(C2^2xC4).D10 - GroupNames

G = (C₂²×C₄).D₁₀ order 320 = 2⁶·5

14^th non-split extension by C₂²×C₄ of D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂²×C₄).₁₄D₁₀, C₂.₅(C₄²⋊₂D₅), C₅⋊(C₂³.₈₄C₂³), C₂.C₄².₅D₅, C₂².₈₈(C₄○D₂₀), (C₂²×C₂₀).₁₁C₂², C₂³.₃₅₈(C₂²×D₅), C₁₀.₁₉(C₄²⋊₂C₂), C₂².₈₇(D₄⋊₂D₅), C₁₀.₁₀C₄².₈C₂, (C₂²×C₁₀).₂₈₇C₂³, C₂².₄₃(Q₈⋊₂D₅), C₂.₁₁(C₂³.D₁₀), (C₂²×Dic₅).₁₂C₂², C₂.₉(C₄⋊C₄⋊D₅), (C₂×C₁₀).₁₃₀(C₄○D₄), (C₅×C₂.C₄²).₂C₂, SmallGroup(320,289)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — (C₂²×C₄).D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×Dic₅ — C₁₀.₁₀C₄² — (C₂²×C₄).D₁₀

Lower central

C₅ — C₂²×C₁₀ — (C₂²×C₄).D₁₀

Upper central

C₁ — C₂³ — C₂.C₄²

Generators and relations for (C₂²×C₄).D₁₀
G = < a,b,c,d,e | a²=b²=c⁴=1, d¹⁰=ba=ab, e²=bc², ac=ca, ad=da, ae=ea, dcd^-1=bc=cb, bd=db, be=eb, ece^-1=abc, ede^-1=abc²d⁹ >

Subgroups: 406 in 118 conjugacy classes, 51 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂³, C₁₀, C₁₀, C₂²×C₄, C₂²×C₄, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂.C₄², C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, C₂³.₈₄C₂³, C₂²×Dic₅, C₂²×Dic₅, C₂²×C₂₀, C₁₀.₁₀C₄², C₅×C₂.C₄², (C₂²×C₄).D₁₀
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄○D₄, D₁₀, C₄²⋊₂C₂, C₂²×D₅, C₂³.₈₄C₂³, C₄○D₂₀, D₄⋊₂D₅, Q₈⋊₂D₅, C₄²⋊₂D₅, C₂³.D₁₀, C₄⋊C₄⋊D₅, (C₂²×C₄).D₁₀

Smallest permutation representation of (C₂²×C₄).D₁₀
►Regular action on 320 points

Generators in S₃₂₀

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

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

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

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

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

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

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

62 conjugacy classes

class	1	2A	···	2G	4A	···	4F	4G	···	4N	5A	5B	10A	···	10N	20A	···	20X
order	1	2	···	2	4	···	4	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	4	···	4	20	···	20	2	2	2	···	2	4	···	4

62 irreducible representations

dim	1	1	1	2	2	2	2	4	4
type	+	+	+	+		+		-	+
image	C₁	C₂	C₂	D₅	C₄○D₄	D₁₀	C₄○D₂₀	D₄⋊₂D₅	Q₈⋊₂D₅
kernel	(C₂²×C₄).D₁₀	C₁₀.₁₀C₄²	C₅×C₂.C₄²	C₂.C₄²	C₂×C₁₀	C₂²×C₄	C₂²	C₂²	C₂²
# reps	1	6	1	2	14	6	24	6	2

Matrix representation of (C₂²×C₄).D₁₀ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	0	0	0	0	0
0	32	0	0	0	0
0	0	2	6	0	0
0	0	6	39	0	0
0	0	0	0	18	6
0	0	0	0	35	23

33	0	0	0	0	0
0	36	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	0	0	39	16
0	0	0	0	25	25

0	36	0	0	0	0
33	0	0	0	0	0
0	0	0	9	0	0
0	0	32	0	0	0
0	0	0	0	0	40
0	0	0	0	40	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,2,6,0,0,0,0,6,39,0,0,0,0,0,0,18,35,0,0,0,0,6,23],[33,0,0,0,0,0,0,36,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,39,25,0,0,0,0,16,25],[0,33,0,0,0,0,36,0,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

(C₂²×C₄).D₁₀ in GAP, Magma, Sage, TeX

(C_2^2\times C_4).D_{10}

% in TeX

G:=Group("(C2^2xC4).D10");

// GroupNames label

G:=SmallGroup(320,289);

// by ID

G=gap.SmallGroup(320,289);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,64,1262,387,268,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^10=b*a=a*b,e^2=b*c^2,a*c=c*a,a*d=d*a,a*e=e*a,d*c*d^-1=b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=a*b*c,e*d*e^-1=a*b*c^2*d^9>;

// generators/relations

G = (C22×C4).D10 order 320 = 26·5

14th non-split extension by C22×C4 of D10 acting via D10/C5=C22

G = (C₂²×C₄).D₁₀ order 320 = 2⁶·5

14^th non-split extension by C₂²×C₄ of D₁₀ acting via D₁₀/C₅=C₂²