direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C16×Dic5, C80⋊16C4, C20.46C42, C5⋊5(C4×C16), C4.20(C8×D5), C8.40(C4×D5), C2.2(D5×C16), C5⋊2C16⋊17C4, C5⋊2C8.10C8, C20.59(C2×C8), C10.18(C4×C8), C40.98(C2×C4), (C2×C80).14C2, (C2×C16).10D5, C2.2(C8×Dic5), C22.8(C8×D5), C10.13(C2×C16), (C2×C8).332D10, C8.23(C2×Dic5), C4.15(C4×Dic5), (C4×Dic5).49C4, (C8×Dic5).28C2, (C2×Dic5).15C8, (C2×C40).398C22, (C2×C5⋊2C8).38C4, (C2×C10).37(C2×C8), (C2×C4).166(C4×D5), (C2×C5⋊2C16).14C2, (C2×C20).413(C2×C4), SmallGroup(320,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C16×Dic5 |
Generators and relations for C16×Dic5
G = < a,b,c | a16=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C16×Dic5
►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)
(1 214 40 138 165 25 193 122 303 97)(2 215 41 139 166 26 194 123 304 98)(3 216 42 140 167 27 195 124 289 99)(4 217 43 141 168 28 196 125 290 100)(5 218 44 142 169 29 197 126 291 101)(6 219 45 143 170 30 198 127 292 102)(7 220 46 144 171 31 199 128 293 103)(8 221 47 129 172 32 200 113 294 104)(9 222 48 130 173 17 201 114 295 105)(10 223 33 131 174 18 202 115 296 106)(11 224 34 132 175 19 203 116 297 107)(12 209 35 133 176 20 204 117 298 108)(13 210 36 134 161 21 205 118 299 109)(14 211 37 135 162 22 206 119 300 110)(15 212 38 136 163 23 207 120 301 111)(16 213 39 137 164 24 208 121 302 112)(49 310 251 257 286 191 74 155 237 96)(50 311 252 258 287 192 75 156 238 81)(51 312 253 259 288 177 76 157 239 82)(52 313 254 260 273 178 77 158 240 83)(53 314 255 261 274 179 78 159 225 84)(54 315 256 262 275 180 79 160 226 85)(55 316 241 263 276 181 80 145 227 86)(56 317 242 264 277 182 65 146 228 87)(57 318 243 265 278 183 66 147 229 88)(58 319 244 266 279 184 67 148 230 89)(59 320 245 267 280 185 68 149 231 90)(60 305 246 268 281 186 69 150 232 91)(61 306 247 269 282 187 70 151 233 92)(62 307 248 270 283 188 71 152 234 93)(63 308 249 271 284 189 72 153 235 94)(64 309 250 272 285 190 73 154 236 95)
(1 182 25 56)(2 183 26 57)(3 184 27 58)(4 185 28 59)(5 186 29 60)(6 187 30 61)(7 188 31 62)(8 189 32 63)(9 190 17 64)(10 191 18 49)(11 192 19 50)(12 177 20 51)(13 178 21 52)(14 179 22 53)(15 180 23 54)(16 181 24 55)(33 257 115 237)(34 258 116 238)(35 259 117 239)(36 260 118 240)(37 261 119 225)(38 262 120 226)(39 263 121 227)(40 264 122 228)(41 265 123 229)(42 266 124 230)(43 267 125 231)(44 268 126 232)(45 269 127 233)(46 270 128 234)(47 271 113 235)(48 272 114 236)(65 165 317 97)(66 166 318 98)(67 167 319 99)(68 168 320 100)(69 169 305 101)(70 170 306 102)(71 171 307 103)(72 172 308 104)(73 173 309 105)(74 174 310 106)(75 175 311 107)(76 176 312 108)(77 161 313 109)(78 162 314 110)(79 163 315 111)(80 164 316 112)(81 224 287 203)(82 209 288 204)(83 210 273 205)(84 211 274 206)(85 212 275 207)(86 213 276 208)(87 214 277 193)(88 215 278 194)(89 216 279 195)(90 217 280 196)(91 218 281 197)(92 219 282 198)(93 220 283 199)(94 221 284 200)(95 222 285 201)(96 223 286 202)(129 249 294 153)(130 250 295 154)(131 251 296 155)(132 252 297 156)(133 253 298 157)(134 254 299 158)(135 255 300 159)(136 256 301 160)(137 241 302 145)(138 242 303 146)(139 243 304 147)(140 244 289 148)(141 245 290 149)(142 246 291 150)(143 247 292 151)(144 248 293 152)
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), (1,214,40,138,165,25,193,122,303,97)(2,215,41,139,166,26,194,123,304,98)(3,216,42,140,167,27,195,124,289,99)(4,217,43,141,168,28,196,125,290,100)(5,218,44,142,169,29,197,126,291,101)(6,219,45,143,170,30,198,127,292,102)(7,220,46,144,171,31,199,128,293,103)(8,221,47,129,172,32,200,113,294,104)(9,222,48,130,173,17,201,114,295,105)(10,223,33,131,174,18,202,115,296,106)(11,224,34,132,175,19,203,116,297,107)(12,209,35,133,176,20,204,117,298,108)(13,210,36,134,161,21,205,118,299,109)(14,211,37,135,162,22,206,119,300,110)(15,212,38,136,163,23,207,120,301,111)(16,213,39,137,164,24,208,121,302,112)(49,310,251,257,286,191,74,155,237,96)(50,311,252,258,287,192,75,156,238,81)(51,312,253,259,288,177,76,157,239,82)(52,313,254,260,273,178,77,158,240,83)(53,314,255,261,274,179,78,159,225,84)(54,315,256,262,275,180,79,160,226,85)(55,316,241,263,276,181,80,145,227,86)(56,317,242,264,277,182,65,146,228,87)(57,318,243,265,278,183,66,147,229,88)(58,319,244,266,279,184,67,148,230,89)(59,320,245,267,280,185,68,149,231,90)(60,305,246,268,281,186,69,150,232,91)(61,306,247,269,282,187,70,151,233,92)(62,307,248,270,283,188,71,152,234,93)(63,308,249,271,284,189,72,153,235,94)(64,309,250,272,285,190,73,154,236,95), (1,182,25,56)(2,183,26,57)(3,184,27,58)(4,185,28,59)(5,186,29,60)(6,187,30,61)(7,188,31,62)(8,189,32,63)(9,190,17,64)(10,191,18,49)(11,192,19,50)(12,177,20,51)(13,178,21,52)(14,179,22,53)(15,180,23,54)(16,181,24,55)(33,257,115,237)(34,258,116,238)(35,259,117,239)(36,260,118,240)(37,261,119,225)(38,262,120,226)(39,263,121,227)(40,264,122,228)(41,265,123,229)(42,266,124,230)(43,267,125,231)(44,268,126,232)(45,269,127,233)(46,270,128,234)(47,271,113,235)(48,272,114,236)(65,165,317,97)(66,166,318,98)(67,167,319,99)(68,168,320,100)(69,169,305,101)(70,170,306,102)(71,171,307,103)(72,172,308,104)(73,173,309,105)(74,174,310,106)(75,175,311,107)(76,176,312,108)(77,161,313,109)(78,162,314,110)(79,163,315,111)(80,164,316,112)(81,224,287,203)(82,209,288,204)(83,210,273,205)(84,211,274,206)(85,212,275,207)(86,213,276,208)(87,214,277,193)(88,215,278,194)(89,216,279,195)(90,217,280,196)(91,218,281,197)(92,219,282,198)(93,220,283,199)(94,221,284,200)(95,222,285,201)(96,223,286,202)(129,249,294,153)(130,250,295,154)(131,251,296,155)(132,252,297,156)(133,253,298,157)(134,254,299,158)(135,255,300,159)(136,256,301,160)(137,241,302,145)(138,242,303,146)(139,243,304,147)(140,244,289,148)(141,245,290,149)(142,246,291,150)(143,247,292,151)(144,248,293,152)>;
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), (1,214,40,138,165,25,193,122,303,97)(2,215,41,139,166,26,194,123,304,98)(3,216,42,140,167,27,195,124,289,99)(4,217,43,141,168,28,196,125,290,100)(5,218,44,142,169,29,197,126,291,101)(6,219,45,143,170,30,198,127,292,102)(7,220,46,144,171,31,199,128,293,103)(8,221,47,129,172,32,200,113,294,104)(9,222,48,130,173,17,201,114,295,105)(10,223,33,131,174,18,202,115,296,106)(11,224,34,132,175,19,203,116,297,107)(12,209,35,133,176,20,204,117,298,108)(13,210,36,134,161,21,205,118,299,109)(14,211,37,135,162,22,206,119,300,110)(15,212,38,136,163,23,207,120,301,111)(16,213,39,137,164,24,208,121,302,112)(49,310,251,257,286,191,74,155,237,96)(50,311,252,258,287,192,75,156,238,81)(51,312,253,259,288,177,76,157,239,82)(52,313,254,260,273,178,77,158,240,83)(53,314,255,261,274,179,78,159,225,84)(54,315,256,262,275,180,79,160,226,85)(55,316,241,263,276,181,80,145,227,86)(56,317,242,264,277,182,65,146,228,87)(57,318,243,265,278,183,66,147,229,88)(58,319,244,266,279,184,67,148,230,89)(59,320,245,267,280,185,68,149,231,90)(60,305,246,268,281,186,69,150,232,91)(61,306,247,269,282,187,70,151,233,92)(62,307,248,270,283,188,71,152,234,93)(63,308,249,271,284,189,72,153,235,94)(64,309,250,272,285,190,73,154,236,95), (1,182,25,56)(2,183,26,57)(3,184,27,58)(4,185,28,59)(5,186,29,60)(6,187,30,61)(7,188,31,62)(8,189,32,63)(9,190,17,64)(10,191,18,49)(11,192,19,50)(12,177,20,51)(13,178,21,52)(14,179,22,53)(15,180,23,54)(16,181,24,55)(33,257,115,237)(34,258,116,238)(35,259,117,239)(36,260,118,240)(37,261,119,225)(38,262,120,226)(39,263,121,227)(40,264,122,228)(41,265,123,229)(42,266,124,230)(43,267,125,231)(44,268,126,232)(45,269,127,233)(46,270,128,234)(47,271,113,235)(48,272,114,236)(65,165,317,97)(66,166,318,98)(67,167,319,99)(68,168,320,100)(69,169,305,101)(70,170,306,102)(71,171,307,103)(72,172,308,104)(73,173,309,105)(74,174,310,106)(75,175,311,107)(76,176,312,108)(77,161,313,109)(78,162,314,110)(79,163,315,111)(80,164,316,112)(81,224,287,203)(82,209,288,204)(83,210,273,205)(84,211,274,206)(85,212,275,207)(86,213,276,208)(87,214,277,193)(88,215,278,194)(89,216,279,195)(90,217,280,196)(91,218,281,197)(92,219,282,198)(93,220,283,199)(94,221,284,200)(95,222,285,201)(96,223,286,202)(129,249,294,153)(130,250,295,154)(131,251,296,155)(132,252,297,156)(133,253,298,157)(134,254,299,158)(135,255,300,159)(136,256,301,160)(137,241,302,145)(138,242,303,146)(139,243,304,147)(140,244,289,148)(141,245,290,149)(142,246,291,150)(143,247,292,151)(144,248,293,152) );
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)], [(1,214,40,138,165,25,193,122,303,97),(2,215,41,139,166,26,194,123,304,98),(3,216,42,140,167,27,195,124,289,99),(4,217,43,141,168,28,196,125,290,100),(5,218,44,142,169,29,197,126,291,101),(6,219,45,143,170,30,198,127,292,102),(7,220,46,144,171,31,199,128,293,103),(8,221,47,129,172,32,200,113,294,104),(9,222,48,130,173,17,201,114,295,105),(10,223,33,131,174,18,202,115,296,106),(11,224,34,132,175,19,203,116,297,107),(12,209,35,133,176,20,204,117,298,108),(13,210,36,134,161,21,205,118,299,109),(14,211,37,135,162,22,206,119,300,110),(15,212,38,136,163,23,207,120,301,111),(16,213,39,137,164,24,208,121,302,112),(49,310,251,257,286,191,74,155,237,96),(50,311,252,258,287,192,75,156,238,81),(51,312,253,259,288,177,76,157,239,82),(52,313,254,260,273,178,77,158,240,83),(53,314,255,261,274,179,78,159,225,84),(54,315,256,262,275,180,79,160,226,85),(55,316,241,263,276,181,80,145,227,86),(56,317,242,264,277,182,65,146,228,87),(57,318,243,265,278,183,66,147,229,88),(58,319,244,266,279,184,67,148,230,89),(59,320,245,267,280,185,68,149,231,90),(60,305,246,268,281,186,69,150,232,91),(61,306,247,269,282,187,70,151,233,92),(62,307,248,270,283,188,71,152,234,93),(63,308,249,271,284,189,72,153,235,94),(64,309,250,272,285,190,73,154,236,95)], [(1,182,25,56),(2,183,26,57),(3,184,27,58),(4,185,28,59),(5,186,29,60),(6,187,30,61),(7,188,31,62),(8,189,32,63),(9,190,17,64),(10,191,18,49),(11,192,19,50),(12,177,20,51),(13,178,21,52),(14,179,22,53),(15,180,23,54),(16,181,24,55),(33,257,115,237),(34,258,116,238),(35,259,117,239),(36,260,118,240),(37,261,119,225),(38,262,120,226),(39,263,121,227),(40,264,122,228),(41,265,123,229),(42,266,124,230),(43,267,125,231),(44,268,126,232),(45,269,127,233),(46,270,128,234),(47,271,113,235),(48,272,114,236),(65,165,317,97),(66,166,318,98),(67,167,319,99),(68,168,320,100),(69,169,305,101),(70,170,306,102),(71,171,307,103),(72,172,308,104),(73,173,309,105),(74,174,310,106),(75,175,311,107),(76,176,312,108),(77,161,313,109),(78,162,314,110),(79,163,315,111),(80,164,316,112),(81,224,287,203),(82,209,288,204),(83,210,273,205),(84,211,274,206),(85,212,275,207),(86,213,276,208),(87,214,277,193),(88,215,278,194),(89,216,279,195),(90,217,280,196),(91,218,281,197),(92,219,282,198),(93,220,283,199),(94,221,284,200),(95,222,285,201),(96,223,286,202),(129,249,294,153),(130,250,295,154),(131,251,296,155),(132,252,297,156),(133,253,298,157),(134,254,299,158),(135,255,300,159),(136,256,301,160),(137,241,302,145),(138,242,303,146),(139,243,304,147),(140,244,289,148),(141,245,290,149),(142,246,291,150),(143,247,292,151),(144,248,293,152)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 16A | ··· | 16P | 16Q | ··· | 16AF | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | C16 | D5 | Dic5 | D10 | C4×D5 | C4×D5 | C8×D5 | C8×D5 | D5×C16 |
| kernel | C16×Dic5 | C2×C5⋊2C16 | C8×Dic5 | C2×C80 | C5⋊2C16 | C80 | C2×C5⋊2C8 | C4×Dic5 | C5⋊2C8 | C2×Dic5 | Dic5 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 32 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C16×Dic5 ►in GL4(𝔽241) generated by
| 165 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 233 | 0 |
| 0 | 0 | 0 | 233 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 189 |
| 1 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 81 | 210 |
| 0 | 0 | 95 | 160 |
G:=sub<GL(4,GF(241))| [165,0,0,0,0,177,0,0,0,0,233,0,0,0,0,233],[1,0,0,0,0,240,0,0,0,0,0,240,0,0,1,189],[1,0,0,0,0,177,0,0,0,0,81,95,0,0,210,160] >;
C16×Dic5 in GAP, Magma, Sage, TeX
C_{16}\times {\rm Dic}_5 % in TeX
G:=Group("C16xDic5"); // GroupNames label
G:=SmallGroup(320,58);
// by ID
G=gap.SmallGroup(320,58);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,80,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^16=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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