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G = C5×C22.4Q16order 320 = 26·5

Direct product of C5 and C22.4Q16

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

Aliases: C5×C22.4Q16, C20.58C42, C4⋊C44C20, (C2×C8)⋊4C20, (C2×C40)⋊18C4, C4.2(C4×C20), (C2×C10).48D8, C20.83(C4⋊C4), (C2×C20).70Q8, C22.7(C5×D8), (C2×C20).507D4, (C22×C40).5C2, (C22×C8).2C10, (C2×C10).19Q16, C23.52(C5×D4), C22.4(C5×Q16), (C2×C10).43SD16, C10.19(C2.D8), C10.14(C4.Q8), C22.9(C5×SD16), (C22×C10).213D4, C10.51(D4⋊C4), C20.150(C22⋊C4), C10.25(Q8⋊C4), (C22×C20).571C22, C10.43(C2.C42), C4.3(C5×C4⋊C4), (C5×C4⋊C4)⋊18C4, (C2×C4⋊C4).2C10, C2.2(C5×C2.D8), C2.2(C5×C4.Q8), (C10×C4⋊C4).29C2, (C2×C4).13(C5×Q8), (C2×C4).40(C2×C20), (C2×C4).112(C5×D4), C2.2(C5×D4⋊C4), C4.19(C5×C22⋊C4), C22.16(C5×C4⋊C4), C2.2(C5×Q8⋊C4), (C2×C10).87(C4⋊C4), (C2×C20).434(C2×C4), C22.29(C5×C22⋊C4), C2.5(C5×C2.C42), (C22×C4).104(C2×C10), (C2×C10).194(C22⋊C4), SmallGroup(320,145)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C22.4Q16
C1C2C4C2×C4C22×C4C22×C20C10×C4⋊C4 — C5×C22.4Q16
C1C2C4 — C5×C22.4Q16
C1C22×C10C22×C20 — C5×C22.4Q16

Generators and relations for C5×C22.4Q16
 G = < a,b,c,d,e | a5=b2=c2=d8=1, e2=cd4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 186 in 114 conjugacy classes, 74 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22 [×3], C22 [×4], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2×C4⋊C4 [×2], C22×C8, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×8], C22×C10, C22.4Q16, C5×C4⋊C4 [×4], C5×C4⋊C4 [×2], C2×C40 [×2], C2×C40 [×2], C22×C20, C22×C20 [×2], C10×C4⋊C4 [×2], C22×C40, C5×C22.4Q16
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C20 [×6], C2×C10, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C2×C20 [×3], C5×D4 [×3], C5×Q8, C22.4Q16, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×D8, C5×SD16 [×2], C5×Q16, C5×C2.C42, C5×D4⋊C4 [×2], C5×Q8⋊C4 [×2], C5×C4.Q8, C5×C2.D8, C5×C22.4Q16

Smallest permutation representation of C5×C22.4Q16
Regular action on 320 points
Generators in S320
(1 12 113 105 97)(2 13 114 106 98)(3 14 115 107 99)(4 15 116 108 100)(5 16 117 109 101)(6 9 118 110 102)(7 10 119 111 103)(8 11 120 112 104)(17 91 41 33 25)(18 92 42 34 26)(19 93 43 35 27)(20 94 44 36 28)(21 95 45 37 29)(22 96 46 38 30)(23 89 47 39 31)(24 90 48 40 32)(49 176 73 65 57)(50 169 74 66 58)(51 170 75 67 59)(52 171 76 68 60)(53 172 77 69 61)(54 173 78 70 62)(55 174 79 71 63)(56 175 80 72 64)(81 229 236 213 220)(82 230 237 214 221)(83 231 238 215 222)(84 232 239 216 223)(85 225 240 209 224)(86 226 233 210 217)(87 227 234 211 218)(88 228 235 212 219)(121 153 145 137 129)(122 154 146 138 130)(123 155 147 139 131)(124 156 148 140 132)(125 157 149 141 133)(126 158 150 142 134)(127 159 151 143 135)(128 160 152 144 136)(161 196 203 180 187)(162 197 204 181 188)(163 198 205 182 189)(164 199 206 183 190)(165 200 207 184 191)(166 193 208 177 192)(167 194 201 178 185)(168 195 202 179 186)(241 274 266 258 250)(242 275 267 259 251)(243 276 268 260 252)(244 277 269 261 253)(245 278 270 262 254)(246 279 271 263 255)(247 280 272 264 256)(248 273 265 257 249)(281 313 305 297 289)(282 314 306 298 290)(283 315 307 299 291)(284 316 308 300 292)(285 317 309 301 293)(286 318 310 302 294)(287 319 311 303 295)(288 320 312 304 296)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 97)(32 98)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 105)(40 106)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 113)(48 114)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 129)(64 130)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 137)(72 138)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 145)(80 146)(81 315)(82 316)(83 317)(84 318)(85 319)(86 320)(87 313)(88 314)(153 174)(154 175)(155 176)(156 169)(157 170)(158 171)(159 172)(160 173)(161 243)(162 244)(163 245)(164 246)(165 247)(166 248)(167 241)(168 242)(177 257)(178 258)(179 259)(180 260)(181 261)(182 262)(183 263)(184 264)(185 250)(186 251)(187 252)(188 253)(189 254)(190 255)(191 256)(192 249)(193 273)(194 274)(195 275)(196 276)(197 277)(198 278)(199 279)(200 280)(201 266)(202 267)(203 268)(204 269)(205 270)(206 271)(207 272)(208 265)(209 295)(210 296)(211 289)(212 290)(213 291)(214 292)(215 293)(216 294)(217 288)(218 281)(219 282)(220 283)(221 284)(222 285)(223 286)(224 287)(225 311)(226 312)(227 305)(228 306)(229 307)(230 308)(231 309)(232 310)(233 304)(234 297)(235 298)(236 299)(237 300)(238 301)(239 302)(240 303)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 121)(8 122)(9 160)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(81 246)(82 247)(83 248)(84 241)(85 242)(86 243)(87 244)(88 245)(89 176)(90 169)(91 170)(92 171)(93 172)(94 173)(95 174)(96 175)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)(103 129)(104 130)(105 139)(106 140)(107 141)(108 142)(109 143)(110 144)(111 137)(112 138)(113 147)(114 148)(115 149)(116 150)(117 151)(118 152)(119 145)(120 146)(161 320)(162 313)(163 314)(164 315)(165 316)(166 317)(167 318)(168 319)(177 293)(178 294)(179 295)(180 296)(181 289)(182 290)(183 291)(184 292)(185 286)(186 287)(187 288)(188 281)(189 282)(190 283)(191 284)(192 285)(193 309)(194 310)(195 311)(196 312)(197 305)(198 306)(199 307)(200 308)(201 302)(202 303)(203 304)(204 297)(205 298)(206 299)(207 300)(208 301)(209 259)(210 260)(211 261)(212 262)(213 263)(214 264)(215 257)(216 258)(217 252)(218 253)(219 254)(220 255)(221 256)(222 249)(223 250)(224 251)(225 275)(226 276)(227 277)(228 278)(229 279)(230 280)(231 273)(232 274)(233 268)(234 269)(235 270)(236 271)(237 272)(238 265)(239 266)(240 267)
(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)
(1 218 127 249)(2 288 128 191)(3 224 121 255)(4 286 122 189)(5 222 123 253)(6 284 124 187)(7 220 125 251)(8 282 126 185)(9 316 156 161)(10 81 157 242)(11 314 158 167)(12 87 159 248)(13 320 160 165)(14 85 153 246)(15 318 154 163)(16 83 155 244)(17 287 55 190)(18 223 56 254)(19 285 49 188)(20 221 50 252)(21 283 51 186)(22 219 52 250)(23 281 53 192)(24 217 54 256)(25 295 63 183)(26 216 64 262)(27 293 57 181)(28 214 58 260)(29 291 59 179)(30 212 60 258)(31 289 61 177)(32 210 62 264)(33 303 71 206)(34 239 72 270)(35 301 65 204)(36 237 66 268)(37 299 67 202)(38 235 68 266)(39 297 69 208)(40 233 70 272)(41 311 79 199)(42 232 80 278)(43 309 73 197)(44 230 74 276)(45 307 75 195)(46 228 76 274)(47 305 77 193)(48 226 78 280)(82 169 243 94)(84 175 245 92)(86 173 247 90)(88 171 241 96)(89 313 172 166)(91 319 174 164)(93 317 176 162)(95 315 170 168)(97 211 135 257)(98 296 136 184)(99 209 129 263)(100 294 130 182)(101 215 131 261)(102 292 132 180)(103 213 133 259)(104 290 134 178)(105 234 143 265)(106 304 144 207)(107 240 137 271)(108 302 138 205)(109 238 139 269)(110 300 140 203)(111 236 141 267)(112 298 142 201)(113 227 151 273)(114 312 152 200)(115 225 145 279)(116 310 146 198)(117 231 147 277)(118 308 148 196)(119 229 149 275)(120 306 150 194)

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

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

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

140 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L5A5B5C5D8A···8H10A···10AB20A···20P20Q···20AV40A···40AF
order12···244444···455558···810···1020···2020···2040···40
size11···122224···411112···21···12···24···42···2

140 irreducible representations

dim1111111111222222222222
type++++-++-
imageC1C2C2C4C4C5C10C10C20C20D4Q8D4D8SD16Q16C5×D4C5×Q8C5×D4C5×D8C5×SD16C5×Q16
kernelC5×C22.4Q16C10×C4⋊C4C22×C40C5×C4⋊C4C2×C40C22.4Q16C2×C4⋊C4C22×C8C4⋊C4C2×C8C2×C20C2×C20C22×C10C2×C10C2×C10C2×C10C2×C4C2×C4C23C22C22C22
# reps1218448432162112428448168

Matrix representation of C5×C22.4Q16 in GL4(𝔽41) generated by

1000
0100
00370
00037
,
40000
04000
00400
00040
,
1000
04000
00400
00040
,
9000
03200
001526
001515
,
1000
03200
00193
00322
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,32,0,0,0,0,15,15,0,0,26,15],[1,0,0,0,0,32,0,0,0,0,19,3,0,0,3,22] >;

C5×C22.4Q16 in GAP, Magma, Sage, TeX

C_5\times C_2^2._4Q_{16}
% in TeX

G:=Group("C5xC2^2.4Q16");
// GroupNames label

G:=SmallGroup(320,145);
// by ID

G=gap.SmallGroup(320,145);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,248,10085,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=1,e^2=c*d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations

׿
×
𝔽