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## G = C22×C4×C20order 320 = 26·5

### Abelian group of type [2,2,4,20]

Aliases: C22×C4×C20, SmallGroup(320,1513)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C4×C20
G = < a,b,c,d | a2=b2=c4=d20=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 498, all normal (8 characteristic)
C1, C2 [×15], C4 [×24], C22, C22 [×34], C5, C2×C4 [×84], C23 [×15], C10 [×15], C42 [×16], C22×C4 [×42], C24, C20 [×24], C2×C10, C2×C10 [×34], C2×C42 [×12], C23×C4 [×3], C2×C20 [×84], C22×C10 [×15], C22×C42, C4×C20 [×16], C22×C20 [×42], C23×C10, C2×C4×C20 [×12], C23×C20 [×3], C22×C4×C20
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C5, C2×C4 [×84], C23 [×15], C10 [×15], C42 [×16], C22×C4 [×42], C24, C20 [×24], C2×C10 [×35], C2×C42 [×12], C23×C4 [×3], C2×C20 [×84], C22×C10 [×15], C22×C42, C4×C20 [×16], C22×C20 [×42], C23×C10, C2×C4×C20 [×12], C23×C20 [×3], C22×C4×C20

Smallest permutation representation of C22×C4×C20
Regular action on 320 points
Generators in S320
(1 259)(2 260)(3 241)(4 242)(5 243)(6 244)(7 245)(8 246)(9 247)(10 248)(11 249)(12 250)(13 251)(14 252)(15 253)(16 254)(17 255)(18 256)(19 257)(20 258)(21 226)(22 227)(23 228)(24 229)(25 230)(26 231)(27 232)(28 233)(29 234)(30 235)(31 236)(32 237)(33 238)(34 239)(35 240)(36 221)(37 222)(38 223)(39 224)(40 225)(41 294)(42 295)(43 296)(44 297)(45 298)(46 299)(47 300)(48 281)(49 282)(50 283)(51 284)(52 285)(53 286)(54 287)(55 288)(56 289)(57 290)(58 291)(59 292)(60 293)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 101)(80 102)(81 140)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 130)(92 131)(93 132)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)(141 276)(142 277)(143 278)(144 279)(145 280)(146 261)(147 262)(148 263)(149 264)(150 265)(151 266)(152 267)(153 268)(154 269)(155 270)(156 271)(157 272)(158 273)(159 274)(160 275)(161 315)(162 316)(163 317)(164 318)(165 319)(166 320)(167 301)(168 302)(169 303)(170 304)(171 305)(172 306)(173 307)(174 308)(175 309)(176 310)(177 311)(178 312)(179 313)(180 314)(181 213)(182 214)(183 215)(184 216)(185 217)(186 218)(187 219)(188 220)(189 201)(190 202)(191 203)(192 204)(193 205)(194 206)(195 207)(196 208)(197 209)(198 210)(199 211)(200 212)
(1 293)(2 294)(3 295)(4 296)(5 297)(6 298)(7 299)(8 300)(9 281)(10 282)(11 283)(12 284)(13 285)(14 286)(15 287)(16 288)(17 289)(18 290)(19 291)(20 292)(21 120)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 260)(42 241)(43 242)(44 243)(45 244)(46 245)(47 246)(48 247)(49 248)(50 249)(51 250)(52 251)(53 252)(54 253)(55 254)(56 255)(57 256)(58 257)(59 258)(60 259)(61 229)(62 230)(63 231)(64 232)(65 233)(66 234)(67 235)(68 236)(69 237)(70 238)(71 239)(72 240)(73 221)(74 222)(75 223)(76 224)(77 225)(78 226)(79 227)(80 228)(81 200)(82 181)(83 182)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 191)(93 192)(94 193)(95 194)(96 195)(97 196)(98 197)(99 198)(100 199)(121 213)(122 214)(123 215)(124 216)(125 217)(126 218)(127 219)(128 220)(129 201)(130 202)(131 203)(132 204)(133 205)(134 206)(135 207)(136 208)(137 209)(138 210)(139 211)(140 212)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 161)(154 162)(155 163)(156 164)(157 165)(158 166)(159 167)(160 168)(261 308)(262 309)(263 310)(264 311)(265 312)(266 313)(267 314)(268 315)(269 316)(270 317)(271 318)(272 319)(273 320)(274 301)(275 302)(276 303)(277 304)(278 305)(279 306)(280 307)
(1 139 171 79)(2 140 172 80)(3 121 173 61)(4 122 174 62)(5 123 175 63)(6 124 176 64)(7 125 177 65)(8 126 178 66)(9 127 179 67)(10 128 180 68)(11 129 161 69)(12 130 162 70)(13 131 163 71)(14 132 164 72)(15 133 165 73)(16 134 166 74)(17 135 167 75)(18 136 168 76)(19 137 169 77)(20 138 170 78)(21 59 198 277)(22 60 199 278)(23 41 200 279)(24 42 181 280)(25 43 182 261)(26 44 183 262)(27 45 184 263)(28 46 185 264)(29 47 186 265)(30 48 187 266)(31 49 188 267)(32 50 189 268)(33 51 190 269)(34 52 191 270)(35 53 192 271)(36 54 193 272)(37 55 194 273)(38 56 195 274)(39 57 196 275)(40 58 197 276)(81 306 102 260)(82 307 103 241)(83 308 104 242)(84 309 105 243)(85 310 106 244)(86 311 107 245)(87 312 108 246)(88 313 109 247)(89 314 110 248)(90 315 111 249)(91 316 112 250)(92 317 113 251)(93 318 114 252)(94 319 115 253)(95 320 116 254)(96 301 117 255)(97 302 118 256)(98 303 119 257)(99 304 120 258)(100 305 101 259)(141 225 291 209)(142 226 292 210)(143 227 293 211)(144 228 294 212)(145 229 295 213)(146 230 296 214)(147 231 297 215)(148 232 298 216)(149 233 299 217)(150 234 300 218)(151 235 281 219)(152 236 282 220)(153 237 283 201)(154 238 284 202)(155 239 285 203)(156 240 286 204)(157 221 287 205)(158 222 288 206)(159 223 289 207)(160 224 290 208)
(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)

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

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

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

320 conjugacy classes

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

320 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 kernel C22×C4×C20 C2×C4×C20 C23×C20 C22×C20 C22×C42 C2×C42 C23×C4 C22×C4 # reps 1 12 3 48 4 48 12 192

Matrix representation of C22×C4×C20 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 40 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 9 0 0 0 0 32 0 0 0 0 32
,
 39 0 0 0 0 8 0 0 0 0 39 0 0 0 0 20
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,9,0,0,0,0,32,0,0,0,0,32],[39,0,0,0,0,8,0,0,0,0,39,0,0,0,0,20] >;

C22×C4×C20 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times C_{20}
% in TeX

G:=Group("C2^2xC4xC20");
// GroupNames label

G:=SmallGroup(320,1513);
// by ID

G=gap.SmallGroup(320,1513);
# by ID

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

G:=Group<a,b,c,d|a^2=b^2=c^4=d^20=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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