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G = C22×C10.D4  order 320 = 26·5

Direct product of C22 and C10.D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C10.D4, C24.76D10, C23.18Dic10, (C23×C4).6D5, (C23×C20).9C2, C23.68(C4×D5), C10.58(C23×C4), Dic5⋊6(C22×C4), (C22×C10).27Q8, C10.18(C22×Q8), (C2×C20).702C23, (C2×C10).281C24, (C22×Dic5)⋊15C4, (C22×C4).411D10, C10.129(C22×D4), (C22×C10).203D4, (C23×Dic5).9C2, C2.2(C22×Dic10), C22.39(C23×D5), C23.102(C5⋊D4), C22.38(C2×Dic10), C23.333(C22×D5), (C22×C20).506C22, (C23×C10).103C22, (C22×C10).410C23, (C2×Dic5).286C23, (C22×Dic5).250C22, C10⋊3(C2×C4⋊C4), C5⋊3(C22×C4⋊C4), (C2×C10)⋊11(C4⋊C4), C22.78(C2×C4×D5), C2.37(D5×C22×C4), (C2×C10).53(C2×Q8), C2.1(C22×C5⋊D4), (C2×Dic5)⋊33(C2×C4), (C2×C10).570(C2×D4), C22.99(C2×C5⋊D4), (C2×C4).655(C22×D5), (C2×C10).258(C22×C4), (C22×C10).172(C2×C4), SmallGroup(320,1455)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C22×C10.D4
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C22×C10.D4
Lower central
C5 — C10 — C22×C10.D4

Subgroups: 1022 in 418 conjugacy classes, 247 normal (17 characteristic)
C1, C2 [×3], C2 [×12], C4 [×16], C22, C22 [×34], C5, C2×C4 [×4], C2×C4 [×56], C23 [×15], C10 [×3], C10 [×12], C4⋊C4 [×16], C22×C4 [×6], C22×C4 [×28], C24, Dic5 [×8], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×34], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C2×Dic5 [×32], C2×Dic5 [×12], C2×C20 [×4], C2×C20 [×12], C22×C10 [×15], C22×C4⋊C4, C10.D4 [×16], C22×Dic5 [×20], C22×Dic5 [×4], C22×C20 [×6], C22×C20 [×4], C23×C10, C2×C10.D4 [×12], C23×Dic5 [×2], C23×C20, C22×C10.D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], D5, C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, Dic10 [×4], C4×D5 [×4], C5⋊D4 [×4], C22×D5 [×7], C22×C4⋊C4, C10.D4 [×16], C2×Dic10 [×6], C2×C4×D5 [×6], C2×C5⋊D4 [×6], C23×D5, C2×C10.D4 [×12], C22×Dic10, D5×C22×C4, C22×C5⋊D4, C22×C10.D4

Generators and relations
 G = < a,b,c,d,e | a2=b2=c10=d4=1, e2=c5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=d-1 >

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

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

(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)

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
040000
004000
00010
00001
,
400000
01000
004000
000400
000040
,
400000
01000
00100
000135
00066
,
400000
040000
00100
000623
0001835
,
90000
01000
00100
000032
000320

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,6,0,0,0,35,6],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,6,18,0,0,0,23,35],[9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,32,0] >;
 

104 conjugacy classes

class 1 2A···2O4A···4H4I···4X5A5B10A···10AD20A···20AF
order12···24···44···45510···1020···20
size11···12···210···10222···22···2

104 irreducible representations

dim1111122222222
type+++++-+++-
imageC1C2C2C2C4D4Q8D5D10D10Dic10C4×D5C5⋊D4
kernelC22×C10.D4C2×C10.D4C23×Dic5C23×C20C22×Dic5C22×C10C22×C10C23×C4C22×C4C24C23C23C23
# reps1122116442122161616

In GAP, Magma, Sage, TeX 

C_2^2\times C_{10}.D_4
 
% in TeX
 
G:=Group("C2^2xC10.D4");
 
// GroupNames label
 
G:=SmallGroup(320,1455);
 
// by ID
 
G=gap.SmallGroup(320,1455);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,1123,80,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^4=1,e^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀