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G = C20.39C42order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.39C42, C23.54D20, C22.11D40, C22.4Dic20, (C2×C40)⋊16C4, (C2×C8)⋊4Dic5, C4⋊Dic513C4, (C2×C10).17D8, C20.61(C4⋊C4), (C2×C20).51Q8, (C2×C10).8Q16, C4.9(C4×Dic5), (C22×C8).3D5, (C2×C20).467D4, (C22×C40).2C2, C2.2(C406C4), C2.2(C405C4), C54(C22.4Q16), (C2×C10).19SD16, (C2×C4).43Dic10, C10.12(C4.Q8), C10.17(C2.D8), C2.2(D205C4), (C22×C10).133D4, (C22×C4).413D10, C22.9(C40⋊C2), C4.15(C23.D5), C10.35(D4⋊C4), C20.132(C22⋊C4), C10.16(Q8⋊C4), C4.23(C10.D4), C2.2(C20.44D4), C22.18(C4⋊Dic5), (C22×C20).512C22, C22.40(D10⋊C4), C10.30(C2.C42), C2.11(C10.10C42), (C2×C4).99(C4×D5), (C2×C4⋊Dic5).1C2, (C2×C10).66(C4⋊C4), (C2×C20).466(C2×C4), (C2×C4).73(C2×Dic5), (C2×C4).233(C5⋊D4), (C2×C10).116(C22⋊C4), SmallGroup(320,109)

Series: Derived Chief Lower central Upper central

C1C20 — C20.39C42
C1C5C10C2×C10C2×C20C22×C20C2×C4⋊Dic5 — C20.39C42
C5C10C20 — C20.39C42
C1C23C22×C4C22×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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22 [×3], C22 [×4], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C10 [×3], C10 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×2], C20 [×2], C2×C10 [×3], C2×C10 [×4], C2×C4⋊C4 [×2], C22×C8, C40 [×2], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×4], C22×C10, C22.4Q16, C4⋊Dic5 [×4], C4⋊Dic5 [×2], C2×C40 [×2], C2×C40 [×2], C22×Dic5 [×2], C22×C20, C2×C4⋊Dic5 [×2], C22×C40, C20.39C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, Dic5 [×2], D10, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C22.4Q16, C40⋊C2 [×2], D40, Dic20, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C20.44D4 [×2], C406C4, C405C4, D205C4 [×2], 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 57 267 260)(2 56 268 259)(3 55 269 258)(4 54 270 257)(5 53 271 256)(6 52 272 255)(7 51 273 254)(8 50 274 253)(9 49 275 252)(10 48 276 251)(11 47 277 250)(12 46 278 249)(13 45 279 248)(14 44 280 247)(15 43 261 246)(16 42 262 245)(17 41 263 244)(18 60 264 243)(19 59 265 242)(20 58 266 241)(21 305 237 109)(22 304 238 108)(23 303 239 107)(24 302 240 106)(25 301 221 105)(26 320 222 104)(27 319 223 103)(28 318 224 102)(29 317 225 101)(30 316 226 120)(31 315 227 119)(32 314 228 118)(33 313 229 117)(34 312 230 116)(35 311 231 115)(36 310 232 114)(37 309 233 113)(38 308 234 112)(39 307 235 111)(40 306 236 110)(61 122 216 197)(62 121 217 196)(63 140 218 195)(64 139 219 194)(65 138 220 193)(66 137 201 192)(67 136 202 191)(68 135 203 190)(69 134 204 189)(70 133 205 188)(71 132 206 187)(72 131 207 186)(73 130 208 185)(74 129 209 184)(75 128 210 183)(76 127 211 182)(77 126 212 181)(78 125 213 200)(79 124 214 199)(80 123 215 198)(81 160 284 172)(82 159 285 171)(83 158 286 170)(84 157 287 169)(85 156 288 168)(86 155 289 167)(87 154 290 166)(88 153 291 165)(89 152 292 164)(90 151 293 163)(91 150 294 162)(92 149 295 161)(93 148 296 180)(94 147 297 179)(95 146 298 178)(96 145 299 177)(97 144 300 176)(98 143 281 175)(99 142 282 174)(100 141 283 173)
(1 313 188 152 11 303 198 142)(2 314 189 153 12 304 199 143)(3 315 190 154 13 305 200 144)(4 316 191 155 14 306 181 145)(5 317 192 156 15 307 182 146)(6 318 193 157 16 308 183 147)(7 319 194 158 17 309 184 148)(8 320 195 159 18 310 185 149)(9 301 196 160 19 311 186 150)(10 302 197 141 20 312 187 151)(21 208 87 243 31 218 97 253)(22 209 88 244 32 219 98 254)(23 210 89 245 33 220 99 255)(24 211 90 246 34 201 100 256)(25 212 91 247 35 202 81 257)(26 213 92 248 36 203 82 258)(27 214 93 249 37 204 83 259)(28 215 94 250 38 205 84 260)(29 216 95 251 39 206 85 241)(30 217 96 252 40 207 86 242)(41 228 64 281 51 238 74 291)(42 229 65 282 52 239 75 292)(43 230 66 283 53 240 76 293)(44 231 67 284 54 221 77 294)(45 232 68 285 55 222 78 295)(46 233 69 286 56 223 79 296)(47 234 70 287 57 224 80 297)(48 235 71 288 58 225 61 298)(49 236 72 289 59 226 62 299)(50 237 73 290 60 227 63 300)(101 137 168 261 111 127 178 271)(102 138 169 262 112 128 179 272)(103 139 170 263 113 129 180 273)(104 140 171 264 114 130 161 274)(105 121 172 265 115 131 162 275)(106 122 173 266 116 132 163 276)(107 123 174 267 117 133 164 277)(108 124 175 268 118 134 165 278)(109 125 176 269 119 135 166 279)(110 126 177 270 120 136 167 280)

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

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

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

92 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L5A5B8A···8H10A···10N20A···20P40A···40AF
order12···244444···4558···810···1020···2040···40
size11···1222220···20222···22···22···22···2

92 irreducible representations

dim111112222222222222222
type++++-+++--+-++-
imageC1C2C2C4C4D4Q8D4D5D8SD16Q16Dic5D10Dic10C4×D5C5⋊D4D20C40⋊C2D40Dic20
kernelC20.39C42C2×C4⋊Dic5C22×C40C4⋊Dic5C2×C40C2×C20C2×C20C22×C10C22×C8C2×C10C2×C10C2×C10C2×C8C22×C4C2×C4C2×C4C2×C4C23C22C22C22
# reps1218421122424248841688

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

1000
0100
002711
003032
,
1000
0900
00639
003835
,
9000
03200
002126
00153
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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𝔽