metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9⋊C32, C18.C16, C72.2C4, C36.2C8, C48.6S3, C16.2D9, C144.2C2, C8.3Dic9, C24.5Dic3, C3.(C3⋊C32), C2.(C9⋊C16), C4.2(C9⋊C8), C6.1(C3⋊C16), C12.4(C3⋊C8), SmallGroup(288,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C32 |
Generators and relations for C9⋊C32
G = < a,b | a9=b32=1, bab-1=a-1 >
Smallest permutation representation of C9⋊C32
►Regular action on 288 points
Generators in S288
(1 261 50 165 128 196 229 80 133)(2 134 81 230 197 97 166 51 262)(3 263 52 167 98 198 231 82 135)(4 136 83 232 199 99 168 53 264)(5 265 54 169 100 200 233 84 137)(6 138 85 234 201 101 170 55 266)(7 267 56 171 102 202 235 86 139)(8 140 87 236 203 103 172 57 268)(9 269 58 173 104 204 237 88 141)(10 142 89 238 205 105 174 59 270)(11 271 60 175 106 206 239 90 143)(12 144 91 240 207 107 176 61 272)(13 273 62 177 108 208 241 92 145)(14 146 93 242 209 109 178 63 274)(15 275 64 179 110 210 243 94 147)(16 148 95 244 211 111 180 33 276)(17 277 34 181 112 212 245 96 149)(18 150 65 246 213 113 182 35 278)(19 279 36 183 114 214 247 66 151)(20 152 67 248 215 115 184 37 280)(21 281 38 185 116 216 249 68 153)(22 154 69 250 217 117 186 39 282)(23 283 40 187 118 218 251 70 155)(24 156 71 252 219 119 188 41 284)(25 285 42 189 120 220 253 72 157)(26 158 73 254 221 121 190 43 286)(27 287 44 191 122 222 255 74 159)(28 160 75 256 223 123 192 45 288)(29 257 46 161 124 224 225 76 129)(30 130 77 226 193 125 162 47 258)(31 259 48 163 126 194 227 78 131)(32 132 79 228 195 127 164 49 260)
(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)
G:=sub<Sym(288)| (1,261,50,165,128,196,229,80,133)(2,134,81,230,197,97,166,51,262)(3,263,52,167,98,198,231,82,135)(4,136,83,232,199,99,168,53,264)(5,265,54,169,100,200,233,84,137)(6,138,85,234,201,101,170,55,266)(7,267,56,171,102,202,235,86,139)(8,140,87,236,203,103,172,57,268)(9,269,58,173,104,204,237,88,141)(10,142,89,238,205,105,174,59,270)(11,271,60,175,106,206,239,90,143)(12,144,91,240,207,107,176,61,272)(13,273,62,177,108,208,241,92,145)(14,146,93,242,209,109,178,63,274)(15,275,64,179,110,210,243,94,147)(16,148,95,244,211,111,180,33,276)(17,277,34,181,112,212,245,96,149)(18,150,65,246,213,113,182,35,278)(19,279,36,183,114,214,247,66,151)(20,152,67,248,215,115,184,37,280)(21,281,38,185,116,216,249,68,153)(22,154,69,250,217,117,186,39,282)(23,283,40,187,118,218,251,70,155)(24,156,71,252,219,119,188,41,284)(25,285,42,189,120,220,253,72,157)(26,158,73,254,221,121,190,43,286)(27,287,44,191,122,222,255,74,159)(28,160,75,256,223,123,192,45,288)(29,257,46,161,124,224,225,76,129)(30,130,77,226,193,125,162,47,258)(31,259,48,163,126,194,227,78,131)(32,132,79,228,195,127,164,49,260), (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)>;
G:=Group( (1,261,50,165,128,196,229,80,133)(2,134,81,230,197,97,166,51,262)(3,263,52,167,98,198,231,82,135)(4,136,83,232,199,99,168,53,264)(5,265,54,169,100,200,233,84,137)(6,138,85,234,201,101,170,55,266)(7,267,56,171,102,202,235,86,139)(8,140,87,236,203,103,172,57,268)(9,269,58,173,104,204,237,88,141)(10,142,89,238,205,105,174,59,270)(11,271,60,175,106,206,239,90,143)(12,144,91,240,207,107,176,61,272)(13,273,62,177,108,208,241,92,145)(14,146,93,242,209,109,178,63,274)(15,275,64,179,110,210,243,94,147)(16,148,95,244,211,111,180,33,276)(17,277,34,181,112,212,245,96,149)(18,150,65,246,213,113,182,35,278)(19,279,36,183,114,214,247,66,151)(20,152,67,248,215,115,184,37,280)(21,281,38,185,116,216,249,68,153)(22,154,69,250,217,117,186,39,282)(23,283,40,187,118,218,251,70,155)(24,156,71,252,219,119,188,41,284)(25,285,42,189,120,220,253,72,157)(26,158,73,254,221,121,190,43,286)(27,287,44,191,122,222,255,74,159)(28,160,75,256,223,123,192,45,288)(29,257,46,161,124,224,225,76,129)(30,130,77,226,193,125,162,47,258)(31,259,48,163,126,194,227,78,131)(32,132,79,228,195,127,164,49,260), (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) );
G=PermutationGroup([[(1,261,50,165,128,196,229,80,133),(2,134,81,230,197,97,166,51,262),(3,263,52,167,98,198,231,82,135),(4,136,83,232,199,99,168,53,264),(5,265,54,169,100,200,233,84,137),(6,138,85,234,201,101,170,55,266),(7,267,56,171,102,202,235,86,139),(8,140,87,236,203,103,172,57,268),(9,269,58,173,104,204,237,88,141),(10,142,89,238,205,105,174,59,270),(11,271,60,175,106,206,239,90,143),(12,144,91,240,207,107,176,61,272),(13,273,62,177,108,208,241,92,145),(14,146,93,242,209,109,178,63,274),(15,275,64,179,110,210,243,94,147),(16,148,95,244,211,111,180,33,276),(17,277,34,181,112,212,245,96,149),(18,150,65,246,213,113,182,35,278),(19,279,36,183,114,214,247,66,151),(20,152,67,248,215,115,184,37,280),(21,281,38,185,116,216,249,68,153),(22,154,69,250,217,117,186,39,282),(23,283,40,187,118,218,251,70,155),(24,156,71,252,219,119,188,41,284),(25,285,42,189,120,220,253,72,157),(26,158,73,254,221,121,190,43,286),(27,287,44,191,122,222,255,74,159),(28,160,75,256,223,123,192,45,288),(29,257,46,161,124,224,225,76,129),(30,130,77,226,193,125,162,47,258),(31,259,48,163,126,194,227,78,131),(32,132,79,228,195,127,164,49,260)], [(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)]])
96 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 16A | ··· | 16H | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 32A | ··· | 32P | 36A | ··· | 36F | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 16 | ··· | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | S3 | Dic3 | D9 | C3⋊C8 | Dic9 | C3⋊C16 | C9⋊C8 | C3⋊C32 | C9⋊C16 | C9⋊C32 |
| kernel | C9⋊C32 | C144 | C72 | C36 | C18 | C9 | C48 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C9⋊C32 ►in GL2(𝔽17) generated by
| 14 | 5 |
| 4 | 10 |
| 0 | 3 |
| 1 | 0 |
G:=sub<GL(2,GF(17))| [14,4,5,10],[0,1,3,0] >;
C9⋊C32 in GAP, Magma, Sage, TeX
C_9\rtimes C_{32} % in TeX
G:=Group("C9:C32"); // GroupNames label
G:=SmallGroup(288,1);
// by ID
G=gap.SmallGroup(288,1);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,14,36,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b|a^9=b^32=1,b*a*b^-1=a^-1>;
// generators/relations
Export