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G = C5×C22.7C42order 320 = 26·5

Direct product of C5 and C22.7C42

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

Aliases: C5×C22.7C42, (C2×C4)⋊2C40, (C2×C20)⋊9C8, (C2×C8)⋊3C20, (C2×C40)⋊17C4, C2.2(C4×C40), C10.22(C4×C8), C20.97(C4⋊C4), C10.22(C4⋊C8), (C2×C20).81Q8, (C2×C20).532D4, C22.7(C2×C40), (C22×C8).1C10, (C22×C40).4C2, C22.7(C4×C20), (C2×C42).1C10, (C22×C4).7C20, (C2×C10).51C42, (C22×C20).42C4, C23.35(C2×C20), C10.16(C8⋊C4), C10.39(C22⋊C8), (C2×C10).46M4(2), C20.156(C22⋊C4), C22.7(C5×M4(2)), (C22×C20).606C22, C10.39(C2.C42), C2.1(C5×C4⋊C8), (C2×C4×C20).1C2, C4.17(C5×C4⋊C4), C2.2(C5×C8⋊C4), C2.1(C5×C22⋊C8), (C2×C4).23(C5×Q8), (C2×C10).66(C2×C8), (C2×C4).81(C2×C20), (C2×C4).141(C5×D4), C22.15(C5×C4⋊C4), C4.24(C5×C22⋊C4), (C2×C10).86(C4⋊C4), (C2×C20).516(C2×C4), C22.27(C5×C22⋊C4), C2.1(C5×C2.C42), (C22×C4).133(C2×C10), (C22×C10).212(C2×C4), (C2×C10).193(C22⋊C4), SmallGroup(320,141)

Series: Derived Chief Lower central Upper central

C1C2 — C5×C22.7C42
C1C2C22C2×C4C22×C4C22×C20C22×C40 — C5×C22.7C42
C1C2 — C5×C22.7C42
C1C22×C20 — C5×C22.7C42

Generators and relations for C5×C22.7C42
 G = < a,b,c,d,e | a5=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 154 in 118 conjugacy classes, 82 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×4], C5, C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C10 [×3], C10 [×4], C42 [×2], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2×C42, C22×C8 [×2], C40 [×4], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×C10, C22.7C42, C4×C20 [×2], C2×C40 [×4], C2×C40 [×4], C22×C20, C22×C20 [×2], C2×C4×C20, C22×C40 [×2], C5×C22.7C42
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C20 [×6], C2×C10, C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C40 [×4], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22.7C42, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C40 [×2], C5×M4(2) [×2], C5×C2.C42, C4×C40, C5×C8⋊C4, C5×C22⋊C8 [×2], C5×C4⋊C8 [×2], C5×C22.7C42

Smallest permutation representation of C5×C22.7C42
Regular action on 320 points
Generators in S320
(1 55 47 39 31)(2 56 48 40 32)(3 49 41 33 25)(4 50 42 34 26)(5 51 43 35 27)(6 52 44 36 28)(7 53 45 37 29)(8 54 46 38 30)(9 86 78 70 62)(10 87 79 71 63)(11 88 80 72 64)(12 81 73 65 57)(13 82 74 66 58)(14 83 75 67 59)(15 84 76 68 60)(16 85 77 69 61)(17 233 225 217 176)(18 234 226 218 169)(19 235 227 219 170)(20 236 228 220 171)(21 237 229 221 172)(22 238 230 222 173)(23 239 231 223 174)(24 240 232 224 175)(89 121 113 105 97)(90 122 114 106 98)(91 123 115 107 99)(92 124 116 108 100)(93 125 117 109 101)(94 126 118 110 102)(95 127 119 111 103)(96 128 120 112 104)(129 167 153 145 137)(130 168 154 146 138)(131 161 155 147 139)(132 162 156 148 140)(133 163 157 149 141)(134 164 158 150 142)(135 165 159 151 143)(136 166 160 152 144)(177 209 201 193 185)(178 210 202 194 186)(179 211 203 195 187)(180 212 204 196 188)(181 213 205 197 189)(182 214 206 198 190)(183 215 207 199 191)(184 216 208 200 192)(241 273 265 257 249)(242 274 266 258 250)(243 275 267 259 251)(244 276 268 260 252)(245 277 269 261 253)(246 278 270 262 254)(247 279 271 263 255)(248 280 272 264 256)(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)
(1 183)(2 184)(3 177)(4 178)(5 179)(6 180)(7 181)(8 182)(9 173)(10 174)(11 175)(12 176)(13 169)(14 170)(15 171)(16 172)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 185)(26 186)(27 187)(28 188)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 197)(38 198)(39 199)(40 200)(41 201)(42 202)(43 203)(44 204)(45 205)(46 206)(47 207)(48 208)(49 209)(50 210)(51 211)(52 212)(53 213)(54 214)(55 215)(56 216)(57 217)(58 218)(59 219)(60 220)(61 221)(62 222)(63 223)(64 224)(65 225)(66 226)(67 227)(68 228)(69 229)(70 230)(71 231)(72 232)(73 233)(74 234)(75 235)(76 236)(77 237)(78 238)(79 239)(80 240)(89 241)(90 242)(91 243)(92 244)(93 245)(94 246)(95 247)(96 248)(97 249)(98 250)(99 251)(100 252)(101 253)(102 254)(103 255)(104 256)(105 257)(106 258)(107 259)(108 260)(109 261)(110 262)(111 263)(112 264)(113 265)(114 266)(115 267)(116 268)(117 269)(118 270)(119 271)(120 272)(121 273)(122 274)(123 275)(124 276)(125 277)(126 278)(127 279)(128 280)(129 281)(130 282)(131 283)(132 284)(133 285)(134 286)(135 287)(136 288)(137 289)(138 290)(139 291)(140 292)(141 293)(142 294)(143 295)(144 296)(145 297)(146 298)(147 299)(148 300)(149 301)(150 302)(151 303)(152 304)(153 305)(154 306)(155 307)(156 308)(157 309)(158 310)(159 311)(160 312)(161 315)(162 316)(163 317)(164 318)(165 319)(166 320)(167 313)(168 314)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)(225 229)(226 230)(227 231)(228 232)(233 237)(234 238)(235 239)(236 240)(241 245)(242 246)(243 247)(244 248)(249 253)(250 254)(251 255)(252 256)(257 261)(258 262)(259 263)(260 264)(265 269)(266 270)(267 271)(268 272)(273 277)(274 278)(275 279)(276 280)(281 285)(282 286)(283 287)(284 288)(289 293)(290 294)(291 295)(292 296)(297 301)(298 302)(299 303)(300 304)(305 309)(306 310)(307 311)(308 312)(313 317)(314 318)(315 319)(316 320)
(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)
(1 287 247 63)(2 136 248 224)(3 281 241 57)(4 130 242 218)(5 283 243 59)(6 132 244 220)(7 285 245 61)(8 134 246 222)(9 214 318 126)(10 55 319 279)(11 216 320 128)(12 49 313 273)(13 210 314 122)(14 51 315 275)(15 212 316 124)(16 53 317 277)(17 201 153 113)(18 42 154 266)(19 203 155 115)(20 44 156 268)(21 205 157 117)(22 46 158 270)(23 207 159 119)(24 48 160 272)(25 289 249 65)(26 138 250 226)(27 291 251 67)(28 140 252 228)(29 293 253 69)(30 142 254 230)(31 295 255 71)(32 144 256 232)(33 297 257 73)(34 146 258 234)(35 299 259 75)(36 148 260 236)(37 301 261 77)(38 150 262 238)(39 303 263 79)(40 152 264 240)(41 305 265 81)(43 307 267 83)(45 309 269 85)(47 311 271 87)(50 168 274 169)(52 162 276 171)(54 164 278 173)(56 166 280 175)(58 178 282 90)(60 180 284 92)(62 182 286 94)(64 184 288 96)(66 186 290 98)(68 188 292 100)(70 190 294 102)(72 192 296 104)(74 194 298 106)(76 196 300 108)(78 198 302 110)(80 200 304 112)(82 202 306 114)(84 204 308 116)(86 206 310 118)(88 208 312 120)(89 217 177 129)(91 219 179 131)(93 221 181 133)(95 223 183 135)(97 225 185 137)(99 227 187 139)(101 229 189 141)(103 231 191 143)(105 233 193 145)(107 235 195 147)(109 237 197 149)(111 239 199 151)(121 176 209 167)(123 170 211 161)(125 172 213 163)(127 174 215 165)

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

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

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

200 conjugacy classes

class 1 2A···2G4A···4H4I···4P5A5B5C5D8A···8P10A···10AB20A···20AF20AG···20BL40A···40BL
order12···24···44···455558···810···1020···2020···2040···40
size11···11···12···211112···21···11···12···22···2

200 irreducible representations

dim111111111111222222
type++++-
imageC1C2C2C4C4C5C8C10C10C20C20C40D4Q8M4(2)C5×D4C5×Q8C5×M4(2)
kernelC5×C22.7C42C2×C4×C20C22×C40C2×C40C22×C20C22.7C42C2×C20C2×C42C22×C8C2×C8C22×C4C2×C4C2×C20C2×C20C2×C10C2×C4C2×C4C22
# reps112844164832166431412416

Matrix representation of C5×C22.7C42 in GL4(𝔽41) generated by

1000
0100
00100
00010
,
1000
0100
00400
00040
,
40000
0100
0010
0001
,
38000
03200
003913
00282
,
1000
03200
0001
0010
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,0,32,0,0,0,0,39,28,0,0,13,2],[1,0,0,0,0,32,0,0,0,0,0,1,0,0,1,0] >;

C5×C22.7C42 in GAP, Magma, Sage, TeX

C_5\times C_2^2._7C_4^2
% in TeX

G:=Group("C5xC2^2.7C4^2");
// GroupNames label

G:=SmallGroup(320,141);
// by ID

G=gap.SmallGroup(320,141);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,172]);
// Polycyclic

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

׿
×
𝔽