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G = (C22×C4).D10order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C22×C4).14D10, C2.5(C422D5), C5⋊(C23.84C23), C2.C42.5D5, C22.88(C4○D20), (C22×C20).11C22, C23.358(C22×D5), C10.19(C422C2), C22.87(D42D5), C10.10C42.8C2, (C22×C10).287C23, C22.43(Q82D5), 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

C1C22×C10 — (C22×C4).D10
C1C5C10C2×C10C22×C10C22×Dic5C10.10C42 — (C22×C4).D10
C5C22×C10 — (C22×C4).D10
C1C23C2.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 [×6], C4 [×7], C22, C22 [×6], C5, C2×C4 [×21], C23, C10, C10 [×6], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], C20 [×3], C2×C10, C2×C10 [×6], C2.C42, C2.C42 [×6], C2×Dic5 [×12], C2×C20 [×9], C22×C10, C23.84C23, C22×Dic5, C22×Dic5 [×3], C22×C20 [×3], C10.10C42 [×6], C5×C2.C42, (C22×C4).D10
Quotients: C1, C2 [×7], C22 [×7], C23, D5, C4○D4 [×7], D10 [×3], C422C2 [×7], C22×D5, C23.84C23, C4○D20 [×3], D42D5 [×3], Q82D5, C422D5, C23.D10 [×3], C4⋊C4⋊D5 [×3], (C22×C4).D10

Smallest permutation representation of (C22×C4).D10
Regular action on 320 points
Generators in S320
(1 274)(2 275)(3 276)(4 277)(5 278)(6 279)(7 280)(8 261)(9 262)(10 263)(11 264)(12 265)(13 266)(14 267)(15 268)(16 269)(17 270)(18 271)(19 272)(20 273)(21 257)(22 258)(23 259)(24 260)(25 241)(26 242)(27 243)(28 244)(29 245)(30 246)(31 247)(32 248)(33 249)(34 250)(35 251)(36 252)(37 253)(38 254)(39 255)(40 256)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 101)(58 102)(59 103)(60 104)(61 299)(62 300)(63 281)(64 282)(65 283)(66 284)(67 285)(68 286)(69 287)(70 288)(71 289)(72 290)(73 291)(74 292)(75 293)(76 294)(77 295)(78 296)(79 297)(80 298)(81 220)(82 201)(83 202)(84 203)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 211)(93 212)(94 213)(95 214)(96 215)(97 216)(98 217)(99 218)(100 219)(121 198)(122 199)(123 200)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 196)(140 197)(141 239)(142 240)(143 221)(144 222)(145 223)(146 224)(147 225)(148 226)(149 227)(150 228)(151 229)(152 230)(153 231)(154 232)(155 233)(156 234)(157 235)(158 236)(159 237)(160 238)(161 317)(162 318)(163 319)(164 320)(165 301)(166 302)(167 303)(168 304)(169 305)(170 306)(171 307)(172 308)(173 309)(174 310)(175 311)(176 312)(177 313)(178 314)(179 315)(180 316)
(1 264)(2 265)(3 266)(4 267)(5 268)(6 269)(7 270)(8 271)(9 272)(10 273)(11 274)(12 275)(13 276)(14 277)(15 278)(16 279)(17 280)(18 261)(19 262)(20 263)(21 247)(22 248)(23 249)(24 250)(25 251)(26 252)(27 253)(28 254)(29 255)(30 256)(31 257)(32 258)(33 259)(34 260)(35 241)(36 242)(37 243)(38 244)(39 245)(40 246)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 289)(62 290)(63 291)(64 292)(65 293)(66 294)(67 295)(68 296)(69 297)(70 298)(71 299)(72 300)(73 281)(74 282)(75 283)(76 284)(77 285)(78 286)(79 287)(80 288)(81 210)(82 211)(83 212)(84 213)(85 214)(86 215)(87 216)(88 217)(89 218)(90 219)(91 220)(92 201)(93 202)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 209)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 198)(132 199)(133 200)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)(141 229)(142 230)(143 231)(144 232)(145 233)(146 234)(147 235)(148 236)(149 237)(150 238)(151 239)(152 240)(153 221)(154 222)(155 223)(156 224)(157 225)(158 226)(159 227)(160 228)(161 307)(162 308)(163 309)(164 310)(165 311)(166 312)(167 313)(168 314)(169 315)(170 316)(171 317)(172 318)(173 319)(174 320)(175 301)(176 302)(177 303)(178 304)(179 305)(180 306)
(1 48 154 176)(2 103 155 303)(3 50 156 178)(4 105 157 305)(5 52 158 180)(6 107 159 307)(7 54 160 162)(8 109 141 309)(9 56 142 164)(10 111 143 311)(11 58 144 166)(12 113 145 313)(13 60 146 168)(14 115 147 315)(15 42 148 170)(16 117 149 317)(17 44 150 172)(18 119 151 319)(19 46 152 174)(20 101 153 301)(21 188 79 81)(22 122 80 211)(23 190 61 83)(24 124 62 213)(25 192 63 85)(26 126 64 215)(27 194 65 87)(28 128 66 217)(29 196 67 89)(30 130 68 219)(31 198 69 91)(32 132 70 201)(33 200 71 93)(34 134 72 203)(35 182 73 95)(36 136 74 205)(37 184 75 97)(38 138 76 207)(39 186 77 99)(40 140 78 209)(41 235 169 277)(43 237 171 279)(45 239 173 261)(47 221 175 263)(49 223 177 265)(51 225 179 267)(53 227 161 269)(55 229 163 271)(57 231 165 273)(59 233 167 275)(82 248 189 288)(84 250 191 290)(86 252 193 292)(88 254 195 294)(90 256 197 296)(92 258 199 298)(94 260 181 300)(96 242 183 282)(98 244 185 284)(100 246 187 286)(102 222 302 264)(104 224 304 266)(106 226 306 268)(108 228 308 270)(110 230 310 272)(112 232 312 274)(114 234 314 276)(116 236 316 278)(118 238 318 280)(120 240 320 262)(121 287 210 247)(123 289 212 249)(125 291 214 251)(127 293 216 253)(129 295 218 255)(131 297 220 257)(133 299 202 259)(135 281 204 241)(137 283 206 243)(139 285 208 245)
(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 31 222 297)(2 68 223 256)(3 29 224 295)(4 66 225 254)(5 27 226 293)(6 64 227 252)(7 25 228 291)(8 62 229 250)(9 23 230 289)(10 80 231 248)(11 21 232 287)(12 78 233 246)(13 39 234 285)(14 76 235 244)(15 37 236 283)(16 74 237 242)(17 35 238 281)(18 72 239 260)(19 33 240 299)(20 70 221 258)(22 273 288 143)(24 271 290 141)(26 269 292 159)(28 267 294 157)(30 265 296 155)(32 263 298 153)(34 261 300 151)(36 279 282 149)(38 277 284 147)(40 275 286 145)(41 88 315 128)(42 194 316 216)(43 86 317 126)(44 192 318 214)(45 84 319 124)(46 190 320 212)(47 82 301 122)(48 188 302 210)(49 100 303 140)(50 186 304 208)(51 98 305 138)(52 184 306 206)(53 96 307 136)(54 182 308 204)(55 94 309 134)(56 200 310 202)(57 92 311 132)(58 198 312 220)(59 90 313 130)(60 196 314 218)(61 272 249 142)(63 270 251 160)(65 268 253 158)(67 266 255 156)(69 264 257 154)(71 262 259 152)(73 280 241 150)(75 278 243 148)(77 276 245 146)(79 274 247 144)(81 102 121 176)(83 120 123 174)(85 118 125 172)(87 116 127 170)(89 114 129 168)(91 112 131 166)(93 110 133 164)(95 108 135 162)(97 106 137 180)(99 104 139 178)(101 211 175 189)(103 209 177 187)(105 207 179 185)(107 205 161 183)(109 203 163 181)(111 201 165 199)(113 219 167 197)(115 217 169 195)(117 215 171 193)(119 213 173 191)

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim111222244
type+++++-+
imageC1C2C2D5C4○D4D10C4○D20D42D5Q82D5
kernel(C22×C4).D10C10.10C42C5×C2.C42C2.C42C2×C10C22×C4C22C22C22
# reps16121462462

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

4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
000010
000001
,
900000
0320000
002600
0063900
0000186
00003523
,
3300000
0360000
000100
0040000
00003916
00002525
,
0360000
3300000
000900
0032000
0000040
0000400

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

׿
×
𝔽