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G = C20.47D8order 320 = 26·5

1st non-split extension by C20 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.47D8, C20.2Q16, C4.6Dic20, C42.4D10, C20.46SD16, C4⋊C8.4D5, C4⋊Dic5.2C4, C203C8.9C2, C4.19(D4⋊D5), (C2×C20).464D4, (C2×C4).122D20, C52(C4.10D8), C4.11(Q8⋊D5), C202Q8.9C2, C4.10(C40⋊C2), (C4×C20).40C22, C2.4(D206C4), C10.17(D4⋊C4), C10.15(Q8⋊C4), C2.5(C20.44D4), C2.4(C4.12D20), C10.8(C4.10D4), C22.61(D10⋊C4), (C5×C4⋊C8).4C2, (C2×C4).15(C4×D5), (C2×C20).200(C2×C4), (C2×C4).228(C5⋊D4), (C2×C10).108(C22⋊C4), SmallGroup(320,40)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.47D8
C1C5C10C2×C10C2×C20C4×C20C203C8 — C20.47D8
C5C2×C10C2×C20 — C20.47D8
C1C22C42C4⋊C8

Generators and relations for C20.47D8
 G = < a,b,c | a20=b8=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a5b-1 >

Subgroups: 254 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2 [×3], C4 [×4], C4 [×3], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], Q8 [×2], C10 [×3], C42, C4⋊C4 [×3], C2×C8 [×2], C2×Q8, Dic5 [×2], C20 [×4], C20, C2×C10, C4⋊C8, C4⋊C8, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C4.10D8, C2×C52C8, C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C203C8, C5×C4⋊C8, C202Q8, C20.47D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16 [×2], Q16, D10, C4.10D4, D4⋊C4, Q8⋊C4, C4×D5, D20, C5⋊D4, C4.10D8, C40⋊C2, Dic20, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, C20.44D4, C4.12D20, C20.47D8

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

dim11111222222222224444
type+++++++-++--++-
imageC1C2C2C2C4D4D5D8SD16Q16D10C4×D5D20C5⋊D4C40⋊C2Dic20C4.10D4D4⋊D5Q8⋊D5C4.12D20
kernelC20.47D8C203C8C5×C4⋊C8C202Q8C4⋊Dic5C2×C20C4⋊C8C20C20C20C42C2×C4C2×C4C2×C4C4C4C10C4C4C2
# reps11114222422444881224

Matrix representation of C20.47D8 in GL6(𝔽41)

4020000
4010000
000100
0040700
0000400
0000040
,
20280000
34210000
00141400
00302700
00001526
00001515
,
37120000
240000
0034700
0040700
00001318
00001828

G:=sub<GL(6,GF(41))| [40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[20,34,0,0,0,0,28,21,0,0,0,0,0,0,14,30,0,0,0,0,14,27,0,0,0,0,0,0,15,15,0,0,0,0,26,15],[37,2,0,0,0,0,12,4,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,13,18,0,0,0,0,18,28] >;

C20.47D8 in GAP, Magma, Sage, TeX

C_{20}._{47}D_8
% in TeX

G:=Group("C20.47D8");
// GroupNames label

G:=SmallGroup(320,40);
// by ID

G=gap.SmallGroup(320,40);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,85,316,422,387,268,570,136,12550]);
// Polycyclic

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

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