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G = C10.C22≀C2  order 320 = 26·5

8th non-split extension by C10 of C22≀C2 acting via C22≀C2/C24=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic5)⋊3Q8, (C2×C20).305D4, C10.33(C4⋊Q8), C10.76C22≀C2, (C22×Q8).5D5, C22.52(Q8×D5), (C22×C4).158D10, C10.79(C22⋊Q8), C2.16(D10⋊3Q8), C2.10(C24⋊2D5), C23.386(C22×D5), C2.13(Dic5⋊Q8), (C22×C10).369C23, (C22×C20).397C22, C5⋊3(C23.78C23), C22.55(Q8⋊2D5), C10.10C42.39C2, (C22×Dic5).70C22, (Q8×C2×C10).10C2, (C2×C10).87(C2×Q8), (C2×C10).562(C2×D4), (C2×C4).88(C5⋊D4), C22.222(C2×C5⋊D4), (C2×C10).193(C4○D4), (C2×C10.D4).35C2, SmallGroup(320,856)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C10 — C10.C22≀C2
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C10.D4 — C10.C22≀C2
Lower central
C5 — C22×C10 — C10.C22≀C2
Upper central
C1 — C23 — C22×Q8

Generators and relations for C10.C22≀C2
 G = < a,b,c,d,e,f | a10=d2=e2=1, b2=c2=f2=a5, ab=ba, ac=ca, ad=da, ae=ea, faf-1=a-1, cbc-1=a5b, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 534 in 182 conjugacy classes, 69 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C4⋊C4, C22×Q8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C23.78C23, C10.D4, C22×Dic5, C22×Dic5, C22×C20, Q8×C10, C10.10C42, C2×C10.D4, Q8×C2×C10, C10.C22≀C2
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22≀C2, C22⋊Q8, C4⋊Q8, C5⋊D4, C22×D5, C23.78C23, Q8×D5, Q8⋊2D5, C2×C5⋊D4, Dic5⋊Q8, D10⋊3Q8, C24⋊2D5, C10.C22≀C2

Smallest permutation representation of C10.C22≀C2
►Regular action on 320 points
Generators in S320
(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 88 6 83)(2 89 7 84)(3 90 8 85)(4 81 9 86)(5 82 10 87)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)(161 256 166 251)(162 257 167 252)(163 258 168 253)(164 259 169 254)(165 260 170 255)(171 246 176 241)(172 247 177 242)(173 248 178 243)(174 249 179 244)(175 250 180 245)(181 276 186 271)(182 277 187 272)(183 278 188 273)(184 279 189 274)(185 280 190 275)(191 266 196 261)(192 267 197 262)(193 268 198 263)(194 269 199 264)(195 270 200 265)(201 296 206 291)(202 297 207 292)(203 298 208 293)(204 299 209 294)(205 300 210 295)(211 286 216 281)(212 287 217 282)(213 288 218 283)(214 289 219 284)(215 290 220 285)(221 316 226 311)(222 317 227 312)(223 318 228 313)(224 319 229 314)(225 320 230 315)(231 306 236 301)(232 307 237 302)(233 308 238 303)(234 309 239 304)(235 310 240 305)

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

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

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

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

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

dim111122222244
type++++-+++-+
imageC1C2C2C2Q8D4D5C4○D4D10C5⋊D4Q8×D5Q8⋊2D5
kernelC10.C22≀C2C10.10C42C2×C10.D4Q8×C2×C10C2×Dic5C2×C20C22×Q8C2×C10C22×C4C2×C4C22C22
# reps1331662262462

Matrix representation of C10.C22≀C2 ►in GL6(𝔽41)

770000
34400000
007700
00344000
0000400
0000040
,
17400000
1240000
0024100
00401700
0000203
0000321
,
4000000
0400000
00174000
0012400
000001
0000400
,
4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
630000
2350000
00353800
0039600
00002138
00003820

G:=sub<GL(6,GF(41))| [7,34,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[17,1,0,0,0,0,40,24,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,20,3,0,0,0,0,3,21],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,2,0,0,0,0,3,35,0,0,0,0,0,0,35,39,0,0,0,0,38,6,0,0,0,0,0,0,21,38,0,0,0,0,38,20] >;
 

C10.C22≀C2 in GAP, Magma, Sage, TeX 

C_{10}.C_2^2\wr C_2
 
% in TeX
 
G:=Group("C10.C2^2wrC2");
 
// GroupNames label
 
G:=SmallGroup(320,856);
 
// by ID
 
G=gap.SmallGroup(320,856);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,254,387,184,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^10=d^2=e^2=1,b^2=c^2=f^2=a^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^-1,c*b*c^-1=a^5*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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