direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C2×C10, (C22×C4)⋊9C20, C4⋊2(C22×C20), C20⋊13(C22×C4), (C22×C20)⋊31C4, (C23×C4).6C10, C2.2(C23×C20), C23.59(C5×D4), C23.11(C5×Q8), C23.39(C2×C20), C10.75(C23×C4), (C23×C20).10C2, C24.36(C2×C10), C22.58(D4×C10), C10.55(C22×Q8), C22.16(Q8×C10), (C22×C10).30Q8, (C2×C20).958C23, (C2×C10).333C24, C10.178(C22×D4), (C22×C10).220D4, C22.6(C23×C10), C23.66(C22×C10), C22.24(C22×C20), (C22×C20).507C22, (C23×C10).119C22, (C22×C10).466C23, C2.2(D4×C2×C10), C2.1(Q8×C2×C10), (C2×C20)⋊52(C2×C4), (C2×C4)⋊10(C2×C20), (C2×C10).680(C2×D4), (C2×C10).114(C2×Q8), (C2×C4).53(C22×C10), (C22×C10).216(C2×C4), (C22×C4).123(C2×C10), (C2×C10).346(C22×C4), SmallGroup(320,1515)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C4⋊C4, C23×C4, C23×C4, C2×C20, C2×C20, C22×C10, C22×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C23×C10, C10×C4⋊C4, C23×C20, C23×C20, C4⋊C4×C2×C10
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C20, C2×C10, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C4⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C23×C10, C10×C4⋊C4, C23×C20, D4×C2×C10, Q8×C2×C10, 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)(11 285)(12 286)(13 287)(14 288)(15 289)(16 290)(17 281)(18 282)(19 283)(20 284)(21 303)(22 304)(23 305)(24 306)(25 307)(26 308)(27 309)(28 310)(29 301)(30 302)(31 278)(32 279)(33 280)(34 271)(35 272)(36 273)(37 274)(38 275)(39 276)(40 277)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 101)(49 102)(50 103)(51 79)(52 80)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 91)(69 92)(70 93)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 171)(120 172)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 181)(129 182)(130 183)(131 159)(132 160)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)(141 168)(142 169)(143 170)(144 161)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)(191 253)(192 254)(193 255)(194 256)(195 257)(196 258)(197 259)(198 260)(199 251)(200 252)(201 264)(202 265)(203 266)(204 267)(205 268)(206 269)(207 270)(208 261)(209 262)(210 263)(211 239)(212 240)(213 231)(214 232)(215 233)(216 234)(217 235)(218 236)(219 237)(220 238)(221 248)(222 249)(223 250)(224 241)(225 242)(226 243)(227 244)(228 245)(229 246)(230 247)(291 319)(292 320)(293 311)(294 312)(295 313)(296 314)(297 315)(298 316)(299 317)(300 318)
(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 63 171)(2 162 64 172)(3 163 65 173)(4 164 66 174)(5 165 67 175)(6 166 68 176)(7 167 69 177)(8 168 70 178)(9 169 61 179)(10 170 62 180)(11 210 313 220)(12 201 314 211)(13 202 315 212)(14 203 316 213)(15 204 317 214)(16 205 318 215)(17 206 319 216)(18 207 320 217)(19 208 311 218)(20 209 312 219)(21 228 31 193)(22 229 32 194)(23 230 33 195)(24 221 34 196)(25 222 35 197)(26 223 36 198)(27 224 37 199)(28 225 38 200)(29 226 39 191)(30 227 40 192)(41 189 51 154)(42 190 52 155)(43 181 53 156)(44 182 54 157)(45 183 55 158)(46 184 56 159)(47 185 57 160)(48 186 58 151)(49 187 59 152)(50 188 60 153)(71 138 106 128)(72 139 107 129)(73 140 108 130)(74 131 109 121)(75 132 110 122)(76 133 101 123)(77 134 102 124)(78 135 103 125)(79 136 104 126)(80 137 105 127)(81 149 91 114)(82 150 92 115)(83 141 93 116)(84 142 94 117)(85 143 95 118)(86 144 96 119)(87 145 97 120)(88 146 98 111)(89 147 99 112)(90 148 100 113)(231 288 266 298)(232 289 267 299)(233 290 268 300)(234 281 269 291)(235 282 270 292)(236 283 261 293)(237 284 262 294)(238 285 263 295)(239 286 264 296)(240 287 265 297)(241 274 251 309)(242 275 252 310)(243 276 253 301)(244 277 254 302)(245 278 255 303)(246 279 256 304)(247 280 257 305)(248 271 258 306)(249 272 259 307)(250 273 260 308)
(1 251 51 269)(2 252 52 270)(3 253 53 261)(4 254 54 262)(5 255 55 263)(6 256 56 264)(7 257 57 265)(8 258 58 266)(9 259 59 267)(10 260 60 268)(11 148 31 140)(12 149 32 131)(13 150 33 132)(14 141 34 133)(15 142 35 134)(16 143 36 135)(17 144 37 136)(18 145 38 137)(19 146 39 138)(20 147 40 139)(21 130 313 113)(22 121 314 114)(23 122 315 115)(24 123 316 116)(25 124 317 117)(26 125 318 118)(27 126 319 119)(28 127 320 120)(29 128 311 111)(30 129 312 112)(41 234 63 241)(42 235 64 242)(43 236 65 243)(44 237 66 244)(45 238 67 245)(46 239 68 246)(47 240 69 247)(48 231 70 248)(49 232 61 249)(50 233 62 250)(71 208 88 191)(72 209 89 192)(73 210 90 193)(74 201 81 194)(75 202 82 195)(76 203 83 196)(77 204 84 197)(78 205 85 198)(79 206 86 199)(80 207 87 200)(91 229 109 211)(92 230 110 212)(93 221 101 213)(94 222 102 214)(95 223 103 215)(96 224 104 216)(97 225 105 217)(98 226 106 218)(99 227 107 219)(100 228 108 220)(151 288 168 271)(152 289 169 272)(153 290 170 273)(154 281 161 274)(155 282 162 275)(156 283 163 276)(157 284 164 277)(158 285 165 278)(159 286 166 279)(160 287 167 280)(171 309 189 291)(172 310 190 292)(173 301 181 293)(174 302 182 294)(175 303 183 295)(176 304 184 296)(177 305 185 297)(178 306 186 298)(179 307 187 299)(180 308 188 300)
G:=sub<Sym(320)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,285)(12,286)(13,287)(14,288)(15,289)(16,290)(17,281)(18,282)(19,283)(20,284)(21,303)(22,304)(23,305)(24,306)(25,307)(26,308)(27,309)(28,310)(29,301)(30,302)(31,278)(32,279)(33,280)(34,271)(35,272)(36,273)(37,274)(38,275)(39,276)(40,277)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,91)(69,92)(70,93)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,171)(120,172)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,181)(129,182)(130,183)(131,159)(132,160)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158)(141,168)(142,169)(143,170)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(191,253)(192,254)(193,255)(194,256)(195,257)(196,258)(197,259)(198,260)(199,251)(200,252)(201,264)(202,265)(203,266)(204,267)(205,268)(206,269)(207,270)(208,261)(209,262)(210,263)(211,239)(212,240)(213,231)(214,232)(215,233)(216,234)(217,235)(218,236)(219,237)(220,238)(221,248)(222,249)(223,250)(224,241)(225,242)(226,243)(227,244)(228,245)(229,246)(230,247)(291,319)(292,320)(293,311)(294,312)(295,313)(296,314)(297,315)(298,316)(299,317)(300,318), (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), (1,161,63,171)(2,162,64,172)(3,163,65,173)(4,164,66,174)(5,165,67,175)(6,166,68,176)(7,167,69,177)(8,168,70,178)(9,169,61,179)(10,170,62,180)(11,210,313,220)(12,201,314,211)(13,202,315,212)(14,203,316,213)(15,204,317,214)(16,205,318,215)(17,206,319,216)(18,207,320,217)(19,208,311,218)(20,209,312,219)(21,228,31,193)(22,229,32,194)(23,230,33,195)(24,221,34,196)(25,222,35,197)(26,223,36,198)(27,224,37,199)(28,225,38,200)(29,226,39,191)(30,227,40,192)(41,189,51,154)(42,190,52,155)(43,181,53,156)(44,182,54,157)(45,183,55,158)(46,184,56,159)(47,185,57,160)(48,186,58,151)(49,187,59,152)(50,188,60,153)(71,138,106,128)(72,139,107,129)(73,140,108,130)(74,131,109,121)(75,132,110,122)(76,133,101,123)(77,134,102,124)(78,135,103,125)(79,136,104,126)(80,137,105,127)(81,149,91,114)(82,150,92,115)(83,141,93,116)(84,142,94,117)(85,143,95,118)(86,144,96,119)(87,145,97,120)(88,146,98,111)(89,147,99,112)(90,148,100,113)(231,288,266,298)(232,289,267,299)(233,290,268,300)(234,281,269,291)(235,282,270,292)(236,283,261,293)(237,284,262,294)(238,285,263,295)(239,286,264,296)(240,287,265,297)(241,274,251,309)(242,275,252,310)(243,276,253,301)(244,277,254,302)(245,278,255,303)(246,279,256,304)(247,280,257,305)(248,271,258,306)(249,272,259,307)(250,273,260,308), (1,251,51,269)(2,252,52,270)(3,253,53,261)(4,254,54,262)(5,255,55,263)(6,256,56,264)(7,257,57,265)(8,258,58,266)(9,259,59,267)(10,260,60,268)(11,148,31,140)(12,149,32,131)(13,150,33,132)(14,141,34,133)(15,142,35,134)(16,143,36,135)(17,144,37,136)(18,145,38,137)(19,146,39,138)(20,147,40,139)(21,130,313,113)(22,121,314,114)(23,122,315,115)(24,123,316,116)(25,124,317,117)(26,125,318,118)(27,126,319,119)(28,127,320,120)(29,128,311,111)(30,129,312,112)(41,234,63,241)(42,235,64,242)(43,236,65,243)(44,237,66,244)(45,238,67,245)(46,239,68,246)(47,240,69,247)(48,231,70,248)(49,232,61,249)(50,233,62,250)(71,208,88,191)(72,209,89,192)(73,210,90,193)(74,201,81,194)(75,202,82,195)(76,203,83,196)(77,204,84,197)(78,205,85,198)(79,206,86,199)(80,207,87,200)(91,229,109,211)(92,230,110,212)(93,221,101,213)(94,222,102,214)(95,223,103,215)(96,224,104,216)(97,225,105,217)(98,226,106,218)(99,227,107,219)(100,228,108,220)(151,288,168,271)(152,289,169,272)(153,290,170,273)(154,281,161,274)(155,282,162,275)(156,283,163,276)(157,284,164,277)(158,285,165,278)(159,286,166,279)(160,287,167,280)(171,309,189,291)(172,310,190,292)(173,301,181,293)(174,302,182,294)(175,303,183,295)(176,304,184,296)(177,305,185,297)(178,306,186,298)(179,307,187,299)(180,308,188,300)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,285)(12,286)(13,287)(14,288)(15,289)(16,290)(17,281)(18,282)(19,283)(20,284)(21,303)(22,304)(23,305)(24,306)(25,307)(26,308)(27,309)(28,310)(29,301)(30,302)(31,278)(32,279)(33,280)(34,271)(35,272)(36,273)(37,274)(38,275)(39,276)(40,277)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,91)(69,92)(70,93)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,171)(120,172)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,181)(129,182)(130,183)(131,159)(132,160)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158)(141,168)(142,169)(143,170)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(191,253)(192,254)(193,255)(194,256)(195,257)(196,258)(197,259)(198,260)(199,251)(200,252)(201,264)(202,265)(203,266)(204,267)(205,268)(206,269)(207,270)(208,261)(209,262)(210,263)(211,239)(212,240)(213,231)(214,232)(215,233)(216,234)(217,235)(218,236)(219,237)(220,238)(221,248)(222,249)(223,250)(224,241)(225,242)(226,243)(227,244)(228,245)(229,246)(230,247)(291,319)(292,320)(293,311)(294,312)(295,313)(296,314)(297,315)(298,316)(299,317)(300,318), (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), (1,161,63,171)(2,162,64,172)(3,163,65,173)(4,164,66,174)(5,165,67,175)(6,166,68,176)(7,167,69,177)(8,168,70,178)(9,169,61,179)(10,170,62,180)(11,210,313,220)(12,201,314,211)(13,202,315,212)(14,203,316,213)(15,204,317,214)(16,205,318,215)(17,206,319,216)(18,207,320,217)(19,208,311,218)(20,209,312,219)(21,228,31,193)(22,229,32,194)(23,230,33,195)(24,221,34,196)(25,222,35,197)(26,223,36,198)(27,224,37,199)(28,225,38,200)(29,226,39,191)(30,227,40,192)(41,189,51,154)(42,190,52,155)(43,181,53,156)(44,182,54,157)(45,183,55,158)(46,184,56,159)(47,185,57,160)(48,186,58,151)(49,187,59,152)(50,188,60,153)(71,138,106,128)(72,139,107,129)(73,140,108,130)(74,131,109,121)(75,132,110,122)(76,133,101,123)(77,134,102,124)(78,135,103,125)(79,136,104,126)(80,137,105,127)(81,149,91,114)(82,150,92,115)(83,141,93,116)(84,142,94,117)(85,143,95,118)(86,144,96,119)(87,145,97,120)(88,146,98,111)(89,147,99,112)(90,148,100,113)(231,288,266,298)(232,289,267,299)(233,290,268,300)(234,281,269,291)(235,282,270,292)(236,283,261,293)(237,284,262,294)(238,285,263,295)(239,286,264,296)(240,287,265,297)(241,274,251,309)(242,275,252,310)(243,276,253,301)(244,277,254,302)(245,278,255,303)(246,279,256,304)(247,280,257,305)(248,271,258,306)(249,272,259,307)(250,273,260,308), (1,251,51,269)(2,252,52,270)(3,253,53,261)(4,254,54,262)(5,255,55,263)(6,256,56,264)(7,257,57,265)(8,258,58,266)(9,259,59,267)(10,260,60,268)(11,148,31,140)(12,149,32,131)(13,150,33,132)(14,141,34,133)(15,142,35,134)(16,143,36,135)(17,144,37,136)(18,145,38,137)(19,146,39,138)(20,147,40,139)(21,130,313,113)(22,121,314,114)(23,122,315,115)(24,123,316,116)(25,124,317,117)(26,125,318,118)(27,126,319,119)(28,127,320,120)(29,128,311,111)(30,129,312,112)(41,234,63,241)(42,235,64,242)(43,236,65,243)(44,237,66,244)(45,238,67,245)(46,239,68,246)(47,240,69,247)(48,231,70,248)(49,232,61,249)(50,233,62,250)(71,208,88,191)(72,209,89,192)(73,210,90,193)(74,201,81,194)(75,202,82,195)(76,203,83,196)(77,204,84,197)(78,205,85,198)(79,206,86,199)(80,207,87,200)(91,229,109,211)(92,230,110,212)(93,221,101,213)(94,222,102,214)(95,223,103,215)(96,224,104,216)(97,225,105,217)(98,226,106,218)(99,227,107,219)(100,228,108,220)(151,288,168,271)(152,289,169,272)(153,290,170,273)(154,281,161,274)(155,282,162,275)(156,283,163,276)(157,284,164,277)(158,285,165,278)(159,286,166,279)(160,287,167,280)(171,309,189,291)(172,310,190,292)(173,301,181,293)(174,302,182,294)(175,303,183,295)(176,304,184,296)(177,305,185,297)(178,306,186,298)(179,307,187,299)(180,308,188,300) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,285),(12,286),(13,287),(14,288),(15,289),(16,290),(17,281),(18,282),(19,283),(20,284),(21,303),(22,304),(23,305),(24,306),(25,307),(26,308),(27,309),(28,310),(29,301),(30,302),(31,278),(32,279),(33,280),(34,271),(35,272),(36,273),(37,274),(38,275),(39,276),(40,277),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,101),(49,102),(50,103),(51,79),(52,80),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,91),(69,92),(70,93),(111,173),(112,174),(113,175),(114,176),(115,177),(116,178),(117,179),(118,180),(119,171),(120,172),(121,184),(122,185),(123,186),(124,187),(125,188),(126,189),(127,190),(128,181),(129,182),(130,183),(131,159),(132,160),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,157),(140,158),(141,168),(142,169),(143,170),(144,161),(145,162),(146,163),(147,164),(148,165),(149,166),(150,167),(191,253),(192,254),(193,255),(194,256),(195,257),(196,258),(197,259),(198,260),(199,251),(200,252),(201,264),(202,265),(203,266),(204,267),(205,268),(206,269),(207,270),(208,261),(209,262),(210,263),(211,239),(212,240),(213,231),(214,232),(215,233),(216,234),(217,235),(218,236),(219,237),(220,238),(221,248),(222,249),(223,250),(224,241),(225,242),(226,243),(227,244),(228,245),(229,246),(230,247),(291,319),(292,320),(293,311),(294,312),(295,313),(296,314),(297,315),(298,316),(299,317),(300,318)], [(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)], [(1,161,63,171),(2,162,64,172),(3,163,65,173),(4,164,66,174),(5,165,67,175),(6,166,68,176),(7,167,69,177),(8,168,70,178),(9,169,61,179),(10,170,62,180),(11,210,313,220),(12,201,314,211),(13,202,315,212),(14,203,316,213),(15,204,317,214),(16,205,318,215),(17,206,319,216),(18,207,320,217),(19,208,311,218),(20,209,312,219),(21,228,31,193),(22,229,32,194),(23,230,33,195),(24,221,34,196),(25,222,35,197),(26,223,36,198),(27,224,37,199),(28,225,38,200),(29,226,39,191),(30,227,40,192),(41,189,51,154),(42,190,52,155),(43,181,53,156),(44,182,54,157),(45,183,55,158),(46,184,56,159),(47,185,57,160),(48,186,58,151),(49,187,59,152),(50,188,60,153),(71,138,106,128),(72,139,107,129),(73,140,108,130),(74,131,109,121),(75,132,110,122),(76,133,101,123),(77,134,102,124),(78,135,103,125),(79,136,104,126),(80,137,105,127),(81,149,91,114),(82,150,92,115),(83,141,93,116),(84,142,94,117),(85,143,95,118),(86,144,96,119),(87,145,97,120),(88,146,98,111),(89,147,99,112),(90,148,100,113),(231,288,266,298),(232,289,267,299),(233,290,268,300),(234,281,269,291),(235,282,270,292),(236,283,261,293),(237,284,262,294),(238,285,263,295),(239,286,264,296),(240,287,265,297),(241,274,251,309),(242,275,252,310),(243,276,253,301),(244,277,254,302),(245,278,255,303),(246,279,256,304),(247,280,257,305),(248,271,258,306),(249,272,259,307),(250,273,260,308)], [(1,251,51,269),(2,252,52,270),(3,253,53,261),(4,254,54,262),(5,255,55,263),(6,256,56,264),(7,257,57,265),(8,258,58,266),(9,259,59,267),(10,260,60,268),(11,148,31,140),(12,149,32,131),(13,150,33,132),(14,141,34,133),(15,142,35,134),(16,143,36,135),(17,144,37,136),(18,145,38,137),(19,146,39,138),(20,147,40,139),(21,130,313,113),(22,121,314,114),(23,122,315,115),(24,123,316,116),(25,124,317,117),(26,125,318,118),(27,126,319,119),(28,127,320,120),(29,128,311,111),(30,129,312,112),(41,234,63,241),(42,235,64,242),(43,236,65,243),(44,237,66,244),(45,238,67,245),(46,239,68,246),(47,240,69,247),(48,231,70,248),(49,232,61,249),(50,233,62,250),(71,208,88,191),(72,209,89,192),(73,210,90,193),(74,201,81,194),(75,202,82,195),(76,203,83,196),(77,204,84,197),(78,205,85,198),(79,206,86,199),(80,207,87,200),(91,229,109,211),(92,230,110,212),(93,221,101,213),(94,222,102,214),(95,223,103,215),(96,224,104,216),(97,225,105,217),(98,226,106,218),(99,227,107,219),(100,228,108,220),(151,288,168,271),(152,289,169,272),(153,290,170,273),(154,281,161,274),(155,282,162,275),(156,283,163,276),(157,284,164,277),(158,285,165,278),(159,286,166,279),(160,287,167,280),(171,309,189,291),(172,310,190,292),(173,301,181,293),(174,302,182,294),(175,303,183,295),(176,304,184,296),(177,305,185,297),(178,306,186,298),(179,307,187,299),(180,308,188,300)]])
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