Dic54D4), (C2×Dic10).66C22, C4.10(C2×C4×D5), C52C8.1(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×C52C8).26C22, (C2×C4).336(C22×D5), SmallGroup(320,416)

Series: Derived Chief Lower central Upper central

C1C20 — C5⋊Q165C4
C1C5C10C20C2×C20C4×Dic5Dic53Q8 — C5⋊Q165C4
C5C10C20 — C5⋊Q165C4
C1C22C2×C4Q8⋊C4

Generators and relations for C5⋊Q165C4
 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 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×3], C2×C4, C2×C4 [×6], Q8 [×2], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, Q16 [×4], C2×Q8, C2×Q8, Dic5 [×5], C20 [×2], C20 [×3], C2×C10, C8⋊C4, Q8⋊C4, Q8⋊C4, C4.Q8, C4×Q8 [×2], C2×Q16, C52C8 [×2], C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, Q16⋊C4, C2×C52C8, C4×Dic5, C4×Dic5 [×2], C10.D4, C4⋊Dic5, C4⋊Dic5, C5⋊Q16 [×4], C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C20.Q8, C408C4, C20.44D4, C5×Q8⋊C4, Dic53Q8, C2×C5⋊Q16, Q8×Dic5, C5⋊Q165C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8.C22 [×2], C4×D5 [×2], C22×D5, Q16⋊C4, C2×C4×D5, D4×D5, D42D5, Dic54D4, SD16⋊D5, Q16⋊D5, C5⋊Q165C4

Smallest permutation representation of C5⋊Q165C4
Regular action on 320 points
Generators in S320
(1 153 298 29 185)(2 186 30 299 154)(3 155 300 31 187)(4 188 32 301 156)(5 157 302 25 189)(6 190 26 303 158)(7 159 304 27 191)(8 192 28 297 160)(9 320 132 152 36)(10 37 145 133 313)(11 314 134 146 38)(12 39 147 135 315)(13 316 136 148 40)(14 33 149 129 317)(15 318 130 150 34)(16 35 151 131 319)(17 312 165 143 288)(18 281 144 166 305)(19 306 167 137 282)(20 283 138 168 307)(21 308 161 139 284)(22 285 140 162 309)(23 310 163 141 286)(24 287 142 164 311)(41 83 224 107 239)(42 240 108 217 84)(43 85 218 109 233)(44 234 110 219 86)(45 87 220 111 235)(46 236 112 221 88)(47 81 222 105 237)(48 238 106 223 82)(49 99 207 225 251)(50 252 226 208 100)(51 101 201 227 253)(52 254 228 202 102)(53 103 203 229 255)(54 256 230 204 104)(55 97 205 231 249)(56 250 232 206 98)(57 196 117 76 263)(58 264 77 118 197)(59 198 119 78 257)(60 258 79 120 199)(61 200 113 80 259)(62 260 73 114 193)(63 194 115 74 261)(64 262 75 116 195)(65 90 213 269 247)(66 248 270 214 91)(67 92 215 271 241)(68 242 272 216 93)(69 94 209 265 243)(70 244 266 210 95)(71 96 211 267 245)(72 246 268 212 89)(121 178 295 278 169)(122 170 279 296 179)(123 180 289 280 171)(124 172 273 290 181)(125 182 291 274 173)(126 174 275 292 183)(127 184 293 276 175)(128 176 277 294 177)
(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)
(1 128 5 124)(2 127 6 123)(3 126 7 122)(4 125 8 121)(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)(33 283 37 287)(34 282 38 286)(35 281 39 285)(36 288 40 284)(41 250 45 254)(42 249 46 253)(43 256 47 252)(44 255 48 251)(49 234 53 238)(50 233 54 237)(51 240 55 236)(52 239 56 235)(57 71 61 67)(58 70 62 66)(59 69 63 65)(60 68 64 72)(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)(89 199 93 195)(90 198 94 194)(91 197 95 193)(92 196 96 200)(97 112 101 108)(98 111 102 107)(99 110 103 106)(100 109 104 105)(113 215 117 211)(114 214 118 210)(115 213 119 209)(116 212 120 216)(129 168 133 164)(130 167 134 163)(131 166 135 162)(132 165 136 161)(137 146 141 150)(138 145 142 149)(139 152 143 148)(140 151 144 147)(153 176 157 172)(154 175 158 171)(155 174 159 170)(156 173 160 169)(177 189 181 185)(178 188 182 192)(179 187 183 191)(180 186 184 190)(201 217 205 221)(202 224 206 220)(203 223 207 219)(204 222 208 218)(241 263 245 259)(242 262 246 258)(243 261 247 257)(244 260 248 264)(273 298 277 302)(274 297 278 301)(275 304 279 300)(276 303 280 299)(305 315 309 319)(306 314 310 318)(307 313 311 317)(308 320 312 316)
(1 204 22 263)(2 207 23 258)(3 202 24 261)(4 205 17 264)(5 208 18 259)(6 203 19 262)(7 206 20 257)(8 201 21 260)(9 242 121 219)(10 245 122 222)(11 248 123 217)(12 243 124 220)(13 246 125 223)(14 241 126 218)(15 244 127 221)(16 247 128 224)(25 252 166 113)(26 255 167 116)(27 250 168 119)(28 253 161 114)(29 256 162 117)(30 251 163 120)(31 254 164 115)(32 249 165 118)(33 67 174 109)(34 70 175 112)(35 65 176 107)(36 68 169 110)(37 71 170 105)(38 66 171 108)(39 69 172 111)(40 72 173 106)(41 131 213 294)(42 134 214 289)(43 129 215 292)(44 132 216 295)(45 135 209 290)(46 130 210 293)(47 133 211 296)(48 136 212 291)(49 141 199 299)(50 144 200 302)(51 139 193 297)(52 142 194 300)(53 137 195 303)(54 140 196 298)(55 143 197 301)(56 138 198 304)(57 153 104 285)(58 156 97 288)(59 159 98 283)(60 154 99 286)(61 157 100 281)(62 160 101 284)(63 155 102 287)(64 158 103 282)(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)(81 313 267 179)(82 316 268 182)(83 319 269 177)(84 314 270 180)(85 317 271 183)(86 320 272 178)(87 315 265 181)(88 318 266 184)(89 274 238 148)(90 277 239 151)(91 280 240 146)(92 275 233 149)(93 278 234 152)(94 273 235 147)(95 276 236 150)(96 279 237 145)

G:=sub<Sym(320)| (1,153,298,29,185)(2,186,30,299,154)(3,155,300,31,187)(4,188,32,301,156)(5,157,302,25,189)(6,190,26,303,158)(7,159,304,27,191)(8,192,28,297,160)(9,320,132,152,36)(10,37,145,133,313)(11,314,134,146,38)(12,39,147,135,315)(13,316,136,148,40)(14,33,149,129,317)(15,318,130,150,34)(16,35,151,131,319)(17,312,165,143,288)(18,281,144,166,305)(19,306,167,137,282)(20,283,138,168,307)(21,308,161,139,284)(22,285,140,162,309)(23,310,163,141,286)(24,287,142,164,311)(41,83,224,107,239)(42,240,108,217,84)(43,85,218,109,233)(44,234,110,219,86)(45,87,220,111,235)(46,236,112,221,88)(47,81,222,105,237)(48,238,106,223,82)(49,99,207,225,251)(50,252,226,208,100)(51,101,201,227,253)(52,254,228,202,102)(53,103,203,229,255)(54,256,230,204,104)(55,97,205,231,249)(56,250,232,206,98)(57,196,117,76,263)(58,264,77,118,197)(59,198,119,78,257)(60,258,79,120,199)(61,200,113,80,259)(62,260,73,114,193)(63,194,115,74,261)(64,262,75,116,195)(65,90,213,269,247)(66,248,270,214,91)(67,92,215,271,241)(68,242,272,216,93)(69,94,209,265,243)(70,244,266,210,95)(71,96,211,267,245)(72,246,268,212,89)(121,178,295,278,169)(122,170,279,296,179)(123,180,289,280,171)(124,172,273,290,181)(125,182,291,274,173)(126,174,275,292,183)(127,184,293,276,175)(128,176,277,294,177), (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), (1,128,5,124)(2,127,6,123)(3,126,7,122)(4,125,8,121)(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)(33,283,37,287)(34,282,38,286)(35,281,39,285)(36,288,40,284)(41,250,45,254)(42,249,46,253)(43,256,47,252)(44,255,48,251)(49,234,53,238)(50,233,54,237)(51,240,55,236)(52,239,56,235)(57,71,61,67)(58,70,62,66)(59,69,63,65)(60,68,64,72)(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)(89,199,93,195)(90,198,94,194)(91,197,95,193)(92,196,96,200)(97,112,101,108)(98,111,102,107)(99,110,103,106)(100,109,104,105)(113,215,117,211)(114,214,118,210)(115,213,119,209)(116,212,120,216)(129,168,133,164)(130,167,134,163)(131,166,135,162)(132,165,136,161)(137,146,141,150)(138,145,142,149)(139,152,143,148)(140,151,144,147)(153,176,157,172)(154,175,158,171)(155,174,159,170)(156,173,160,169)(177,189,181,185)(178,188,182,192)(179,187,183,191)(180,186,184,190)(201,217,205,221)(202,224,206,220)(203,223,207,219)(204,222,208,218)(241,263,245,259)(242,262,246,258)(243,261,247,257)(244,260,248,264)(273,298,277,302)(274,297,278,301)(275,304,279,300)(276,303,280,299)(305,315,309,319)(306,314,310,318)(307,313,311,317)(308,320,312,316), (1,204,22,263)(2,207,23,258)(3,202,24,261)(4,205,17,264)(5,208,18,259)(6,203,19,262)(7,206,20,257)(8,201,21,260)(9,242,121,219)(10,245,122,222)(11,248,123,217)(12,243,124,220)(13,246,125,223)(14,241,126,218)(15,244,127,221)(16,247,128,224)(25,252,166,113)(26,255,167,116)(27,250,168,119)(28,253,161,114)(29,256,162,117)(30,251,163,120)(31,254,164,115)(32,249,165,118)(33,67,174,109)(34,70,175,112)(35,65,176,107)(36,68,169,110)(37,71,170,105)(38,66,171,108)(39,69,172,111)(40,72,173,106)(41,131,213,294)(42,134,214,289)(43,129,215,292)(44,132,216,295)(45,135,209,290)(46,130,210,293)(47,133,211,296)(48,136,212,291)(49,141,199,299)(50,144,200,302)(51,139,193,297)(52,142,194,300)(53,137,195,303)(54,140,196,298)(55,143,197,301)(56,138,198,304)(57,153,104,285)(58,156,97,288)(59,159,98,283)(60,154,99,286)(61,157,100,281)(62,160,101,284)(63,155,102,287)(64,158,103,282)(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)(81,313,267,179)(82,316,268,182)(83,319,269,177)(84,314,270,180)(85,317,271,183)(86,320,272,178)(87,315,265,181)(88,318,266,184)(89,274,238,148)(90,277,239,151)(91,280,240,146)(92,275,233,149)(93,278,234,152)(94,273,235,147)(95,276,236,150)(96,279,237,145)>;

G:=Group( (1,153,298,29,185)(2,186,30,299,154)(3,155,300,31,187)(4,188,32,301,156)(5,157,302,25,189)(6,190,26,303,158)(7,159,304,27,191)(8,192,28,297,160)(9,320,132,152,36)(10,37,145,133,313)(11,314,134,146,38)(12,39,147,135,315)(13,316,136,148,40)(14,33,149,129,317)(15,318,130,150,34)(16,35,151,131,319)(17,312,165,143,288)(18,281,144,166,305)(19,306,167,137,282)(20,283,138,168,307)(21,308,161,139,284)(22,285,140,162,309)(23,310,163,141,286)(24,287,142,164,311)(41,83,224,107,239)(42,240,108,217,84)(43,85,218,109,233)(44,234,110,219,86)(45,87,220,111,235)(46,236,112,221,88)(47,81,222,105,237)(48,238,106,223,82)(49,99,207,225,251)(50,252,226,208,100)(51,101,201,227,253)(52,254,228,202,102)(53,103,203,229,255)(54,256,230,204,104)(55,97,205,231,249)(56,250,232,206,98)(57,196,117,76,263)(58,264,77,118,197)(59,198,119,78,257)(60,258,79,120,199)(61,200,113,80,259)(62,260,73,114,193)(63,194,115,74,261)(64,262,75,116,195)(65,90,213,269,247)(66,248,270,214,91)(67,92,215,271,241)(68,242,272,216,93)(69,94,209,265,243)(70,244,266,210,95)(71,96,211,267,245)(72,246,268,212,89)(121,178,295,278,169)(122,170,279,296,179)(123,180,289,280,171)(124,172,273,290,181)(125,182,291,274,173)(126,174,275,292,183)(127,184,293,276,175)(128,176,277,294,177), (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), (1,128,5,124)(2,127,6,123)(3,126,7,122)(4,125,8,121)(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)(33,283,37,287)(34,282,38,286)(35,281,39,285)(36,288,40,284)(41,250,45,254)(42,249,46,253)(43,256,47,252)(44,255,48,251)(49,234,53,238)(50,233,54,237)(51,240,55,236)(52,239,56,235)(57,71,61,67)(58,70,62,66)(59,69,63,65)(60,68,64,72)(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)(89,199,93,195)(90,198,94,194)(91,197,95,193)(92,196,96,200)(97,112,101,108)(98,111,102,107)(99,110,103,106)(100,109,104,105)(113,215,117,211)(114,214,118,210)(115,213,119,209)(116,212,120,216)(129,168,133,164)(130,167,134,163)(131,166,135,162)(132,165,136,161)(137,146,141,150)(138,145,142,149)(139,152,143,148)(140,151,144,147)(153,176,157,172)(154,175,158,171)(155,174,159,170)(156,173,160,169)(177,189,181,185)(178,188,182,192)(179,187,183,191)(180,186,184,190)(201,217,205,221)(202,224,206,220)(203,223,207,219)(204,222,208,218)(241,263,245,259)(242,262,246,258)(243,261,247,257)(244,260,248,264)(273,298,277,302)(274,297,278,301)(275,304,279,300)(276,303,280,299)(305,315,309,319)(306,314,310,318)(307,313,311,317)(308,320,312,316), (1,204,22,263)(2,207,23,258)(3,202,24,261)(4,205,17,264)(5,208,18,259)(6,203,19,262)(7,206,20,257)(8,201,21,260)(9,242,121,219)(10,245,122,222)(11,248,123,217)(12,243,124,220)(13,246,125,223)(14,241,126,218)(15,244,127,221)(16,247,128,224)(25,252,166,113)(26,255,167,116)(27,250,168,119)(28,253,161,114)(29,256,162,117)(30,251,163,120)(31,254,164,115)(32,249,165,118)(33,67,174,109)(34,70,175,112)(35,65,176,107)(36,68,169,110)(37,71,170,105)(38,66,171,108)(39,69,172,111)(40,72,173,106)(41,131,213,294)(42,134,214,289)(43,129,215,292)(44,132,216,295)(45,135,209,290)(46,130,210,293)(47,133,211,296)(48,136,212,291)(49,141,199,299)(50,144,200,302)(51,139,193,297)(52,142,194,300)(53,137,195,303)(54,140,196,298)(55,143,197,301)(56,138,198,304)(57,153,104,285)(58,156,97,288)(59,159,98,283)(60,154,99,286)(61,157,100,281)(62,160,101,284)(63,155,102,287)(64,158,103,282)(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)(81,313,267,179)(82,316,268,182)(83,319,269,177)(84,314,270,180)(85,317,271,183)(86,320,272,178)(87,315,265,181)(88,318,266,184)(89,274,238,148)(90,277,239,151)(91,280,240,146)(92,275,233,149)(93,278,234,152)(94,273,235,147)(95,276,236,150)(96,279,237,145) );

G=PermutationGroup([(1,153,298,29,185),(2,186,30,299,154),(3,155,300,31,187),(4,188,32,301,156),(5,157,302,25,189),(6,190,26,303,158),(7,159,304,27,191),(8,192,28,297,160),(9,320,132,152,36),(10,37,145,133,313),(11,314,134,146,38),(12,39,147,135,315),(13,316,136,148,40),(14,33,149,129,317),(15,318,130,150,34),(16,35,151,131,319),(17,312,165,143,288),(18,281,144,166,305),(19,306,167,137,282),(20,283,138,168,307),(21,308,161,139,284),(22,285,140,162,309),(23,310,163,141,286),(24,287,142,164,311),(41,83,224,107,239),(42,240,108,217,84),(43,85,218,109,233),(44,234,110,219,86),(45,87,220,111,235),(46,236,112,221,88),(47,81,222,105,237),(48,238,106,223,82),(49,99,207,225,251),(50,252,226,208,100),(51,101,201,227,253),(52,254,228,202,102),(53,103,203,229,255),(54,256,230,204,104),(55,97,205,231,249),(56,250,232,206,98),(57,196,117,76,263),(58,264,77,118,197),(59,198,119,78,257),(60,258,79,120,199),(61,200,113,80,259),(62,260,73,114,193),(63,194,115,74,261),(64,262,75,116,195),(65,90,213,269,247),(66,248,270,214,91),(67,92,215,271,241),(68,242,272,216,93),(69,94,209,265,243),(70,244,266,210,95),(71,96,211,267,245),(72,246,268,212,89),(121,178,295,278,169),(122,170,279,296,179),(123,180,289,280,171),(124,172,273,290,181),(125,182,291,274,173),(126,174,275,292,183),(127,184,293,276,175),(128,176,277,294,177)], [(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)], [(1,128,5,124),(2,127,6,123),(3,126,7,122),(4,125,8,121),(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),(33,283,37,287),(34,282,38,286),(35,281,39,285),(36,288,40,284),(41,250,45,254),(42,249,46,253),(43,256,47,252),(44,255,48,251),(49,234,53,238),(50,233,54,237),(51,240,55,236),(52,239,56,235),(57,71,61,67),(58,70,62,66),(59,69,63,65),(60,68,64,72),(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),(89,199,93,195),(90,198,94,194),(91,197,95,193),(92,196,96,200),(97,112,101,108),(98,111,102,107),(99,110,103,106),(100,109,104,105),(113,215,117,211),(114,214,118,210),(115,213,119,209),(116,212,120,216),(129,168,133,164),(130,167,134,163),(131,166,135,162),(132,165,136,161),(137,146,141,150),(138,145,142,149),(139,152,143,148),(140,151,144,147),(153,176,157,172),(154,175,158,171),(155,174,159,170),(156,173,160,169),(177,189,181,185),(178,188,182,192),(179,187,183,191),(180,186,184,190),(201,217,205,221),(202,224,206,220),(203,223,207,219),(204,222,208,218),(241,263,245,259),(242,262,246,258),(243,261,247,257),(244,260,248,264),(273,298,277,302),(274,297,278,301),(275,304,279,300),(276,303,280,299),(305,315,309,319),(306,314,310,318),(307,313,311,317),(308,320,312,316)], [(1,204,22,263),(2,207,23,258),(3,202,24,261),(4,205,17,264),(5,208,18,259),(6,203,19,262),(7,206,20,257),(8,201,21,260),(9,242,121,219),(10,245,122,222),(11,248,123,217),(12,243,124,220),(13,246,125,223),(14,241,126,218),(15,244,127,221),(16,247,128,224),(25,252,166,113),(26,255,167,116),(27,250,168,119),(28,253,161,114),(29,256,162,117),(30,251,163,120),(31,254,164,115),(32,249,165,118),(33,67,174,109),(34,70,175,112),(35,65,176,107),(36,68,169,110),(37,71,170,105),(38,66,171,108),(39,69,172,111),(40,72,173,106),(41,131,213,294),(42,134,214,289),(43,129,215,292),(44,132,216,295),(45,135,209,290),(46,130,210,293),(47,133,211,296),(48,136,212,291),(49,141,199,299),(50,144,200,302),(51,139,193,297),(52,142,194,300),(53,137,195,303),(54,140,196,298),(55,143,197,301),(56,138,198,304),(57,153,104,285),(58,156,97,288),(59,159,98,283),(60,154,99,286),(61,157,100,281),(62,160,101,284),(63,155,102,287),(64,158,103,282),(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),(81,313,267,179),(82,316,268,182),(83,319,269,177),(84,314,270,180),(85,317,271,183),(86,320,272,178),(87,315,265,181),(88,318,266,184),(89,274,238,148),(90,277,239,151),(91,280,240,146),(92,275,233,149),(93,278,234,152),(94,273,235,147),(95,276,236,150),(96,279,237,145)])

50 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444444455888810···102020202020···2040···40
size11112244441010101020202020224420202···244448···84···4

50 irreducible representations

dim111111111222222244444
type+++++++++++++--+-
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C4×D5C8.C22D42D5D4×D5SD16⋊D5Q16⋊D5
kernelC5⋊Q165C4C20.Q8C408C4C20.44D4C5×Q8⋊C4Dic53Q8C2×C5⋊Q16Q8×Dic5C5⋊Q16C2×Dic5Q8⋊C4C20C4⋊C4C2×C8C2×Q8Q8C10C4C22C2C2
# reps111111118222222822244

Matrix representation of C5⋊Q165C4 in GL6(𝔽41)

100000
010000
0016000
0001600
0000180
0000018
,
3870000
2230000
0000140
0000038
0014000
0003800
,
40370000
010000
000100
0040000
0000040
000010
,
2770000
7140000
000300
0014000
000003
0000140

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⋊Q165C4 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

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