Copied to
clipboard

## G = C4⋊C4×C2×C10order 320 = 26·5

### Direct product of C2×C10 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4⋊C4×C2×C10
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — C10×C4⋊C4 — C4⋊C4×C2×C10
 Lower central C1 — C2 — C4⋊C4×C2×C10
 Upper central C1 — C23×C10 — C4⋊C4×C2×C10

Generators and relations for C4⋊C4×C2×C10
G = < a,b,c,d | a2=b10=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 498 in 418 conjugacy classes, 338 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×8], C22, C22 [×34], C5, C2×C4 [×36], C2×C4 [×24], C23 [×15], C10 [×3], C10 [×12], C4⋊C4 [×16], C22×C4 [×26], C22×C4 [×8], C24, C20 [×8], C20 [×8], C2×C10, C2×C10 [×34], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C2×C20 [×36], C2×C20 [×24], C22×C10 [×15], C22×C4⋊C4, C5×C4⋊C4 [×16], C22×C20 [×26], C22×C20 [×8], C23×C10, C10×C4⋊C4 [×12], C23×C20, C23×C20 [×2], C4⋊C4×C2×C10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C20 [×8], C2×C10 [×35], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C2×C20 [×28], C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], C22×C4⋊C4, C5×C4⋊C4 [×16], C22×C20 [×14], D4×C10 [×6], Q8×C10 [×6], C23×C10, C10×C4⋊C4 [×12], C23×C20, D4×C2×C10, Q8×C2×C10, C4⋊C4×C2×C10

Smallest permutation representation of C4⋊C4×C2×C10
Regular action on 320 points
Generators in S320
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 293)(12 294)(13 295)(14 296)(15 297)(16 298)(17 299)(18 300)(19 291)(20 292)(21 272)(22 273)(23 274)(24 275)(25 276)(26 277)(27 278)(28 279)(29 280)(30 271)(31 309)(32 310)(33 301)(34 302)(35 303)(36 304)(37 305)(38 306)(39 307)(40 308)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)(51 110)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 91)(111 169)(112 170)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 151)(130 152)(131 190)(132 181)(133 182)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 172)(142 173)(143 174)(144 175)(145 176)(146 177)(147 178)(148 179)(149 180)(150 171)(191 249)(192 250)(193 241)(194 242)(195 243)(196 244)(197 245)(198 246)(199 247)(200 248)(201 233)(202 234)(203 235)(204 236)(205 237)(206 238)(207 239)(208 240)(209 231)(210 232)(211 270)(212 261)(213 262)(214 263)(215 264)(216 265)(217 266)(218 267)(219 268)(220 269)(221 252)(222 253)(223 254)(224 255)(225 256)(226 257)(227 258)(228 259)(229 260)(230 251)(281 313)(282 314)(283 315)(284 316)(285 317)(286 318)(287 319)(288 320)(289 311)(290 312)
(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)
(1 161 65 171)(2 162 66 172)(3 163 67 173)(4 164 68 174)(5 165 69 175)(6 166 70 176)(7 167 61 177)(8 168 62 178)(9 169 63 179)(10 170 64 180)(11 218 315 208)(12 219 316 209)(13 220 317 210)(14 211 318 201)(15 212 319 202)(16 213 320 203)(17 214 311 204)(18 215 312 205)(19 216 313 206)(20 217 314 207)(21 197 31 224)(22 198 32 225)(23 199 33 226)(24 200 34 227)(25 191 35 228)(26 192 36 229)(27 193 37 230)(28 194 38 221)(29 195 39 222)(30 196 40 223)(41 158 51 185)(42 159 52 186)(43 160 53 187)(44 151 54 188)(45 152 55 189)(46 153 56 190)(47 154 57 181)(48 155 58 182)(49 156 59 183)(50 157 60 184)(71 124 108 134)(72 125 109 135)(73 126 110 136)(74 127 101 137)(75 128 102 138)(76 129 103 139)(77 130 104 140)(78 121 105 131)(79 122 106 132)(80 123 107 133)(81 118 91 145)(82 119 92 146)(83 120 93 147)(84 111 94 148)(85 112 95 149)(86 113 96 150)(87 114 97 141)(88 115 98 142)(89 116 99 143)(90 117 100 144)(231 294 268 284)(232 295 269 285)(233 296 270 286)(234 297 261 287)(235 298 262 288)(236 299 263 289)(237 300 264 290)(238 291 265 281)(239 292 266 282)(240 293 267 283)(241 305 251 278)(242 306 252 279)(243 307 253 280)(244 308 254 271)(245 309 255 272)(246 310 256 273)(247 301 257 274)(248 302 258 275)(249 303 259 276)(250 304 260 277)
(1 251 51 238)(2 252 52 239)(3 253 53 240)(4 254 54 231)(5 255 55 232)(6 256 56 233)(7 257 57 234)(8 258 58 235)(9 259 59 236)(10 260 60 237)(11 142 29 128)(12 143 30 129)(13 144 21 130)(14 145 22 121)(15 146 23 122)(16 147 24 123)(17 148 25 124)(18 149 26 125)(19 150 27 126)(20 141 28 127)(31 140 317 117)(32 131 318 118)(33 132 319 119)(34 133 320 120)(35 134 311 111)(36 135 312 112)(37 136 313 113)(38 137 314 114)(39 138 315 115)(40 139 316 116)(41 265 65 241)(42 266 66 242)(43 267 67 243)(44 268 68 244)(45 269 69 245)(46 270 70 246)(47 261 61 247)(48 262 62 248)(49 263 63 249)(50 264 64 250)(71 214 94 191)(72 215 95 192)(73 216 96 193)(74 217 97 194)(75 218 98 195)(76 219 99 196)(77 220 100 197)(78 211 91 198)(79 212 92 199)(80 213 93 200)(81 225 105 201)(82 226 106 202)(83 227 107 203)(84 228 108 204)(85 229 109 205)(86 230 110 206)(87 221 101 207)(88 222 102 208)(89 223 103 209)(90 224 104 210)(151 294 174 271)(152 295 175 272)(153 296 176 273)(154 297 177 274)(155 298 178 275)(156 299 179 276)(157 300 180 277)(158 291 171 278)(159 292 172 279)(160 293 173 280)(161 305 185 281)(162 306 186 282)(163 307 187 283)(164 308 188 284)(165 309 189 285)(166 310 190 286)(167 301 181 287)(168 302 182 288)(169 303 183 289)(170 304 184 290)

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

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

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

200 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4X 5A 5B 5C 5D 10A ··· 10BH 20A ··· 20CR order 1 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C4 C5 C10 C10 C20 D4 Q8 C5×D4 C5×Q8 kernel C4⋊C4×C2×C10 C10×C4⋊C4 C23×C20 C22×C20 C22×C4⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C22×C10 C22×C10 C23 C23 # reps 1 12 3 16 4 48 12 64 4 4 16 16

Matrix representation of C4⋊C4×C2×C10 in GL5(𝔽41)

 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 1 0
,
 9 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 17 9 0 0 0 9 24

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,40,0],[9,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,17,9,0,0,0,9,24] >;

C4⋊C4×C2×C10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_2\times C_{10}
% in TeX

G:=Group("C4:C4xC2xC10");
// GroupNames label

G:=SmallGroup(320,1515);
// by ID

G=gap.SmallGroup(320,1515);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568]);
// Polycyclic

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

׿
×
𝔽