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G = C42.2D10order 320 = 26·5

2nd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.2D10, C10.18C4≀C2, C8⋊C4.3D5, C4⋊Dic5.1C4, (C2×C20).224D4, (C2×C4).106D20, (C4×C20).11C22, C20.6Q8.4C2, C2.7(D204C4), C2.5(D207C4), C42.D5.1C2, C2.3(C4.12D20), C10.6(C4.10D4), C52(C42.2C22), C22.57(D10⋊C4), (C2×C4).11(C4×D5), (C5×C8⋊C4).7C2, (C2×C20).196(C2×C4), (C2×C4).207(C5⋊D4), (C2×C10).102(C22⋊C4), SmallGroup(320,23)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.2D10
C1C5C10C2×C10C2×C20C4×C20C20.6Q8 — C42.2D10
C5C2×C10C2×C20 — C42.2D10
C1C22C42C8⋊C4

Generators and relations for C42.2D10
 G = < a,b,c,d | a4=b4=1, c10=a, d2=ba=ab, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a2bc9 >

Subgroups: 206 in 60 conjugacy classes, 25 normal (all characteristic)
C1, C2 [×3], C4 [×5], C22, C5, C8 [×4], C2×C4 [×3], C2×C4 [×2], C10 [×3], C42, C4⋊C4 [×4], C2×C8 [×2], Dic5 [×2], C20 [×3], C2×C10, C8⋊C4, C8⋊C4, C42.C2, C52C8 [×2], C40 [×2], C2×Dic5 [×2], C2×C20 [×3], C42.2C22, C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C2×C40, C42.D5, C5×C8⋊C4, C20.6Q8, C42.2D10
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C4.10D4, C4≀C2 [×2], C4×D5, D20, C5⋊D4, C42.2C22, D10⋊C4, D204C4, C4.12D20, D207C4, C42.2D10

Smallest permutation representation of C42.2D10
Regular action on 320 points
Generators in S320
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)(161 171 181 191)(162 172 182 192)(163 173 183 193)(164 174 184 194)(165 175 185 195)(166 176 186 196)(167 177 187 197)(168 178 188 198)(169 179 189 199)(170 180 190 200)(201 211 221 231)(202 212 222 232)(203 213 223 233)(204 214 224 234)(205 215 225 235)(206 216 226 236)(207 217 227 237)(208 218 228 238)(209 219 229 239)(210 220 230 240)(241 251 261 271)(242 252 262 272)(243 253 263 273)(244 254 264 274)(245 255 265 275)(246 256 266 276)(247 257 267 277)(248 258 268 278)(249 259 269 279)(250 260 270 280)(281 291 301 311)(282 292 302 312)(283 293 303 313)(284 294 304 314)(285 295 305 315)(286 296 306 316)(287 297 307 317)(288 298 308 318)(289 299 309 319)(290 300 310 320)
(1 167 204 295)(2 188 205 316)(3 169 206 297)(4 190 207 318)(5 171 208 299)(6 192 209 320)(7 173 210 301)(8 194 211 282)(9 175 212 303)(10 196 213 284)(11 177 214 305)(12 198 215 286)(13 179 216 307)(14 200 217 288)(15 181 218 309)(16 162 219 290)(17 183 220 311)(18 164 221 292)(19 185 222 313)(20 166 223 294)(21 187 224 315)(22 168 225 296)(23 189 226 317)(24 170 227 298)(25 191 228 319)(26 172 229 300)(27 193 230 281)(28 174 231 302)(29 195 232 283)(30 176 233 304)(31 197 234 285)(32 178 235 306)(33 199 236 287)(34 180 237 308)(35 161 238 289)(36 182 239 310)(37 163 240 291)(38 184 201 312)(39 165 202 293)(40 186 203 314)(41 150 112 258)(42 131 113 279)(43 152 114 260)(44 133 115 241)(45 154 116 262)(46 135 117 243)(47 156 118 264)(48 137 119 245)(49 158 120 266)(50 139 81 247)(51 160 82 268)(52 141 83 249)(53 122 84 270)(54 143 85 251)(55 124 86 272)(56 145 87 253)(57 126 88 274)(58 147 89 255)(59 128 90 276)(60 149 91 257)(61 130 92 278)(62 151 93 259)(63 132 94 280)(64 153 95 261)(65 134 96 242)(66 155 97 263)(67 136 98 244)(68 157 99 265)(69 138 100 246)(70 159 101 267)(71 140 102 248)(72 121 103 269)(73 142 104 250)(74 123 105 271)(75 144 106 252)(76 125 107 273)(77 146 108 254)(78 127 109 275)(79 148 110 256)(80 129 111 277)
(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)
(1 147 177 99 224 275 285 48)(2 98 198 254 225 47 306 126)(3 253 179 66 226 125 287 117)(4 65 200 144 227 116 308 272)(5 143 181 95 228 271 289 44)(6 94 162 250 229 43 310 122)(7 249 183 62 230 121 291 113)(8 61 164 140 231 112 312 268)(9 139 185 91 232 267 293 80)(10 90 166 246 233 79 314 158)(11 245 187 58 234 157 295 109)(12 57 168 136 235 108 316 264)(13 135 189 87 236 263 297 76)(14 86 170 242 237 75 318 154)(15 241 191 54 238 153 299 105)(16 53 172 132 239 104 320 260)(17 131 193 83 240 259 301 72)(18 82 174 278 201 71 282 150)(19 277 195 50 202 149 303 101)(20 49 176 128 203 100 284 256)(21 127 197 119 204 255 305 68)(22 118 178 274 205 67 286 146)(23 273 199 46 206 145 307 97)(24 45 180 124 207 96 288 252)(25 123 161 115 208 251 309 64)(26 114 182 270 209 63 290 142)(27 269 163 42 210 141 311 93)(28 41 184 160 211 92 292 248)(29 159 165 111 212 247 313 60)(30 110 186 266 213 59 294 138)(31 265 167 78 214 137 315 89)(32 77 188 156 215 88 296 244)(33 155 169 107 216 243 317 56)(34 106 190 262 217 55 298 134)(35 261 171 74 218 133 319 85)(36 73 192 152 219 84 300 280)(37 151 173 103 220 279 281 52)(38 102 194 258 221 51 302 130)(39 257 175 70 222 129 283 81)(40 69 196 148 223 120 304 276)

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

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

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

59 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order12224444444558888888810···1020···2020···2040···40
size1111222244040224444202020202···22···24···44···4

59 irreducible representations

dim1111122222222444
type++++++++--
imageC1C2C2C2C4D4D5D10C4≀C2C4×D5D20C5⋊D4D204C4C4.10D4C4.12D20D207C4
kernelC42.2D10C42.D5C5×C8⋊C4C20.6Q8C4⋊Dic5C2×C20C8⋊C4C42C10C2×C4C2×C4C2×C4C2C10C2C2
# reps11114222844416144

Matrix representation of C42.2D10 in GL4(𝔽41) generated by

9000
0900
00241
004017
,
13900
14000
00119
003230
,
241300
101700
00623
00189
,
03300
43300
0003
0030
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,24,40,0,0,1,17],[1,1,0,0,39,40,0,0,0,0,11,32,0,0,9,30],[24,10,0,0,13,17,0,0,0,0,6,18,0,0,23,9],[0,4,0,0,33,33,0,0,0,0,0,3,0,0,3,0] >;

C42.2D10 in GAP, Magma, Sage, TeX

C_4^2._2D_{10}
% in TeX

G:=Group("C4^2.2D10");
// GroupNames label

G:=SmallGroup(320,23);
// by ID

G=gap.SmallGroup(320,23);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,1571,570,192,12550]);
// Polycyclic

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

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