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G = C20.31C42order 320 = 26·5

1st non-split extension by C20 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.31C42, C4⋊C43Dic5, (C2×C20).4Q8, C4⋊Dic517C4, (C2×C10).35D8, C20.37(C4⋊C4), C4.1(C4×Dic5), (C2×C20).102D4, (C2×C4).125D20, C4.1(C4⋊Dic5), (C2×C10).14Q16, C10.9(C4.Q8), C53(C22.4Q16), (C2×C10).38SD16, (C2×C4).23Dic10, C10.12(C2.D8), C22.9(Q8⋊D5), C2.2(D206C4), C20.54(C22⋊C4), C22.16(D4⋊D5), C2.2(C10.Q16), C2.1(Q8⋊Dic5), C4.6(C10.D4), C2.1(D4⋊Dic5), C2.2(C10.D8), (C22×C10).178D4, (C22×C4).323D10, C23.94(C5⋊D4), C22.9(D4.D5), C2.2(C20.Q8), C4.31(D10⋊C4), C10.36(D4⋊C4), C22.6(C5⋊Q16), C10.11(Q8⋊C4), (C22×C20).118C22, C22.26(C23.D5), C22.38(D10⋊C4), C2.6(C10.10C42), C10.24(C2.C42), C22.20(C10.D4), (C5×C4⋊C4)⋊12C4, (C2×C52C8)⋊6C4, (C2×C4⋊C4).2D5, (C10×C4⋊C4).1C2, (C2×C4).138(C4×D5), (C2×C10).63(C4⋊C4), (C2×C20).227(C2×C4), (C2×C4⋊Dic5).28C2, (C22×C52C8).1C2, (C2×C4).35(C2×Dic5), (C2×C4).174(C5⋊D4), (C2×C10).151(C22⋊C4), SmallGroup(320,87)

Series: Derived Chief Lower central Upper central

C1C20 — C20.31C42
C1C5C10C2×C10C2×C20C22×C20C22×C52C8 — C20.31C42
C5C10C20 — C20.31C42
C1C23C22×C4C2×C4⋊C4

Generators and relations for C20.31C42
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a9, cac-1=a11, cbc-1=a15b >

Subgroups: 342 in 114 conjugacy classes, 67 normal (59 characteristic)
C1, C2 [×7], C4 [×4], C4 [×4], C22 [×7], C5, C8 [×2], C2×C4 [×6], C2×C4 [×8], C23, C10 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], Dic5 [×2], C20 [×4], C20 [×2], C2×C10 [×7], C2×C4⋊C4, C2×C4⋊C4, C22×C8, C52C8 [×2], C2×Dic5 [×4], C2×C20 [×6], C2×C20 [×4], C22×C10, C22.4Q16, C2×C52C8 [×2], C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C5×C4⋊C4 [×2], C5×C4⋊C4, C22×Dic5, C22×C20, C22×C20, C22×C52C8, C2×C4⋊Dic5, C10×C4⋊C4, C20.31C42
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, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C23.D5, C10.D8, C20.Q8, D206C4, C10.Q16, C10.10C42, D4⋊Dic5, Q8⋊Dic5, C20.31C42

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

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10N20A···20X
order12···2444444444444558···810···1020···20
size11···122224444202020202210···102···24···4

68 irreducible representations

dim1111111222222222222224444
type+++++-+++--+-++-+-
imageC1C2C2C2C4C4C4D4Q8D4D5D8SD16Q16Dic5D10Dic10C4×D5D20C5⋊D4C5⋊D4D4⋊D5D4.D5Q8⋊D5C5⋊Q16
kernelC20.31C42C22×C52C8C2×C4⋊Dic5C10×C4⋊C4C2×C52C8C4⋊Dic5C5×C4⋊C4C2×C20C2×C20C22×C10C2×C4⋊C4C2×C10C2×C10C2×C10C4⋊C4C22×C4C2×C4C2×C4C2×C4C2×C4C23C22C22C22C22
# reps1111444211224242484442222

Matrix representation of C20.31C42 in GL6(𝔽41)

6400000
100000
001100
00333400
000012
00004040
,
28280000
32130000
00141000
0092700
00003030
0000260
,
100000
010000
00111300
00193000
00001111
00001530

G:=sub<GL(6,GF(41))| [6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,33,0,0,0,0,1,34,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[28,32,0,0,0,0,28,13,0,0,0,0,0,0,14,9,0,0,0,0,10,27,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,19,0,0,0,0,13,30,0,0,0,0,0,0,11,15,0,0,0,0,11,30] >;

C20.31C42 in GAP, Magma, Sage, TeX

C_{20}._{31}C_4^2
% in TeX

G:=Group("C20.31C4^2");
// GroupNames label

G:=SmallGroup(320,87);
// by ID

G=gap.SmallGroup(320,87);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,570,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

׿
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𝔽