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G = C10×C2.C42order 320 = 26·5

Direct product of C10 and C2.C42

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

Aliases: C10×C2.C42, (C22×C4)⋊5C20, (C22×C20)⋊22C4, (C23×C4).1C10, C22.8(C4×C20), (C23×C20).4C2, C23.53(C5×D4), C23.10(C5×Q8), C22.7(Q8×C10), C10.49(C2×C42), C24.35(C2×C10), (C2×C10).52C42, C23.36(C2×C20), C22.26(D4×C10), (C22×C10).29Q8, (C22×C10).214D4, C23.48(C22×C10), C22.12(C22×C20), (C22×C20).486C22, (C22×C10).439C23, (C23×C10).118C22, C2.1(C2×C4×C20), (C2×C4)⋊9(C2×C20), C2.1(C10×C4⋊C4), (C2×C20)⋊48(C2×C4), C10.80(C2×C4⋊C4), C2.1(C10×C22⋊C4), C22.18(C5×C4⋊C4), (C2×C10).99(C2×Q8), (C2×C10).89(C4⋊C4), (C2×C10).593(C2×D4), C10.130(C2×C22⋊C4), (C22×C4).80(C2×C10), C22.31(C5×C22⋊C4), (C22×C10).213(C2×C4), (C2×C10).312(C22×C4), (C2×C10).197(C22⋊C4), SmallGroup(320,876)

Series: Derived Chief Lower central Upper central

C1C2 — C10×C2.C42
C1C2C22C23C22×C10C22×C20C5×C2.C42 — C10×C2.C42
C1C2 — C10×C2.C42
C1C23×C10 — C10×C2.C42

Generators and relations for C10×C2.C42
 G = < a,b,c,d | a10=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 450 in 330 conjugacy classes, 210 normal (12 characteristic)
C1, C2, C2 [×14], C4 [×12], C22, C22 [×34], C5, C2×C4 [×12], C2×C4 [×36], C23, C23 [×14], C10, C10 [×14], C22×C4 [×18], C22×C4 [×12], C24, C20 [×12], C2×C10, C2×C10 [×34], C2.C42 [×4], C23×C4 [×3], C2×C20 [×12], C2×C20 [×36], C22×C10, C22×C10 [×14], C2×C2.C42, C22×C20 [×18], C22×C20 [×12], C23×C10, C5×C2.C42 [×4], C23×C20 [×3], C10×C2.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×18], D4 [×6], Q8 [×2], C23, C10 [×7], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C20 [×12], C2×C10 [×7], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C20 [×18], C5×D4 [×6], C5×Q8 [×2], C22×C10, C2×C2.C42, C4×C20 [×4], C5×C22⋊C4 [×12], C5×C4⋊C4 [×12], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C2.C42 [×8], C2×C4×C20, C10×C22⋊C4 [×3], C10×C4⋊C4 [×3], C10×C2.C42

Smallest permutation representation of C10×C2.C42
Regular action on 320 points
Generators in S320
(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)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 101)(9 102)(10 103)(11 78)(12 79)(13 80)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 312)(22 313)(23 314)(24 315)(25 316)(26 317)(27 318)(28 319)(29 320)(30 311)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 51)(48 52)(49 53)(50 54)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 131)(90 132)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(111 147)(112 148)(113 149)(114 150)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(151 198)(152 199)(153 200)(154 191)(155 192)(156 193)(157 194)(158 195)(159 196)(160 197)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)(169 181)(170 182)(171 206)(172 207)(173 208)(174 209)(175 210)(176 201)(177 202)(178 203)(179 204)(180 205)(211 225)(212 226)(213 227)(214 228)(215 229)(216 230)(217 221)(218 222)(219 223)(220 224)(231 278)(232 279)(233 280)(234 271)(235 272)(236 273)(237 274)(238 275)(239 276)(240 277)(241 263)(242 264)(243 265)(244 266)(245 267)(246 268)(247 269)(248 270)(249 261)(250 262)(251 286)(252 287)(253 288)(254 289)(255 290)(256 281)(257 282)(258 283)(259 284)(260 285)(291 305)(292 306)(293 307)(294 308)(295 309)(296 310)(297 301)(298 302)(299 303)(300 304)
(1 274 129 299)(2 275 130 300)(3 276 121 291)(4 277 122 292)(5 278 123 293)(6 279 124 294)(7 280 125 295)(8 271 126 296)(9 272 127 297)(10 273 128 298)(11 154 39 230)(12 155 40 221)(13 156 31 222)(14 157 32 223)(15 158 33 224)(16 159 34 225)(17 160 35 226)(18 151 36 227)(19 152 37 228)(20 153 38 229)(21 204 46 167)(22 205 47 168)(23 206 48 169)(24 207 49 170)(25 208 50 161)(26 209 41 162)(27 210 42 163)(28 201 43 164)(29 202 44 165)(30 203 45 166)(51 190 313 180)(52 181 314 171)(53 182 315 172)(54 183 316 173)(55 184 317 174)(56 185 318 175)(57 186 319 176)(58 187 320 177)(59 188 311 178)(60 189 312 179)(61 213 75 198)(62 214 76 199)(63 215 77 200)(64 216 78 191)(65 217 79 192)(66 218 80 193)(67 219 71 194)(68 220 72 195)(69 211 73 196)(70 212 74 197)(81 258 144 268)(82 259 145 269)(83 260 146 270)(84 251 147 261)(85 252 148 262)(86 253 149 263)(87 254 150 264)(88 255 141 265)(89 256 142 266)(90 257 143 267)(91 310 101 234)(92 301 102 235)(93 302 103 236)(94 303 104 237)(95 304 105 238)(96 305 106 239)(97 306 107 240)(98 307 108 231)(99 308 109 232)(100 309 110 233)(111 249 136 286)(112 250 137 287)(113 241 138 288)(114 242 139 289)(115 243 140 290)(116 244 131 281)(117 245 132 282)(118 246 133 283)(119 247 134 284)(120 248 135 285)
(1 179 145 219)(2 180 146 220)(3 171 147 211)(4 172 148 212)(5 173 149 213)(6 174 150 214)(7 175 141 215)(8 176 142 216)(9 177 143 217)(10 178 144 218)(11 271 43 266)(12 272 44 267)(13 273 45 268)(14 274 46 269)(15 275 47 270)(16 276 48 261)(17 277 49 262)(18 278 50 263)(19 279 41 264)(20 280 42 265)(21 259 32 299)(22 260 33 300)(23 251 34 291)(24 252 35 292)(25 253 36 293)(26 254 37 294)(27 255 38 295)(28 256 39 296)(29 257 40 297)(30 258 31 298)(51 248 72 238)(52 249 73 239)(53 250 74 240)(54 241 75 231)(55 242 76 232)(56 243 77 233)(57 244 78 234)(58 245 79 235)(59 246 80 236)(60 247 71 237)(61 307 316 288)(62 308 317 289)(63 309 318 290)(64 310 319 281)(65 301 320 282)(66 302 311 283)(67 303 312 284)(68 304 313 285)(69 305 314 286)(70 306 315 287)(81 193 128 188)(82 194 129 189)(83 195 130 190)(84 196 121 181)(85 197 122 182)(86 198 123 183)(87 199 124 184)(88 200 125 185)(89 191 126 186)(90 192 127 187)(91 164 131 154)(92 165 132 155)(93 166 133 156)(94 167 134 157)(95 168 135 158)(96 169 136 159)(97 170 137 160)(98 161 138 151)(99 162 139 152)(100 163 140 153)(101 201 116 230)(102 202 117 221)(103 203 118 222)(104 204 119 223)(105 205 120 224)(106 206 111 225)(107 207 112 226)(108 208 113 227)(109 209 114 228)(110 210 115 229)

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

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

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

200 conjugacy classes

class 1 2A···2O4A···4X5A5B5C5D10A···10BH20A···20CR
order12···24···4555510···1020···20
size11···12···211111···12···2

200 irreducible representations

dim111111112222
type++++-
imageC1C2C2C4C5C10C10C20D4Q8C5×D4C5×Q8
kernelC10×C2.C42C5×C2.C42C23×C20C22×C20C2×C2.C42C2.C42C23×C4C22×C4C22×C10C22×C10C23C23
# reps14324416129662248

Matrix representation of C10×C2.C42 in GL5(𝔽41)

400000
040000
004000
000230
000023
,
10000
01000
00100
000400
000040
,
90000
09000
00100
0001416
0001627
,
10000
09000
00100
000040
00010

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,14,16,0,0,0,16,27],[1,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,40,0] >;

C10×C2.C42 in GAP, Magma, Sage, TeX

C_{10}\times C_2.C_4^2
% in TeX

G:=Group("C10xC2.C4^2");
// GroupNames label

G:=SmallGroup(320,876);
// by ID

G=gap.SmallGroup(320,876);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations

׿
×
𝔽