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G = C20.M4(2)  order 320 = 26·5

5th non-split extension by C20 of M4(2) acting via M4(2)/C2=C2×C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.5M4(2), C5⋊C82Q8, C2.5(Q8×F5), C52(C84Q8), C4⋊C4.10F5, C10.3(C4×Q8), C20⋊C8.4C2, C4.3(C4.F5), C10.13(C8○D4), Dic5.30(C2×Q8), C10.D4.4C4, C2.14(D4.F5), (C2×Dic10).12C4, C10.16(C2×M4(2)), Dic5.69(C4○D4), Dic53Q8.17C2, C22.81(C22×F5), C10.C42.4C2, (C4×Dic5).193C22, (C2×Dic5).336C23, (C4×C5⋊C8).3C2, (C5×C4⋊C4).13C4, (C2×C4).29(C2×F5), C2.10(C2×C4.F5), (C2×C20).87(C2×C4), (C2×C5⋊C8).32C22, (C2×C10).47(C22×C4), (C2×Dic5).62(C2×C4), SmallGroup(320,1047)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — C20.M4(2)
C5C2×C10 — C20.M4(2)
C1C22C4⋊C4

Generators and relations for C20.M4(2)
 G = < a,b,c | a20=b8=1, c2=a10, bab-1=a7, cac-1=a-1, cbc-1=b5 >

Subgroups: 282 in 94 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×2], C10 [×3], C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×2], C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C84Q8, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×Dic10, C4×C5⋊C8, C20⋊C8, C20⋊C8 [×2], C10.C42 [×2], Dic53Q8, C20.M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, M4(2) [×2], C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C2×M4(2), C8○D4, C2×F5 [×3], C84Q8, C4.F5 [×2], C22×F5, C2×C4.F5, D4.F5, Q8×F5, C20.M4(2)

Smallest permutation representation of C20.M4(2)
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)
(1 163 151 68 54 246 83 102)(2 166 160 75 55 249 92 109)(3 169 149 62 56 252 81 116)(4 172 158 69 57 255 90 103)(5 175 147 76 58 258 99 110)(6 178 156 63 59 241 88 117)(7 161 145 70 60 244 97 104)(8 164 154 77 41 247 86 111)(9 167 143 64 42 250 95 118)(10 170 152 71 43 253 84 105)(11 173 141 78 44 256 93 112)(12 176 150 65 45 259 82 119)(13 179 159 72 46 242 91 106)(14 162 148 79 47 245 100 113)(15 165 157 66 48 248 89 120)(16 168 146 73 49 251 98 107)(17 171 155 80 50 254 87 114)(18 174 144 67 51 257 96 101)(19 177 153 74 52 260 85 108)(20 180 142 61 53 243 94 115)(21 261 204 199 137 230 304 284)(22 264 213 186 138 233 313 291)(23 267 202 193 139 236 302 298)(24 270 211 200 140 239 311 285)(25 273 220 187 121 222 320 292)(26 276 209 194 122 225 309 299)(27 279 218 181 123 228 318 286)(28 262 207 188 124 231 307 293)(29 265 216 195 125 234 316 300)(30 268 205 182 126 237 305 287)(31 271 214 189 127 240 314 294)(32 274 203 196 128 223 303 281)(33 277 212 183 129 226 312 288)(34 280 201 190 130 229 301 295)(35 263 210 197 131 232 310 282)(36 266 219 184 132 235 319 289)(37 269 208 191 133 238 308 296)(38 272 217 198 134 221 317 283)(39 275 206 185 135 224 306 290)(40 278 215 192 136 227 315 297)
(1 131 11 121)(2 130 12 140)(3 129 13 139)(4 128 14 138)(5 127 15 137)(6 126 16 136)(7 125 17 135)(8 124 18 134)(9 123 19 133)(10 122 20 132)(21 58 31 48)(22 57 32 47)(23 56 33 46)(24 55 34 45)(25 54 35 44)(26 53 36 43)(27 52 37 42)(28 51 38 41)(29 50 39 60)(30 49 40 59)(61 184 71 194)(62 183 72 193)(63 182 73 192)(64 181 74 191)(65 200 75 190)(66 199 76 189)(67 198 77 188)(68 197 78 187)(69 196 79 186)(70 195 80 185)(81 212 91 202)(82 211 92 201)(83 210 93 220)(84 209 94 219)(85 208 95 218)(86 207 96 217)(87 206 97 216)(88 205 98 215)(89 204 99 214)(90 203 100 213)(101 283 111 293)(102 282 112 292)(103 281 113 291)(104 300 114 290)(105 299 115 289)(106 298 116 288)(107 297 117 287)(108 296 118 286)(109 295 119 285)(110 294 120 284)(141 320 151 310)(142 319 152 309)(143 318 153 308)(144 317 154 307)(145 316 155 306)(146 315 156 305)(147 314 157 304)(148 313 158 303)(149 312 159 302)(150 311 160 301)(161 265 171 275)(162 264 172 274)(163 263 173 273)(164 262 174 272)(165 261 175 271)(166 280 176 270)(167 279 177 269)(168 278 178 268)(169 277 179 267)(170 276 180 266)(221 247 231 257)(222 246 232 256)(223 245 233 255)(224 244 234 254)(225 243 235 253)(226 242 236 252)(227 241 237 251)(228 260 238 250)(229 259 239 249)(230 258 240 248)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122244444444444458···8888810101020···20
size11112244555510102020410···10202020204448···8

38 irreducible representations

dim11111111222244488
type+++++-++--
imageC1C2C2C2C2C4C4C4Q8C4○D4M4(2)C8○D4F5C2×F5C4.F5D4.F5Q8×F5
kernelC20.M4(2)C4×C5⋊C8C20⋊C8C10.C42Dic53Q8C10.D4C5×C4⋊C4C2×Dic10C5⋊C8Dic5C20C10C4⋊C4C2×C4C4C2C2
# reps11321422224413411

Matrix representation of C20.M4(2) in GL6(𝔽41)

26260000
26150000
003710160
003810160
003711160
001482118
,
0400000
100000
00313452
004182422
003213817
002351436
,
0400000
100000
001814311
0054154
002314201
0040391940

G:=sub<GL(6,GF(41))| [26,26,0,0,0,0,26,15,0,0,0,0,0,0,37,38,37,14,0,0,10,10,11,8,0,0,16,16,16,21,0,0,0,0,0,18],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,31,4,32,23,0,0,34,18,1,5,0,0,5,24,38,14,0,0,2,22,17,36],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,18,5,23,40,0,0,14,4,14,39,0,0,3,15,20,19,0,0,11,4,1,40] >;

C20.M4(2) in GAP, Magma, Sage, TeX

C_{20}.M_4(2)
% in TeX

G:=Group("C20.M4(2)");
// GroupNames label

G:=SmallGroup(320,1047);
// by ID

G=gap.SmallGroup(320,1047);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,268,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

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