direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C10×Q32, C20.46D8, C40.74D4, C40.74C23, C80.22C22, C4.8(C5×D8), C4.9(D4×C10), C8.11(C5×D4), C16.5(C2×C10), (C2×C80).10C2, (C2×C16).4C10, (C2×C10).57D8, C2.14(C10×D8), C10.86(C2×D8), (C2×C20).428D4, C20.316(C2×D4), C8.5(C22×C10), Q16.1(C2×C10), (C2×Q16).4C10, C22.16(C5×D8), (C10×Q16).11C2, (C2×C40).428C22, (C5×Q16).13C22, (C2×C4).84(C5×D4), (C2×C8).86(C2×C10), SmallGroup(320,1008)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×Q32
G = < a,b,c | a10=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 146 in 82 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C16, C2×C8, Q16, Q16, C2×Q8, C20, C20, C2×C10, C2×C16, Q32, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, C2×Q32, C80, C2×C40, C5×Q16, C5×Q16, Q8×C10, C2×C80, C5×Q32, C10×Q16, C10×Q32
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, Q32, C2×D8, C5×D4, C22×C10, C2×Q32, C5×D8, D4×C10, C5×Q32, C10×D8, C10×Q32
►Regular action on 320 points
Generators in S320
(1 102 44 154 291 79 26 90 244 130)(2 103 45 155 292 80 27 91 245 131)(3 104 46 156 293 65 28 92 246 132)(4 105 47 157 294 66 29 93 247 133)(5 106 48 158 295 67 30 94 248 134)(6 107 33 159 296 68 31 95 249 135)(7 108 34 160 297 69 32 96 250 136)(8 109 35 145 298 70 17 81 251 137)(9 110 36 146 299 71 18 82 252 138)(10 111 37 147 300 72 19 83 253 139)(11 112 38 148 301 73 20 84 254 140)(12 97 39 149 302 74 21 85 255 141)(13 98 40 150 303 75 22 86 256 142)(14 99 41 151 304 76 23 87 241 143)(15 100 42 152 289 77 24 88 242 144)(16 101 43 153 290 78 25 89 243 129)(49 163 267 196 190 212 316 275 225 125)(50 164 268 197 191 213 317 276 226 126)(51 165 269 198 192 214 318 277 227 127)(52 166 270 199 177 215 319 278 228 128)(53 167 271 200 178 216 320 279 229 113)(54 168 272 201 179 217 305 280 230 114)(55 169 257 202 180 218 306 281 231 115)(56 170 258 203 181 219 307 282 232 116)(57 171 259 204 182 220 308 283 233 117)(58 172 260 205 183 221 309 284 234 118)(59 173 261 206 184 222 310 285 235 119)(60 174 262 207 185 223 311 286 236 120)(61 175 263 208 186 224 312 287 237 121)(62 176 264 193 187 209 313 288 238 122)(63 161 265 194 188 210 314 273 239 123)(64 162 266 195 189 211 315 274 240 124)
(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)
(1 223 9 215)(2 222 10 214)(3 221 11 213)(4 220 12 212)(5 219 13 211)(6 218 14 210)(7 217 15 209)(8 216 16 224)(17 167 25 175)(18 166 26 174)(19 165 27 173)(20 164 28 172)(21 163 29 171)(22 162 30 170)(23 161 31 169)(24 176 32 168)(33 281 41 273)(34 280 42 288)(35 279 43 287)(36 278 44 286)(37 277 45 285)(38 276 46 284)(39 275 47 283)(40 274 48 282)(49 66 57 74)(50 65 58 73)(51 80 59 72)(52 79 60 71)(53 78 61 70)(54 77 62 69)(55 76 63 68)(56 75 64 67)(81 271 89 263)(82 270 90 262)(83 269 91 261)(84 268 92 260)(85 267 93 259)(86 266 94 258)(87 265 95 257)(88 264 96 272)(97 316 105 308)(98 315 106 307)(99 314 107 306)(100 313 108 305)(101 312 109 320)(102 311 110 319)(103 310 111 318)(104 309 112 317)(113 290 121 298)(114 289 122 297)(115 304 123 296)(116 303 124 295)(117 302 125 294)(118 301 126 293)(119 300 127 292)(120 299 128 291)(129 186 137 178)(130 185 138 177)(131 184 139 192)(132 183 140 191)(133 182 141 190)(134 181 142 189)(135 180 143 188)(136 179 144 187)(145 229 153 237)(146 228 154 236)(147 227 155 235)(148 226 156 234)(149 225 157 233)(150 240 158 232)(151 239 159 231)(152 238 160 230)(193 250 201 242)(194 249 202 241)(195 248 203 256)(196 247 204 255)(197 246 205 254)(198 245 206 253)(199 244 207 252)(200 243 208 251)
G:=sub<Sym(320)| (1,102,44,154,291,79,26,90,244,130)(2,103,45,155,292,80,27,91,245,131)(3,104,46,156,293,65,28,92,246,132)(4,105,47,157,294,66,29,93,247,133)(5,106,48,158,295,67,30,94,248,134)(6,107,33,159,296,68,31,95,249,135)(7,108,34,160,297,69,32,96,250,136)(8,109,35,145,298,70,17,81,251,137)(9,110,36,146,299,71,18,82,252,138)(10,111,37,147,300,72,19,83,253,139)(11,112,38,148,301,73,20,84,254,140)(12,97,39,149,302,74,21,85,255,141)(13,98,40,150,303,75,22,86,256,142)(14,99,41,151,304,76,23,87,241,143)(15,100,42,152,289,77,24,88,242,144)(16,101,43,153,290,78,25,89,243,129)(49,163,267,196,190,212,316,275,225,125)(50,164,268,197,191,213,317,276,226,126)(51,165,269,198,192,214,318,277,227,127)(52,166,270,199,177,215,319,278,228,128)(53,167,271,200,178,216,320,279,229,113)(54,168,272,201,179,217,305,280,230,114)(55,169,257,202,180,218,306,281,231,115)(56,170,258,203,181,219,307,282,232,116)(57,171,259,204,182,220,308,283,233,117)(58,172,260,205,183,221,309,284,234,118)(59,173,261,206,184,222,310,285,235,119)(60,174,262,207,185,223,311,286,236,120)(61,175,263,208,186,224,312,287,237,121)(62,176,264,193,187,209,313,288,238,122)(63,161,265,194,188,210,314,273,239,123)(64,162,266,195,189,211,315,274,240,124), (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), (1,223,9,215)(2,222,10,214)(3,221,11,213)(4,220,12,212)(5,219,13,211)(6,218,14,210)(7,217,15,209)(8,216,16,224)(17,167,25,175)(18,166,26,174)(19,165,27,173)(20,164,28,172)(21,163,29,171)(22,162,30,170)(23,161,31,169)(24,176,32,168)(33,281,41,273)(34,280,42,288)(35,279,43,287)(36,278,44,286)(37,277,45,285)(38,276,46,284)(39,275,47,283)(40,274,48,282)(49,66,57,74)(50,65,58,73)(51,80,59,72)(52,79,60,71)(53,78,61,70)(54,77,62,69)(55,76,63,68)(56,75,64,67)(81,271,89,263)(82,270,90,262)(83,269,91,261)(84,268,92,260)(85,267,93,259)(86,266,94,258)(87,265,95,257)(88,264,96,272)(97,316,105,308)(98,315,106,307)(99,314,107,306)(100,313,108,305)(101,312,109,320)(102,311,110,319)(103,310,111,318)(104,309,112,317)(113,290,121,298)(114,289,122,297)(115,304,123,296)(116,303,124,295)(117,302,125,294)(118,301,126,293)(119,300,127,292)(120,299,128,291)(129,186,137,178)(130,185,138,177)(131,184,139,192)(132,183,140,191)(133,182,141,190)(134,181,142,189)(135,180,143,188)(136,179,144,187)(145,229,153,237)(146,228,154,236)(147,227,155,235)(148,226,156,234)(149,225,157,233)(150,240,158,232)(151,239,159,231)(152,238,160,230)(193,250,201,242)(194,249,202,241)(195,248,203,256)(196,247,204,255)(197,246,205,254)(198,245,206,253)(199,244,207,252)(200,243,208,251)>;
G:=Group( (1,102,44,154,291,79,26,90,244,130)(2,103,45,155,292,80,27,91,245,131)(3,104,46,156,293,65,28,92,246,132)(4,105,47,157,294,66,29,93,247,133)(5,106,48,158,295,67,30,94,248,134)(6,107,33,159,296,68,31,95,249,135)(7,108,34,160,297,69,32,96,250,136)(8,109,35,145,298,70,17,81,251,137)(9,110,36,146,299,71,18,82,252,138)(10,111,37,147,300,72,19,83,253,139)(11,112,38,148,301,73,20,84,254,140)(12,97,39,149,302,74,21,85,255,141)(13,98,40,150,303,75,22,86,256,142)(14,99,41,151,304,76,23,87,241,143)(15,100,42,152,289,77,24,88,242,144)(16,101,43,153,290,78,25,89,243,129)(49,163,267,196,190,212,316,275,225,125)(50,164,268,197,191,213,317,276,226,126)(51,165,269,198,192,214,318,277,227,127)(52,166,270,199,177,215,319,278,228,128)(53,167,271,200,178,216,320,279,229,113)(54,168,272,201,179,217,305,280,230,114)(55,169,257,202,180,218,306,281,231,115)(56,170,258,203,181,219,307,282,232,116)(57,171,259,204,182,220,308,283,233,117)(58,172,260,205,183,221,309,284,234,118)(59,173,261,206,184,222,310,285,235,119)(60,174,262,207,185,223,311,286,236,120)(61,175,263,208,186,224,312,287,237,121)(62,176,264,193,187,209,313,288,238,122)(63,161,265,194,188,210,314,273,239,123)(64,162,266,195,189,211,315,274,240,124), (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), (1,223,9,215)(2,222,10,214)(3,221,11,213)(4,220,12,212)(5,219,13,211)(6,218,14,210)(7,217,15,209)(8,216,16,224)(17,167,25,175)(18,166,26,174)(19,165,27,173)(20,164,28,172)(21,163,29,171)(22,162,30,170)(23,161,31,169)(24,176,32,168)(33,281,41,273)(34,280,42,288)(35,279,43,287)(36,278,44,286)(37,277,45,285)(38,276,46,284)(39,275,47,283)(40,274,48,282)(49,66,57,74)(50,65,58,73)(51,80,59,72)(52,79,60,71)(53,78,61,70)(54,77,62,69)(55,76,63,68)(56,75,64,67)(81,271,89,263)(82,270,90,262)(83,269,91,261)(84,268,92,260)(85,267,93,259)(86,266,94,258)(87,265,95,257)(88,264,96,272)(97,316,105,308)(98,315,106,307)(99,314,107,306)(100,313,108,305)(101,312,109,320)(102,311,110,319)(103,310,111,318)(104,309,112,317)(113,290,121,298)(114,289,122,297)(115,304,123,296)(116,303,124,295)(117,302,125,294)(118,301,126,293)(119,300,127,292)(120,299,128,291)(129,186,137,178)(130,185,138,177)(131,184,139,192)(132,183,140,191)(133,182,141,190)(134,181,142,189)(135,180,143,188)(136,179,144,187)(145,229,153,237)(146,228,154,236)(147,227,155,235)(148,226,156,234)(149,225,157,233)(150,240,158,232)(151,239,159,231)(152,238,160,230)(193,250,201,242)(194,249,202,241)(195,248,203,256)(196,247,204,255)(197,246,205,254)(198,245,206,253)(199,244,207,252)(200,243,208,251) );
G=PermutationGroup([[(1,102,44,154,291,79,26,90,244,130),(2,103,45,155,292,80,27,91,245,131),(3,104,46,156,293,65,28,92,246,132),(4,105,47,157,294,66,29,93,247,133),(5,106,48,158,295,67,30,94,248,134),(6,107,33,159,296,68,31,95,249,135),(7,108,34,160,297,69,32,96,250,136),(8,109,35,145,298,70,17,81,251,137),(9,110,36,146,299,71,18,82,252,138),(10,111,37,147,300,72,19,83,253,139),(11,112,38,148,301,73,20,84,254,140),(12,97,39,149,302,74,21,85,255,141),(13,98,40,150,303,75,22,86,256,142),(14,99,41,151,304,76,23,87,241,143),(15,100,42,152,289,77,24,88,242,144),(16,101,43,153,290,78,25,89,243,129),(49,163,267,196,190,212,316,275,225,125),(50,164,268,197,191,213,317,276,226,126),(51,165,269,198,192,214,318,277,227,127),(52,166,270,199,177,215,319,278,228,128),(53,167,271,200,178,216,320,279,229,113),(54,168,272,201,179,217,305,280,230,114),(55,169,257,202,180,218,306,281,231,115),(56,170,258,203,181,219,307,282,232,116),(57,171,259,204,182,220,308,283,233,117),(58,172,260,205,183,221,309,284,234,118),(59,173,261,206,184,222,310,285,235,119),(60,174,262,207,185,223,311,286,236,120),(61,175,263,208,186,224,312,287,237,121),(62,176,264,193,187,209,313,288,238,122),(63,161,265,194,188,210,314,273,239,123),(64,162,266,195,189,211,315,274,240,124)], [(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)], [(1,223,9,215),(2,222,10,214),(3,221,11,213),(4,220,12,212),(5,219,13,211),(6,218,14,210),(7,217,15,209),(8,216,16,224),(17,167,25,175),(18,166,26,174),(19,165,27,173),(20,164,28,172),(21,163,29,171),(22,162,30,170),(23,161,31,169),(24,176,32,168),(33,281,41,273),(34,280,42,288),(35,279,43,287),(36,278,44,286),(37,277,45,285),(38,276,46,284),(39,275,47,283),(40,274,48,282),(49,66,57,74),(50,65,58,73),(51,80,59,72),(52,79,60,71),(53,78,61,70),(54,77,62,69),(55,76,63,68),(56,75,64,67),(81,271,89,263),(82,270,90,262),(83,269,91,261),(84,268,92,260),(85,267,93,259),(86,266,94,258),(87,265,95,257),(88,264,96,272),(97,316,105,308),(98,315,106,307),(99,314,107,306),(100,313,108,305),(101,312,109,320),(102,311,110,319),(103,310,111,318),(104,309,112,317),(113,290,121,298),(114,289,122,297),(115,304,123,296),(116,303,124,295),(117,302,125,294),(118,301,126,293),(119,300,127,292),(120,299,128,291),(129,186,137,178),(130,185,138,177),(131,184,139,192),(132,183,140,191),(133,182,141,190),(134,181,142,189),(135,180,143,188),(136,179,144,187),(145,229,153,237),(146,228,154,236),(147,227,155,235),(148,226,156,234),(149,225,157,233),(150,240,158,232),(151,239,159,231),(152,238,160,230),(193,250,201,242),(194,249,202,241),(195,248,203,256),(196,247,204,255),(197,246,205,254),(198,245,206,253),(199,244,207,252),(200,243,208,251)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20X | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | Q32 | C5×D4 | C5×D4 | C5×D8 | C5×D8 | C5×Q32 |
| kernel | C10×Q32 | C2×C80 | C5×Q32 | C10×Q16 | C2×Q32 | C2×C16 | Q32 | C2×Q16 | C40 | C2×C20 | C20 | C2×C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 1 | 1 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C10×Q32 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 87 | 0 |
| 0 | 0 | 87 |
| 1 | 0 | 0 |
| 0 | 85 | 214 |
| 0 | 27 | 85 |
| 1 | 0 | 0 |
| 0 | 31 | 22 |
| 0 | 22 | 210 |
G:=sub<GL(3,GF(241))| [240,0,0,0,87,0,0,0,87],[1,0,0,0,85,27,0,214,85],[1,0,0,0,31,22,0,22,210] >;
C10×Q32 in GAP, Magma, Sage, TeX
C_{10}\times Q_{32} % in TeX
G:=Group("C10xQ32"); // GroupNames label
G:=SmallGroup(320,1008);
// by ID
G=gap.SmallGroup(320,1008);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations