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## G = C20.39C42order 320 = 26·5

### 2nd non-split extension by C20 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.39C42
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C2×C4⋊Dic5 — C20.39C42
 Lower central C5 — C10 — C20 — C20.39C42
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C20.39C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, ac=ca, cbc-1=a15b >

Subgroups: 406 in 114 conjugacy classes, 67 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C4⋊C4, C22×C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C22.4Q16, C4⋊Dic5, C4⋊Dic5, C2×C40, C2×C40, C22×Dic5, C22×C20, C2×C4⋊Dic5, C22×C40, C20.39C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, Dic5, D10, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C22.4Q16, C40⋊C2, D40, Dic20, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C20.44D4, C406C4, C405C4, D205C4, C10.10C42, C20.39C42

Smallest permutation representation of C20.39C42
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)
(1 154 92 38)(2 153 93 37)(3 152 94 36)(4 151 95 35)(5 150 96 34)(6 149 97 33)(7 148 98 32)(8 147 99 31)(9 146 100 30)(10 145 81 29)(11 144 82 28)(12 143 83 27)(13 142 84 26)(14 141 85 25)(15 160 86 24)(16 159 87 23)(17 158 88 22)(18 157 89 21)(19 156 90 40)(20 155 91 39)(41 112 133 69)(42 111 134 68)(43 110 135 67)(44 109 136 66)(45 108 137 65)(46 107 138 64)(47 106 139 63)(48 105 140 62)(49 104 121 61)(50 103 122 80)(51 102 123 79)(52 101 124 78)(53 120 125 77)(54 119 126 76)(55 118 127 75)(56 117 128 74)(57 116 129 73)(58 115 130 72)(59 114 131 71)(60 113 132 70)(161 203 254 185)(162 202 255 184)(163 201 256 183)(164 220 257 182)(165 219 258 181)(166 218 259 200)(167 217 260 199)(168 216 241 198)(169 215 242 197)(170 214 243 196)(171 213 244 195)(172 212 245 194)(173 211 246 193)(174 210 247 192)(175 209 248 191)(176 208 249 190)(177 207 250 189)(178 206 251 188)(179 205 252 187)(180 204 253 186)(221 315 299 262)(222 314 300 261)(223 313 281 280)(224 312 282 279)(225 311 283 278)(226 310 284 277)(227 309 285 276)(228 308 286 275)(229 307 287 274)(230 306 288 273)(231 305 289 272)(232 304 290 271)(233 303 291 270)(234 302 292 269)(235 301 293 268)(236 320 294 267)(237 319 295 266)(238 318 296 265)(239 317 297 264)(240 316 298 263)
(1 308 251 79 11 318 241 69)(2 309 252 80 12 319 242 70)(3 310 253 61 13 320 243 71)(4 311 254 62 14 301 244 72)(5 312 255 63 15 302 245 73)(6 313 256 64 16 303 246 74)(7 314 257 65 17 304 247 75)(8 315 258 66 18 305 248 76)(9 316 259 67 19 306 249 77)(10 317 260 68 20 307 250 78)(21 226 219 131 31 236 209 121)(22 227 220 132 32 237 210 122)(23 228 201 133 33 238 211 123)(24 229 202 134 34 239 212 124)(25 230 203 135 35 240 213 125)(26 231 204 136 36 221 214 126)(27 232 205 137 37 222 215 127)(28 233 206 138 38 223 216 128)(29 234 207 139 39 224 217 129)(30 235 208 140 40 225 218 130)(41 149 296 193 51 159 286 183)(42 150 297 194 52 160 287 184)(43 151 298 195 53 141 288 185)(44 152 299 196 54 142 289 186)(45 153 300 197 55 143 290 187)(46 154 281 198 56 144 291 188)(47 155 282 199 57 145 292 189)(48 156 283 200 58 146 293 190)(49 157 284 181 59 147 294 191)(50 158 285 182 60 148 295 192)(81 264 167 111 91 274 177 101)(82 265 168 112 92 275 178 102)(83 266 169 113 93 276 179 103)(84 267 170 114 94 277 180 104)(85 268 171 115 95 278 161 105)(86 269 172 116 96 279 162 106)(87 270 173 117 97 280 163 107)(88 271 174 118 98 261 164 108)(89 272 175 119 99 262 165 109)(90 273 176 120 100 263 166 110)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 2 2 2 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + + - - + - + + - image C1 C2 C2 C4 C4 D4 Q8 D4 D5 D8 SD16 Q16 Dic5 D10 Dic10 C4×D5 C5⋊D4 D20 C40⋊C2 D40 Dic20 kernel C20.39C42 C2×C4⋊Dic5 C22×C40 C4⋊Dic5 C2×C40 C2×C20 C2×C20 C22×C10 C22×C8 C2×C10 C2×C10 C2×C10 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 C22 # reps 1 2 1 8 4 2 1 1 2 2 4 2 4 2 4 8 8 4 16 8 8

Matrix representation of C20.39C42 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 27 11 0 0 30 32
,
 1 0 0 0 0 9 0 0 0 0 6 39 0 0 38 35
,
 9 0 0 0 0 32 0 0 0 0 21 26 0 0 15 3
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,27,30,0,0,11,32],[1,0,0,0,0,9,0,0,0,0,6,38,0,0,39,35],[9,0,0,0,0,32,0,0,0,0,21,15,0,0,26,3] >;

C20.39C42 in GAP, Magma, Sage, TeX

C_{20}._{39}C_4^2
% in TeX

G:=Group("C20.39C4^2");
// GroupNames label

G:=SmallGroup(320,109);
// by ID

G=gap.SmallGroup(320,109);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations

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