metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊4Q16, C8.9D20, C4⋊1Dic20, C40.59D4, C42.266D10, (C4×C8).9D5, (C4×C40).11C2, C5⋊1(C4⋊Q16), (C2×C4).82D20, C4.33(C2×D20), C10.4(C2×Q16), (C2×C8).302D10, (C2×C20).379D4, C20.276(C2×D4), C20⋊2Q8.5C2, C2.6(C2×Dic20), C10.6(C4⋊1D4), C2.8(C20⋊4D4), (C2×Dic20).2C2, C22.94(C2×D20), (C2×C20).728C23, (C4×C20).311C22, (C2×C40).375C22, (C2×Dic10).7C22, (C2×C10).111(C2×D4), (C2×C4).671(C22×D5), SmallGroup(320,326)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊4Q16
G = < a,b,c | a20=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 494 in 122 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C4×C8, C4⋊Q8, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C4⋊Q16, Dic20, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C4×C40, C20⋊2Q8, C2×Dic20, C20⋊4Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C4⋊1D4, C2×Q16, D20, C22×D5, C4⋊Q16, Dic20, C2×D20, C20⋊4D4, C2×Dic20, C20⋊4Q16
►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)
(1 311 278 34 248 182 69 94)(2 312 279 35 249 183 70 95)(3 313 280 36 250 184 71 96)(4 314 261 37 251 185 72 97)(5 315 262 38 252 186 73 98)(6 316 263 39 253 187 74 99)(7 317 264 40 254 188 75 100)(8 318 265 21 255 189 76 81)(9 319 266 22 256 190 77 82)(10 320 267 23 257 191 78 83)(11 301 268 24 258 192 79 84)(12 302 269 25 259 193 80 85)(13 303 270 26 260 194 61 86)(14 304 271 27 241 195 62 87)(15 305 272 28 242 196 63 88)(16 306 273 29 243 197 64 89)(17 307 274 30 244 198 65 90)(18 308 275 31 245 199 66 91)(19 309 276 32 246 200 67 92)(20 310 277 33 247 181 68 93)(41 121 289 174 237 104 203 158)(42 122 290 175 238 105 204 159)(43 123 291 176 239 106 205 160)(44 124 292 177 240 107 206 141)(45 125 293 178 221 108 207 142)(46 126 294 179 222 109 208 143)(47 127 295 180 223 110 209 144)(48 128 296 161 224 111 210 145)(49 129 297 162 225 112 211 146)(50 130 298 163 226 113 212 147)(51 131 299 164 227 114 213 148)(52 132 300 165 228 115 214 149)(53 133 281 166 229 116 215 150)(54 134 282 167 230 117 216 151)(55 135 283 168 231 118 217 152)(56 136 284 169 232 119 218 153)(57 137 285 170 233 120 219 154)(58 138 286 171 234 101 220 155)(59 139 287 172 235 102 201 156)(60 140 288 173 236 103 202 157)
(1 215 248 281)(2 214 249 300)(3 213 250 299)(4 212 251 298)(5 211 252 297)(6 210 253 296)(7 209 254 295)(8 208 255 294)(9 207 256 293)(10 206 257 292)(11 205 258 291)(12 204 259 290)(13 203 260 289)(14 202 241 288)(15 201 242 287)(16 220 243 286)(17 219 244 285)(18 218 245 284)(19 217 246 283)(20 216 247 282)(21 179 81 143)(22 178 82 142)(23 177 83 141)(24 176 84 160)(25 175 85 159)(26 174 86 158)(27 173 87 157)(28 172 88 156)(29 171 89 155)(30 170 90 154)(31 169 91 153)(32 168 92 152)(33 167 93 151)(34 166 94 150)(35 165 95 149)(36 164 96 148)(37 163 97 147)(38 162 98 146)(39 161 99 145)(40 180 100 144)(41 270 237 61)(42 269 238 80)(43 268 239 79)(44 267 240 78)(45 266 221 77)(46 265 222 76)(47 264 223 75)(48 263 224 74)(49 262 225 73)(50 261 226 72)(51 280 227 71)(52 279 228 70)(53 278 229 69)(54 277 230 68)(55 276 231 67)(56 275 232 66)(57 274 233 65)(58 273 234 64)(59 272 235 63)(60 271 236 62)(101 197 138 306)(102 196 139 305)(103 195 140 304)(104 194 121 303)(105 193 122 302)(106 192 123 301)(107 191 124 320)(108 190 125 319)(109 189 126 318)(110 188 127 317)(111 187 128 316)(112 186 129 315)(113 185 130 314)(114 184 131 313)(115 183 132 312)(116 182 133 311)(117 181 134 310)(118 200 135 309)(119 199 136 308)(120 198 137 307)
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), (1,311,278,34,248,182,69,94)(2,312,279,35,249,183,70,95)(3,313,280,36,250,184,71,96)(4,314,261,37,251,185,72,97)(5,315,262,38,252,186,73,98)(6,316,263,39,253,187,74,99)(7,317,264,40,254,188,75,100)(8,318,265,21,255,189,76,81)(9,319,266,22,256,190,77,82)(10,320,267,23,257,191,78,83)(11,301,268,24,258,192,79,84)(12,302,269,25,259,193,80,85)(13,303,270,26,260,194,61,86)(14,304,271,27,241,195,62,87)(15,305,272,28,242,196,63,88)(16,306,273,29,243,197,64,89)(17,307,274,30,244,198,65,90)(18,308,275,31,245,199,66,91)(19,309,276,32,246,200,67,92)(20,310,277,33,247,181,68,93)(41,121,289,174,237,104,203,158)(42,122,290,175,238,105,204,159)(43,123,291,176,239,106,205,160)(44,124,292,177,240,107,206,141)(45,125,293,178,221,108,207,142)(46,126,294,179,222,109,208,143)(47,127,295,180,223,110,209,144)(48,128,296,161,224,111,210,145)(49,129,297,162,225,112,211,146)(50,130,298,163,226,113,212,147)(51,131,299,164,227,114,213,148)(52,132,300,165,228,115,214,149)(53,133,281,166,229,116,215,150)(54,134,282,167,230,117,216,151)(55,135,283,168,231,118,217,152)(56,136,284,169,232,119,218,153)(57,137,285,170,233,120,219,154)(58,138,286,171,234,101,220,155)(59,139,287,172,235,102,201,156)(60,140,288,173,236,103,202,157), (1,215,248,281)(2,214,249,300)(3,213,250,299)(4,212,251,298)(5,211,252,297)(6,210,253,296)(7,209,254,295)(8,208,255,294)(9,207,256,293)(10,206,257,292)(11,205,258,291)(12,204,259,290)(13,203,260,289)(14,202,241,288)(15,201,242,287)(16,220,243,286)(17,219,244,285)(18,218,245,284)(19,217,246,283)(20,216,247,282)(21,179,81,143)(22,178,82,142)(23,177,83,141)(24,176,84,160)(25,175,85,159)(26,174,86,158)(27,173,87,157)(28,172,88,156)(29,171,89,155)(30,170,90,154)(31,169,91,153)(32,168,92,152)(33,167,93,151)(34,166,94,150)(35,165,95,149)(36,164,96,148)(37,163,97,147)(38,162,98,146)(39,161,99,145)(40,180,100,144)(41,270,237,61)(42,269,238,80)(43,268,239,79)(44,267,240,78)(45,266,221,77)(46,265,222,76)(47,264,223,75)(48,263,224,74)(49,262,225,73)(50,261,226,72)(51,280,227,71)(52,279,228,70)(53,278,229,69)(54,277,230,68)(55,276,231,67)(56,275,232,66)(57,274,233,65)(58,273,234,64)(59,272,235,63)(60,271,236,62)(101,197,138,306)(102,196,139,305)(103,195,140,304)(104,194,121,303)(105,193,122,302)(106,192,123,301)(107,191,124,320)(108,190,125,319)(109,189,126,318)(110,188,127,317)(111,187,128,316)(112,186,129,315)(113,185,130,314)(114,184,131,313)(115,183,132,312)(116,182,133,311)(117,181,134,310)(118,200,135,309)(119,199,136,308)(120,198,137,307)>;
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), (1,311,278,34,248,182,69,94)(2,312,279,35,249,183,70,95)(3,313,280,36,250,184,71,96)(4,314,261,37,251,185,72,97)(5,315,262,38,252,186,73,98)(6,316,263,39,253,187,74,99)(7,317,264,40,254,188,75,100)(8,318,265,21,255,189,76,81)(9,319,266,22,256,190,77,82)(10,320,267,23,257,191,78,83)(11,301,268,24,258,192,79,84)(12,302,269,25,259,193,80,85)(13,303,270,26,260,194,61,86)(14,304,271,27,241,195,62,87)(15,305,272,28,242,196,63,88)(16,306,273,29,243,197,64,89)(17,307,274,30,244,198,65,90)(18,308,275,31,245,199,66,91)(19,309,276,32,246,200,67,92)(20,310,277,33,247,181,68,93)(41,121,289,174,237,104,203,158)(42,122,290,175,238,105,204,159)(43,123,291,176,239,106,205,160)(44,124,292,177,240,107,206,141)(45,125,293,178,221,108,207,142)(46,126,294,179,222,109,208,143)(47,127,295,180,223,110,209,144)(48,128,296,161,224,111,210,145)(49,129,297,162,225,112,211,146)(50,130,298,163,226,113,212,147)(51,131,299,164,227,114,213,148)(52,132,300,165,228,115,214,149)(53,133,281,166,229,116,215,150)(54,134,282,167,230,117,216,151)(55,135,283,168,231,118,217,152)(56,136,284,169,232,119,218,153)(57,137,285,170,233,120,219,154)(58,138,286,171,234,101,220,155)(59,139,287,172,235,102,201,156)(60,140,288,173,236,103,202,157), (1,215,248,281)(2,214,249,300)(3,213,250,299)(4,212,251,298)(5,211,252,297)(6,210,253,296)(7,209,254,295)(8,208,255,294)(9,207,256,293)(10,206,257,292)(11,205,258,291)(12,204,259,290)(13,203,260,289)(14,202,241,288)(15,201,242,287)(16,220,243,286)(17,219,244,285)(18,218,245,284)(19,217,246,283)(20,216,247,282)(21,179,81,143)(22,178,82,142)(23,177,83,141)(24,176,84,160)(25,175,85,159)(26,174,86,158)(27,173,87,157)(28,172,88,156)(29,171,89,155)(30,170,90,154)(31,169,91,153)(32,168,92,152)(33,167,93,151)(34,166,94,150)(35,165,95,149)(36,164,96,148)(37,163,97,147)(38,162,98,146)(39,161,99,145)(40,180,100,144)(41,270,237,61)(42,269,238,80)(43,268,239,79)(44,267,240,78)(45,266,221,77)(46,265,222,76)(47,264,223,75)(48,263,224,74)(49,262,225,73)(50,261,226,72)(51,280,227,71)(52,279,228,70)(53,278,229,69)(54,277,230,68)(55,276,231,67)(56,275,232,66)(57,274,233,65)(58,273,234,64)(59,272,235,63)(60,271,236,62)(101,197,138,306)(102,196,139,305)(103,195,140,304)(104,194,121,303)(105,193,122,302)(106,192,123,301)(107,191,124,320)(108,190,125,319)(109,189,126,318)(110,188,127,317)(111,187,128,316)(112,186,129,315)(113,185,130,314)(114,184,131,313)(115,183,132,312)(116,182,133,311)(117,181,134,310)(118,200,135,309)(119,199,136,308)(120,198,137,307) );
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)], [(1,311,278,34,248,182,69,94),(2,312,279,35,249,183,70,95),(3,313,280,36,250,184,71,96),(4,314,261,37,251,185,72,97),(5,315,262,38,252,186,73,98),(6,316,263,39,253,187,74,99),(7,317,264,40,254,188,75,100),(8,318,265,21,255,189,76,81),(9,319,266,22,256,190,77,82),(10,320,267,23,257,191,78,83),(11,301,268,24,258,192,79,84),(12,302,269,25,259,193,80,85),(13,303,270,26,260,194,61,86),(14,304,271,27,241,195,62,87),(15,305,272,28,242,196,63,88),(16,306,273,29,243,197,64,89),(17,307,274,30,244,198,65,90),(18,308,275,31,245,199,66,91),(19,309,276,32,246,200,67,92),(20,310,277,33,247,181,68,93),(41,121,289,174,237,104,203,158),(42,122,290,175,238,105,204,159),(43,123,291,176,239,106,205,160),(44,124,292,177,240,107,206,141),(45,125,293,178,221,108,207,142),(46,126,294,179,222,109,208,143),(47,127,295,180,223,110,209,144),(48,128,296,161,224,111,210,145),(49,129,297,162,225,112,211,146),(50,130,298,163,226,113,212,147),(51,131,299,164,227,114,213,148),(52,132,300,165,228,115,214,149),(53,133,281,166,229,116,215,150),(54,134,282,167,230,117,216,151),(55,135,283,168,231,118,217,152),(56,136,284,169,232,119,218,153),(57,137,285,170,233,120,219,154),(58,138,286,171,234,101,220,155),(59,139,287,172,235,102,201,156),(60,140,288,173,236,103,202,157)], [(1,215,248,281),(2,214,249,300),(3,213,250,299),(4,212,251,298),(5,211,252,297),(6,210,253,296),(7,209,254,295),(8,208,255,294),(9,207,256,293),(10,206,257,292),(11,205,258,291),(12,204,259,290),(13,203,260,289),(14,202,241,288),(15,201,242,287),(16,220,243,286),(17,219,244,285),(18,218,245,284),(19,217,246,283),(20,216,247,282),(21,179,81,143),(22,178,82,142),(23,177,83,141),(24,176,84,160),(25,175,85,159),(26,174,86,158),(27,173,87,157),(28,172,88,156),(29,171,89,155),(30,170,90,154),(31,169,91,153),(32,168,92,152),(33,167,93,151),(34,166,94,150),(35,165,95,149),(36,164,96,148),(37,163,97,147),(38,162,98,146),(39,161,99,145),(40,180,100,144),(41,270,237,61),(42,269,238,80),(43,268,239,79),(44,267,240,78),(45,266,221,77),(46,265,222,76),(47,264,223,75),(48,263,224,74),(49,262,225,73),(50,261,226,72),(51,280,227,71),(52,279,228,70),(53,278,229,69),(54,277,230,68),(55,276,231,67),(56,275,232,66),(57,274,233,65),(58,273,234,64),(59,272,235,63),(60,271,236,62),(101,197,138,306),(102,196,139,305),(103,195,140,304),(104,194,121,303),(105,193,122,302),(106,192,123,301),(107,191,124,320),(108,190,125,319),(109,189,126,318),(110,188,127,317),(111,187,128,316),(112,186,129,315),(113,185,130,314),(114,184,131,313),(115,183,132,312),(116,182,133,311),(117,181,134,310),(118,200,135,309),(119,199,136,308),(120,198,137,307)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D5 | Q16 | D10 | D10 | D20 | D20 | Dic20 |
| kernel | C20⋊4Q16 | C4×C40 | C20⋊2Q8 | C2×Dic20 | C40 | C2×C20 | C4×C8 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 8 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of C20⋊4Q16 ►in GL4(𝔽41) generated by
| 34 | 1 | 0 | 0 |
| 33 | 1 | 0 | 0 |
| 0 | 0 | 14 | 30 |
| 0 | 0 | 11 | 9 |
| 20 | 33 | 0 | 0 |
| 23 | 38 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 32 | 30 |
| 14 | 37 | 0 | 0 |
| 39 | 27 | 0 | 0 |
| 0 | 0 | 9 | 11 |
| 0 | 0 | 30 | 32 |
G:=sub<GL(4,GF(41))| [34,33,0,0,1,1,0,0,0,0,14,11,0,0,30,9],[20,23,0,0,33,38,0,0,0,0,11,32,0,0,9,30],[14,39,0,0,37,27,0,0,0,0,9,30,0,0,11,32] >;
C20⋊4Q16 in GAP, Magma, Sage, TeX
C_{20}\rtimes_4Q_{16} % in TeX
G:=Group("C20:4Q16"); // GroupNames label
G:=SmallGroup(320,326);
// by ID
G=gap.SmallGroup(320,326);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,226,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations