metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C9⋊2Q32, Q16.D9, C36.6D4, C24.9D6, C8.7D18, C18.11D8, C72.5C22, Dic36.2C2, C9⋊C16.C2, C3.(C3⋊Q32), C2.7(D4⋊D9), C4.4(C9⋊D4), (C9×Q16).1C2, (C3×Q16).2S3, C6.18(D4⋊S3), C12.4(C3⋊D4), SmallGroup(288,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊Q32
G = < a,b,c | a9=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9⋊Q32
►Regular action on 288 points
Generators in S288
(1 156 89 53 99 69 179 261 274)(2 275 262 180 70 100 54 90 157)(3 158 91 55 101 71 181 263 276)(4 277 264 182 72 102 56 92 159)(5 160 93 57 103 73 183 265 278)(6 279 266 184 74 104 58 94 145)(7 146 95 59 105 75 185 267 280)(8 281 268 186 76 106 60 96 147)(9 148 81 61 107 77 187 269 282)(10 283 270 188 78 108 62 82 149)(11 150 83 63 109 79 189 271 284)(12 285 272 190 80 110 64 84 151)(13 152 85 49 111 65 191 257 286)(14 287 258 192 66 112 50 86 153)(15 154 87 51 97 67 177 259 288)(16 273 260 178 68 98 52 88 155)(17 113 240 214 244 40 194 165 136)(18 137 166 195 41 245 215 225 114)(19 115 226 216 246 42 196 167 138)(20 139 168 197 43 247 217 227 116)(21 117 228 218 248 44 198 169 140)(22 141 170 199 45 249 219 229 118)(23 119 230 220 250 46 200 171 142)(24 143 172 201 47 251 221 231 120)(25 121 232 222 252 48 202 173 144)(26 129 174 203 33 253 223 233 122)(27 123 234 224 254 34 204 175 130)(28 131 176 205 35 255 209 235 124)(29 125 236 210 256 36 206 161 132)(30 133 162 207 37 241 211 237 126)(31 127 238 212 242 38 208 163 134)(32 135 164 193 39 243 213 239 128)
(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 45 9 37)(2 44 10 36)(3 43 11 35)(4 42 12 34)(5 41 13 33)(6 40 14 48)(7 39 15 47)(8 38 16 46)(17 66 25 74)(18 65 26 73)(19 80 27 72)(20 79 28 71)(21 78 29 70)(22 77 30 69)(23 76 31 68)(24 75 32 67)(49 233 57 225)(50 232 58 240)(51 231 59 239)(52 230 60 238)(53 229 61 237)(54 228 62 236)(55 227 63 235)(56 226 64 234)(81 211 89 219)(82 210 90 218)(83 209 91 217)(84 224 92 216)(85 223 93 215)(86 222 94 214)(87 221 95 213)(88 220 96 212)(97 120 105 128)(98 119 106 127)(99 118 107 126)(100 117 108 125)(101 116 109 124)(102 115 110 123)(103 114 111 122)(104 113 112 121)(129 183 137 191)(130 182 138 190)(131 181 139 189)(132 180 140 188)(133 179 141 187)(134 178 142 186)(135 177 143 185)(136 192 144 184)(145 244 153 252)(146 243 154 251)(147 242 155 250)(148 241 156 249)(149 256 157 248)(150 255 158 247)(151 254 159 246)(152 253 160 245)(161 262 169 270)(162 261 170 269)(163 260 171 268)(164 259 172 267)(165 258 173 266)(166 257 174 265)(167 272 175 264)(168 271 176 263)(193 288 201 280)(194 287 202 279)(195 286 203 278)(196 285 204 277)(197 284 205 276)(198 283 206 275)(199 282 207 274)(200 281 208 273)
G:=sub<Sym(288)| (1,156,89,53,99,69,179,261,274)(2,275,262,180,70,100,54,90,157)(3,158,91,55,101,71,181,263,276)(4,277,264,182,72,102,56,92,159)(5,160,93,57,103,73,183,265,278)(6,279,266,184,74,104,58,94,145)(7,146,95,59,105,75,185,267,280)(8,281,268,186,76,106,60,96,147)(9,148,81,61,107,77,187,269,282)(10,283,270,188,78,108,62,82,149)(11,150,83,63,109,79,189,271,284)(12,285,272,190,80,110,64,84,151)(13,152,85,49,111,65,191,257,286)(14,287,258,192,66,112,50,86,153)(15,154,87,51,97,67,177,259,288)(16,273,260,178,68,98,52,88,155)(17,113,240,214,244,40,194,165,136)(18,137,166,195,41,245,215,225,114)(19,115,226,216,246,42,196,167,138)(20,139,168,197,43,247,217,227,116)(21,117,228,218,248,44,198,169,140)(22,141,170,199,45,249,219,229,118)(23,119,230,220,250,46,200,171,142)(24,143,172,201,47,251,221,231,120)(25,121,232,222,252,48,202,173,144)(26,129,174,203,33,253,223,233,122)(27,123,234,224,254,34,204,175,130)(28,131,176,205,35,255,209,235,124)(29,125,236,210,256,36,206,161,132)(30,133,162,207,37,241,211,237,126)(31,127,238,212,242,38,208,163,134)(32,135,164,193,39,243,213,239,128), (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), (1,45,9,37)(2,44,10,36)(3,43,11,35)(4,42,12,34)(5,41,13,33)(6,40,14,48)(7,39,15,47)(8,38,16,46)(17,66,25,74)(18,65,26,73)(19,80,27,72)(20,79,28,71)(21,78,29,70)(22,77,30,69)(23,76,31,68)(24,75,32,67)(49,233,57,225)(50,232,58,240)(51,231,59,239)(52,230,60,238)(53,229,61,237)(54,228,62,236)(55,227,63,235)(56,226,64,234)(81,211,89,219)(82,210,90,218)(83,209,91,217)(84,224,92,216)(85,223,93,215)(86,222,94,214)(87,221,95,213)(88,220,96,212)(97,120,105,128)(98,119,106,127)(99,118,107,126)(100,117,108,125)(101,116,109,124)(102,115,110,123)(103,114,111,122)(104,113,112,121)(129,183,137,191)(130,182,138,190)(131,181,139,189)(132,180,140,188)(133,179,141,187)(134,178,142,186)(135,177,143,185)(136,192,144,184)(145,244,153,252)(146,243,154,251)(147,242,155,250)(148,241,156,249)(149,256,157,248)(150,255,158,247)(151,254,159,246)(152,253,160,245)(161,262,169,270)(162,261,170,269)(163,260,171,268)(164,259,172,267)(165,258,173,266)(166,257,174,265)(167,272,175,264)(168,271,176,263)(193,288,201,280)(194,287,202,279)(195,286,203,278)(196,285,204,277)(197,284,205,276)(198,283,206,275)(199,282,207,274)(200,281,208,273)>;
G:=Group( (1,156,89,53,99,69,179,261,274)(2,275,262,180,70,100,54,90,157)(3,158,91,55,101,71,181,263,276)(4,277,264,182,72,102,56,92,159)(5,160,93,57,103,73,183,265,278)(6,279,266,184,74,104,58,94,145)(7,146,95,59,105,75,185,267,280)(8,281,268,186,76,106,60,96,147)(9,148,81,61,107,77,187,269,282)(10,283,270,188,78,108,62,82,149)(11,150,83,63,109,79,189,271,284)(12,285,272,190,80,110,64,84,151)(13,152,85,49,111,65,191,257,286)(14,287,258,192,66,112,50,86,153)(15,154,87,51,97,67,177,259,288)(16,273,260,178,68,98,52,88,155)(17,113,240,214,244,40,194,165,136)(18,137,166,195,41,245,215,225,114)(19,115,226,216,246,42,196,167,138)(20,139,168,197,43,247,217,227,116)(21,117,228,218,248,44,198,169,140)(22,141,170,199,45,249,219,229,118)(23,119,230,220,250,46,200,171,142)(24,143,172,201,47,251,221,231,120)(25,121,232,222,252,48,202,173,144)(26,129,174,203,33,253,223,233,122)(27,123,234,224,254,34,204,175,130)(28,131,176,205,35,255,209,235,124)(29,125,236,210,256,36,206,161,132)(30,133,162,207,37,241,211,237,126)(31,127,238,212,242,38,208,163,134)(32,135,164,193,39,243,213,239,128), (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), (1,45,9,37)(2,44,10,36)(3,43,11,35)(4,42,12,34)(5,41,13,33)(6,40,14,48)(7,39,15,47)(8,38,16,46)(17,66,25,74)(18,65,26,73)(19,80,27,72)(20,79,28,71)(21,78,29,70)(22,77,30,69)(23,76,31,68)(24,75,32,67)(49,233,57,225)(50,232,58,240)(51,231,59,239)(52,230,60,238)(53,229,61,237)(54,228,62,236)(55,227,63,235)(56,226,64,234)(81,211,89,219)(82,210,90,218)(83,209,91,217)(84,224,92,216)(85,223,93,215)(86,222,94,214)(87,221,95,213)(88,220,96,212)(97,120,105,128)(98,119,106,127)(99,118,107,126)(100,117,108,125)(101,116,109,124)(102,115,110,123)(103,114,111,122)(104,113,112,121)(129,183,137,191)(130,182,138,190)(131,181,139,189)(132,180,140,188)(133,179,141,187)(134,178,142,186)(135,177,143,185)(136,192,144,184)(145,244,153,252)(146,243,154,251)(147,242,155,250)(148,241,156,249)(149,256,157,248)(150,255,158,247)(151,254,159,246)(152,253,160,245)(161,262,169,270)(162,261,170,269)(163,260,171,268)(164,259,172,267)(165,258,173,266)(166,257,174,265)(167,272,175,264)(168,271,176,263)(193,288,201,280)(194,287,202,279)(195,286,203,278)(196,285,204,277)(197,284,205,276)(198,283,206,275)(199,282,207,274)(200,281,208,273) );
G=PermutationGroup([[(1,156,89,53,99,69,179,261,274),(2,275,262,180,70,100,54,90,157),(3,158,91,55,101,71,181,263,276),(4,277,264,182,72,102,56,92,159),(5,160,93,57,103,73,183,265,278),(6,279,266,184,74,104,58,94,145),(7,146,95,59,105,75,185,267,280),(8,281,268,186,76,106,60,96,147),(9,148,81,61,107,77,187,269,282),(10,283,270,188,78,108,62,82,149),(11,150,83,63,109,79,189,271,284),(12,285,272,190,80,110,64,84,151),(13,152,85,49,111,65,191,257,286),(14,287,258,192,66,112,50,86,153),(15,154,87,51,97,67,177,259,288),(16,273,260,178,68,98,52,88,155),(17,113,240,214,244,40,194,165,136),(18,137,166,195,41,245,215,225,114),(19,115,226,216,246,42,196,167,138),(20,139,168,197,43,247,217,227,116),(21,117,228,218,248,44,198,169,140),(22,141,170,199,45,249,219,229,118),(23,119,230,220,250,46,200,171,142),(24,143,172,201,47,251,221,231,120),(25,121,232,222,252,48,202,173,144),(26,129,174,203,33,253,223,233,122),(27,123,234,224,254,34,204,175,130),(28,131,176,205,35,255,209,235,124),(29,125,236,210,256,36,206,161,132),(30,133,162,207,37,241,211,237,126),(31,127,238,212,242,38,208,163,134),(32,135,164,193,39,243,213,239,128)], [(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)], [(1,45,9,37),(2,44,10,36),(3,43,11,35),(4,42,12,34),(5,41,13,33),(6,40,14,48),(7,39,15,47),(8,38,16,46),(17,66,25,74),(18,65,26,73),(19,80,27,72),(20,79,28,71),(21,78,29,70),(22,77,30,69),(23,76,31,68),(24,75,32,67),(49,233,57,225),(50,232,58,240),(51,231,59,239),(52,230,60,238),(53,229,61,237),(54,228,62,236),(55,227,63,235),(56,226,64,234),(81,211,89,219),(82,210,90,218),(83,209,91,217),(84,224,92,216),(85,223,93,215),(86,222,94,214),(87,221,95,213),(88,220,96,212),(97,120,105,128),(98,119,106,127),(99,118,107,126),(100,117,108,125),(101,116,109,124),(102,115,110,123),(103,114,111,122),(104,113,112,121),(129,183,137,191),(130,182,138,190),(131,181,139,189),(132,180,140,188),(133,179,141,187),(134,178,142,186),(135,177,143,185),(136,192,144,184),(145,244,153,252),(146,243,154,251),(147,242,155,250),(148,241,156,249),(149,256,157,248),(150,255,158,247),(151,254,159,246),(152,253,160,245),(161,262,169,270),(162,261,170,269),(163,260,171,268),(164,259,172,267),(165,258,173,266),(166,257,174,265),(167,272,175,264),(168,271,176,263),(193,288,201,280),(194,287,202,279),(195,286,203,278),(196,285,204,277),(197,284,205,276),(198,283,206,275),(199,282,207,274),(200,281,208,273)]])
39 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 18A | 18B | 18C | 24A | 24B | 36A | 36B | 36C | 36D | ··· | 36I | 72A | ··· | 72F |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 24 | 24 | 36 | 36 | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 8 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | D9 | C3⋊D4 | Q32 | D18 | C9⋊D4 | D4⋊S3 | C3⋊Q32 | D4⋊D9 | C9⋊Q32 |
| kernel | C9⋊Q32 | C9⋊C16 | Dic36 | C9×Q16 | C3×Q16 | C36 | C24 | C18 | Q16 | C12 | C9 | C8 | C4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 4 | 3 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of C9⋊Q32 ►in GL4(𝔽433) generated by
| 397 | 350 | 0 | 0 |
| 83 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 19 | 66 | 0 | 0 |
| 47 | 414 | 0 | 0 |
| 0 | 0 | 132 | 316 |
| 0 | 0 | 310 | 214 |
| 421 | 409 | 0 | 0 |
| 24 | 12 | 0 | 0 |
| 0 | 0 | 398 | 177 |
| 0 | 0 | 42 | 35 |
G:=sub<GL(4,GF(433))| [397,83,0,0,350,47,0,0,0,0,1,0,0,0,0,1],[19,47,0,0,66,414,0,0,0,0,132,310,0,0,316,214],[421,24,0,0,409,12,0,0,0,0,398,42,0,0,177,35] >;
C9⋊Q32 in GAP, Magma, Sage, TeX
C_9\rtimes Q_{32} % in TeX
G:=Group("C9:Q32"); // GroupNames label
G:=SmallGroup(288,36);
// by ID
G=gap.SmallGroup(288,36);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,120,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export