direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C9×Q32, C16.C18, C48.3C6, Q16.C18, C144.3C2, C36.39D4, C18.17D8, C72.26C22, C3.(C3×Q32), C4.3(D4×C9), C2.5(C9×D8), (C3×Q32).C3, C8.4(C2×C18), C6.17(C3×D8), C24.23(C2×C6), C12.38(C3×D4), (C3×Q16).3C6, (C9×Q16).2C2, SmallGroup(288,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×Q32
G = < a,b,c | a9=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Q32
►Regular action on 288 points
Generators in S288
(1 260 181 57 137 22 107 39 65)(2 261 182 58 138 23 108 40 66)(3 262 183 59 139 24 109 41 67)(4 263 184 60 140 25 110 42 68)(5 264 185 61 141 26 111 43 69)(6 265 186 62 142 27 112 44 70)(7 266 187 63 143 28 97 45 71)(8 267 188 64 144 29 98 46 72)(9 268 189 49 129 30 99 47 73)(10 269 190 50 130 31 100 48 74)(11 270 191 51 131 32 101 33 75)(12 271 192 52 132 17 102 34 76)(13 272 177 53 133 18 103 35 77)(14 257 178 54 134 19 104 36 78)(15 258 179 55 135 20 105 37 79)(16 259 180 56 136 21 106 38 80)(81 234 115 279 152 224 251 199 171)(82 235 116 280 153 209 252 200 172)(83 236 117 281 154 210 253 201 173)(84 237 118 282 155 211 254 202 174)(85 238 119 283 156 212 255 203 175)(86 239 120 284 157 213 256 204 176)(87 240 121 285 158 214 241 205 161)(88 225 122 286 159 215 242 206 162)(89 226 123 287 160 216 243 207 163)(90 227 124 288 145 217 244 208 164)(91 228 125 273 146 218 245 193 165)(92 229 126 274 147 219 246 194 166)(93 230 127 275 148 220 247 195 167)(94 231 128 276 149 221 248 196 168)(95 232 113 277 150 222 249 197 169)(96 233 114 278 151 223 250 198 170)
(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)
(1 214 9 222)(2 213 10 221)(3 212 11 220)(4 211 12 219)(5 210 13 218)(6 209 14 217)(7 224 15 216)(8 223 16 215)(17 229 25 237)(18 228 26 236)(19 227 27 235)(20 226 28 234)(21 225 29 233)(22 240 30 232)(23 239 31 231)(24 238 32 230)(33 275 41 283)(34 274 42 282)(35 273 43 281)(36 288 44 280)(37 287 45 279)(38 286 46 278)(39 285 47 277)(40 284 48 276)(49 169 57 161)(50 168 58 176)(51 167 59 175)(52 166 60 174)(53 165 61 173)(54 164 62 172)(55 163 63 171)(56 162 64 170)(65 158 73 150)(66 157 74 149)(67 156 75 148)(68 155 76 147)(69 154 77 146)(70 153 78 145)(71 152 79 160)(72 151 80 159)(81 135 89 143)(82 134 90 142)(83 133 91 141)(84 132 92 140)(85 131 93 139)(86 130 94 138)(87 129 95 137)(88 144 96 136)(97 115 105 123)(98 114 106 122)(99 113 107 121)(100 128 108 120)(101 127 109 119)(102 126 110 118)(103 125 111 117)(104 124 112 116)(177 193 185 201)(178 208 186 200)(179 207 187 199)(180 206 188 198)(181 205 189 197)(182 204 190 196)(183 203 191 195)(184 202 192 194)(241 268 249 260)(242 267 250 259)(243 266 251 258)(244 265 252 257)(245 264 253 272)(246 263 254 271)(247 262 255 270)(248 261 256 269)
G:=sub<Sym(288)| (1,260,181,57,137,22,107,39,65)(2,261,182,58,138,23,108,40,66)(3,262,183,59,139,24,109,41,67)(4,263,184,60,140,25,110,42,68)(5,264,185,61,141,26,111,43,69)(6,265,186,62,142,27,112,44,70)(7,266,187,63,143,28,97,45,71)(8,267,188,64,144,29,98,46,72)(9,268,189,49,129,30,99,47,73)(10,269,190,50,130,31,100,48,74)(11,270,191,51,131,32,101,33,75)(12,271,192,52,132,17,102,34,76)(13,272,177,53,133,18,103,35,77)(14,257,178,54,134,19,104,36,78)(15,258,179,55,135,20,105,37,79)(16,259,180,56,136,21,106,38,80)(81,234,115,279,152,224,251,199,171)(82,235,116,280,153,209,252,200,172)(83,236,117,281,154,210,253,201,173)(84,237,118,282,155,211,254,202,174)(85,238,119,283,156,212,255,203,175)(86,239,120,284,157,213,256,204,176)(87,240,121,285,158,214,241,205,161)(88,225,122,286,159,215,242,206,162)(89,226,123,287,160,216,243,207,163)(90,227,124,288,145,217,244,208,164)(91,228,125,273,146,218,245,193,165)(92,229,126,274,147,219,246,194,166)(93,230,127,275,148,220,247,195,167)(94,231,128,276,149,221,248,196,168)(95,232,113,277,150,222,249,197,169)(96,233,114,278,151,223,250,198,170), (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), (1,214,9,222)(2,213,10,221)(3,212,11,220)(4,211,12,219)(5,210,13,218)(6,209,14,217)(7,224,15,216)(8,223,16,215)(17,229,25,237)(18,228,26,236)(19,227,27,235)(20,226,28,234)(21,225,29,233)(22,240,30,232)(23,239,31,231)(24,238,32,230)(33,275,41,283)(34,274,42,282)(35,273,43,281)(36,288,44,280)(37,287,45,279)(38,286,46,278)(39,285,47,277)(40,284,48,276)(49,169,57,161)(50,168,58,176)(51,167,59,175)(52,166,60,174)(53,165,61,173)(54,164,62,172)(55,163,63,171)(56,162,64,170)(65,158,73,150)(66,157,74,149)(67,156,75,148)(68,155,76,147)(69,154,77,146)(70,153,78,145)(71,152,79,160)(72,151,80,159)(81,135,89,143)(82,134,90,142)(83,133,91,141)(84,132,92,140)(85,131,93,139)(86,130,94,138)(87,129,95,137)(88,144,96,136)(97,115,105,123)(98,114,106,122)(99,113,107,121)(100,128,108,120)(101,127,109,119)(102,126,110,118)(103,125,111,117)(104,124,112,116)(177,193,185,201)(178,208,186,200)(179,207,187,199)(180,206,188,198)(181,205,189,197)(182,204,190,196)(183,203,191,195)(184,202,192,194)(241,268,249,260)(242,267,250,259)(243,266,251,258)(244,265,252,257)(245,264,253,272)(246,263,254,271)(247,262,255,270)(248,261,256,269)>;
G:=Group( (1,260,181,57,137,22,107,39,65)(2,261,182,58,138,23,108,40,66)(3,262,183,59,139,24,109,41,67)(4,263,184,60,140,25,110,42,68)(5,264,185,61,141,26,111,43,69)(6,265,186,62,142,27,112,44,70)(7,266,187,63,143,28,97,45,71)(8,267,188,64,144,29,98,46,72)(9,268,189,49,129,30,99,47,73)(10,269,190,50,130,31,100,48,74)(11,270,191,51,131,32,101,33,75)(12,271,192,52,132,17,102,34,76)(13,272,177,53,133,18,103,35,77)(14,257,178,54,134,19,104,36,78)(15,258,179,55,135,20,105,37,79)(16,259,180,56,136,21,106,38,80)(81,234,115,279,152,224,251,199,171)(82,235,116,280,153,209,252,200,172)(83,236,117,281,154,210,253,201,173)(84,237,118,282,155,211,254,202,174)(85,238,119,283,156,212,255,203,175)(86,239,120,284,157,213,256,204,176)(87,240,121,285,158,214,241,205,161)(88,225,122,286,159,215,242,206,162)(89,226,123,287,160,216,243,207,163)(90,227,124,288,145,217,244,208,164)(91,228,125,273,146,218,245,193,165)(92,229,126,274,147,219,246,194,166)(93,230,127,275,148,220,247,195,167)(94,231,128,276,149,221,248,196,168)(95,232,113,277,150,222,249,197,169)(96,233,114,278,151,223,250,198,170), (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), (1,214,9,222)(2,213,10,221)(3,212,11,220)(4,211,12,219)(5,210,13,218)(6,209,14,217)(7,224,15,216)(8,223,16,215)(17,229,25,237)(18,228,26,236)(19,227,27,235)(20,226,28,234)(21,225,29,233)(22,240,30,232)(23,239,31,231)(24,238,32,230)(33,275,41,283)(34,274,42,282)(35,273,43,281)(36,288,44,280)(37,287,45,279)(38,286,46,278)(39,285,47,277)(40,284,48,276)(49,169,57,161)(50,168,58,176)(51,167,59,175)(52,166,60,174)(53,165,61,173)(54,164,62,172)(55,163,63,171)(56,162,64,170)(65,158,73,150)(66,157,74,149)(67,156,75,148)(68,155,76,147)(69,154,77,146)(70,153,78,145)(71,152,79,160)(72,151,80,159)(81,135,89,143)(82,134,90,142)(83,133,91,141)(84,132,92,140)(85,131,93,139)(86,130,94,138)(87,129,95,137)(88,144,96,136)(97,115,105,123)(98,114,106,122)(99,113,107,121)(100,128,108,120)(101,127,109,119)(102,126,110,118)(103,125,111,117)(104,124,112,116)(177,193,185,201)(178,208,186,200)(179,207,187,199)(180,206,188,198)(181,205,189,197)(182,204,190,196)(183,203,191,195)(184,202,192,194)(241,268,249,260)(242,267,250,259)(243,266,251,258)(244,265,252,257)(245,264,253,272)(246,263,254,271)(247,262,255,270)(248,261,256,269) );
G=PermutationGroup([[(1,260,181,57,137,22,107,39,65),(2,261,182,58,138,23,108,40,66),(3,262,183,59,139,24,109,41,67),(4,263,184,60,140,25,110,42,68),(5,264,185,61,141,26,111,43,69),(6,265,186,62,142,27,112,44,70),(7,266,187,63,143,28,97,45,71),(8,267,188,64,144,29,98,46,72),(9,268,189,49,129,30,99,47,73),(10,269,190,50,130,31,100,48,74),(11,270,191,51,131,32,101,33,75),(12,271,192,52,132,17,102,34,76),(13,272,177,53,133,18,103,35,77),(14,257,178,54,134,19,104,36,78),(15,258,179,55,135,20,105,37,79),(16,259,180,56,136,21,106,38,80),(81,234,115,279,152,224,251,199,171),(82,235,116,280,153,209,252,200,172),(83,236,117,281,154,210,253,201,173),(84,237,118,282,155,211,254,202,174),(85,238,119,283,156,212,255,203,175),(86,239,120,284,157,213,256,204,176),(87,240,121,285,158,214,241,205,161),(88,225,122,286,159,215,242,206,162),(89,226,123,287,160,216,243,207,163),(90,227,124,288,145,217,244,208,164),(91,228,125,273,146,218,245,193,165),(92,229,126,274,147,219,246,194,166),(93,230,127,275,148,220,247,195,167),(94,231,128,276,149,221,248,196,168),(95,232,113,277,150,222,249,197,169),(96,233,114,278,151,223,250,198,170)], [(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)], [(1,214,9,222),(2,213,10,221),(3,212,11,220),(4,211,12,219),(5,210,13,218),(6,209,14,217),(7,224,15,216),(8,223,16,215),(17,229,25,237),(18,228,26,236),(19,227,27,235),(20,226,28,234),(21,225,29,233),(22,240,30,232),(23,239,31,231),(24,238,32,230),(33,275,41,283),(34,274,42,282),(35,273,43,281),(36,288,44,280),(37,287,45,279),(38,286,46,278),(39,285,47,277),(40,284,48,276),(49,169,57,161),(50,168,58,176),(51,167,59,175),(52,166,60,174),(53,165,61,173),(54,164,62,172),(55,163,63,171),(56,162,64,170),(65,158,73,150),(66,157,74,149),(67,156,75,148),(68,155,76,147),(69,154,77,146),(70,153,78,145),(71,152,79,160),(72,151,80,159),(81,135,89,143),(82,134,90,142),(83,133,91,141),(84,132,92,140),(85,131,93,139),(86,130,94,138),(87,129,95,137),(88,144,96,136),(97,115,105,123),(98,114,106,122),(99,113,107,121),(100,128,108,120),(101,127,109,119),(102,126,110,118),(103,125,111,117),(104,124,112,116),(177,193,185,201),(178,208,186,200),(179,207,187,199),(180,206,188,198),(181,205,189,197),(182,204,190,196),(183,203,191,195),(184,202,192,194),(241,268,249,260),(242,267,250,259),(243,266,251,258),(244,265,252,257),(245,264,253,272),(246,263,254,271),(247,262,255,270),(248,261,256,269)]])
99 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 16A | 16B | 16C | 16D | 18A | ··· | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36R | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | D8 | C3×D4 | Q32 | C3×D8 | D4×C9 | C3×Q32 | C9×D8 | C9×Q32 |
| kernel | C9×Q32 | C144 | C9×Q16 | C3×Q32 | C48 | C3×Q16 | Q32 | C16 | Q16 | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 2 | 2 | 4 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C9×Q32 ►in GL2(𝔽433) generated by
| 153 | 0 |
| 0 | 153 |
| 132 | 82 |
| 392 | 214 |
| 315 | 423 |
| 310 | 118 |
G:=sub<GL(2,GF(433))| [153,0,0,153],[132,392,82,214],[315,310,423,118] >;
C9×Q32 in GAP, Magma, Sage, TeX
C_9\times Q_{32} % in TeX
G:=Group("C9xQ32"); // GroupNames label
G:=SmallGroup(288,63);
// by ID
G=gap.SmallGroup(288,63);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,1008,197,1016,142,2355,1186,528,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export