direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C9⋊C16, C18⋊C16, C72.3C4, C36.3C8, C8.20D18, C24.86D6, C8.4Dic9, C24.6Dic3, C72.21C22, C9⋊2(C2×C16), C4.3(C9⋊C8), (C2×C8).9D9, C12.6(C3⋊C8), C6.2(C3⋊C16), (C2×C72).9C2, C18.8(C2×C8), (C2×C18).2C8, (C2×C24).23S3, (C2×C36).12C4, C36.40(C2×C4), C22.2(C9⋊C8), C4.9(C2×Dic9), (C2×C4).8Dic9, (C2×C12).21Dic3, C12.49(C2×Dic3), C3.(C2×C3⋊C16), C2.2(C2×C9⋊C8), C6.8(C2×C3⋊C8), (C2×C6).3(C3⋊C8), SmallGroup(288,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C2×C9⋊C16 |
Generators and relations for C2×C9⋊C16
G = < a,b,c | a2=b9=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C9⋊C16
►Regular action on 288 points
Generators in S288
(1 159)(2 160)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 142)(45 143)(46 144)(47 129)(48 130)(49 191)(50 192)(51 177)(52 178)(53 179)(54 180)(55 181)(56 182)(57 183)(58 184)(59 185)(60 186)(61 187)(62 188)(63 189)(64 190)(65 265)(66 266)(67 267)(68 268)(69 269)(70 270)(71 271)(72 272)(73 257)(74 258)(75 259)(76 260)(77 261)(78 262)(79 263)(80 264)(81 284)(82 285)(83 286)(84 287)(85 288)(86 273)(87 274)(88 275)(89 276)(90 277)(91 278)(92 279)(93 280)(94 281)(95 282)(96 283)(97 200)(98 201)(99 202)(100 203)(101 204)(102 205)(103 206)(104 207)(105 208)(106 193)(107 194)(108 195)(109 196)(110 197)(111 198)(112 199)(161 218)(162 219)(163 220)(164 221)(165 222)(166 223)(167 224)(168 209)(169 210)(170 211)(171 212)(172 213)(173 214)(174 215)(175 216)(176 217)(225 253)(226 254)(227 255)(228 256)(229 241)(230 242)(231 243)(232 244)(233 245)(234 246)(235 247)(236 248)(237 249)(238 250)(239 251)(240 252)
(1 249 114 221 192 80 36 97 93)(2 94 98 37 65 177 222 115 250)(3 251 116 223 178 66 38 99 95)(4 96 100 39 67 179 224 117 252)(5 253 118 209 180 68 40 101 81)(6 82 102 41 69 181 210 119 254)(7 255 120 211 182 70 42 103 83)(8 84 104 43 71 183 212 121 256)(9 241 122 213 184 72 44 105 85)(10 86 106 45 73 185 214 123 242)(11 243 124 215 186 74 46 107 87)(12 88 108 47 75 187 216 125 244)(13 245 126 217 188 76 48 109 89)(14 90 110 33 77 189 218 127 246)(15 247 128 219 190 78 34 111 91)(16 92 112 35 79 191 220 113 248)(17 230 152 273 193 143 257 59 173)(18 174 60 258 144 194 274 153 231)(19 232 154 275 195 129 259 61 175)(20 176 62 260 130 196 276 155 233)(21 234 156 277 197 131 261 63 161)(22 162 64 262 132 198 278 157 235)(23 236 158 279 199 133 263 49 163)(24 164 50 264 134 200 280 159 237)(25 238 160 281 201 135 265 51 165)(26 166 52 266 136 202 282 145 239)(27 240 146 283 203 137 267 53 167)(28 168 54 268 138 204 284 147 225)(29 226 148 285 205 139 269 55 169)(30 170 56 270 140 206 286 149 227)(31 228 150 287 207 141 271 57 171)(32 172 58 272 142 208 288 151 229)
(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)| (1,159)(2,160)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,142)(45,143)(46,144)(47,129)(48,130)(49,191)(50,192)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,183)(58,184)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(65,265)(66,266)(67,267)(68,268)(69,269)(70,270)(71,271)(72,272)(73,257)(74,258)(75,259)(76,260)(77,261)(78,262)(79,263)(80,264)(81,284)(82,285)(83,286)(84,287)(85,288)(86,273)(87,274)(88,275)(89,276)(90,277)(91,278)(92,279)(93,280)(94,281)(95,282)(96,283)(97,200)(98,201)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,193)(107,194)(108,195)(109,196)(110,197)(111,198)(112,199)(161,218)(162,219)(163,220)(164,221)(165,222)(166,223)(167,224)(168,209)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(225,253)(226,254)(227,255)(228,256)(229,241)(230,242)(231,243)(232,244)(233,245)(234,246)(235,247)(236,248)(237,249)(238,250)(239,251)(240,252), (1,249,114,221,192,80,36,97,93)(2,94,98,37,65,177,222,115,250)(3,251,116,223,178,66,38,99,95)(4,96,100,39,67,179,224,117,252)(5,253,118,209,180,68,40,101,81)(6,82,102,41,69,181,210,119,254)(7,255,120,211,182,70,42,103,83)(8,84,104,43,71,183,212,121,256)(9,241,122,213,184,72,44,105,85)(10,86,106,45,73,185,214,123,242)(11,243,124,215,186,74,46,107,87)(12,88,108,47,75,187,216,125,244)(13,245,126,217,188,76,48,109,89)(14,90,110,33,77,189,218,127,246)(15,247,128,219,190,78,34,111,91)(16,92,112,35,79,191,220,113,248)(17,230,152,273,193,143,257,59,173)(18,174,60,258,144,194,274,153,231)(19,232,154,275,195,129,259,61,175)(20,176,62,260,130,196,276,155,233)(21,234,156,277,197,131,261,63,161)(22,162,64,262,132,198,278,157,235)(23,236,158,279,199,133,263,49,163)(24,164,50,264,134,200,280,159,237)(25,238,160,281,201,135,265,51,165)(26,166,52,266,136,202,282,145,239)(27,240,146,283,203,137,267,53,167)(28,168,54,268,138,204,284,147,225)(29,226,148,285,205,139,269,55,169)(30,170,56,270,140,206,286,149,227)(31,228,150,287,207,141,271,57,171)(32,172,58,272,142,208,288,151,229), (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( (1,159)(2,160)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,142)(45,143)(46,144)(47,129)(48,130)(49,191)(50,192)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,183)(58,184)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(65,265)(66,266)(67,267)(68,268)(69,269)(70,270)(71,271)(72,272)(73,257)(74,258)(75,259)(76,260)(77,261)(78,262)(79,263)(80,264)(81,284)(82,285)(83,286)(84,287)(85,288)(86,273)(87,274)(88,275)(89,276)(90,277)(91,278)(92,279)(93,280)(94,281)(95,282)(96,283)(97,200)(98,201)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,193)(107,194)(108,195)(109,196)(110,197)(111,198)(112,199)(161,218)(162,219)(163,220)(164,221)(165,222)(166,223)(167,224)(168,209)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(225,253)(226,254)(227,255)(228,256)(229,241)(230,242)(231,243)(232,244)(233,245)(234,246)(235,247)(236,248)(237,249)(238,250)(239,251)(240,252), (1,249,114,221,192,80,36,97,93)(2,94,98,37,65,177,222,115,250)(3,251,116,223,178,66,38,99,95)(4,96,100,39,67,179,224,117,252)(5,253,118,209,180,68,40,101,81)(6,82,102,41,69,181,210,119,254)(7,255,120,211,182,70,42,103,83)(8,84,104,43,71,183,212,121,256)(9,241,122,213,184,72,44,105,85)(10,86,106,45,73,185,214,123,242)(11,243,124,215,186,74,46,107,87)(12,88,108,47,75,187,216,125,244)(13,245,126,217,188,76,48,109,89)(14,90,110,33,77,189,218,127,246)(15,247,128,219,190,78,34,111,91)(16,92,112,35,79,191,220,113,248)(17,230,152,273,193,143,257,59,173)(18,174,60,258,144,194,274,153,231)(19,232,154,275,195,129,259,61,175)(20,176,62,260,130,196,276,155,233)(21,234,156,277,197,131,261,63,161)(22,162,64,262,132,198,278,157,235)(23,236,158,279,199,133,263,49,163)(24,164,50,264,134,200,280,159,237)(25,238,160,281,201,135,265,51,165)(26,166,52,266,136,202,282,145,239)(27,240,146,283,203,137,267,53,167)(28,168,54,268,138,204,284,147,225)(29,226,148,285,205,139,269,55,169)(30,170,56,270,140,206,286,149,227)(31,228,150,287,207,141,271,57,171)(32,172,58,272,142,208,288,151,229), (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([[(1,159),(2,160),(3,145),(4,146),(5,147),(6,148),(7,149),(8,150),(9,151),(10,152),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,123),(18,124),(19,125),(20,126),(21,127),(22,128),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,121),(32,122),(33,131),(34,132),(35,133),(36,134),(37,135),(38,136),(39,137),(40,138),(41,139),(42,140),(43,141),(44,142),(45,143),(46,144),(47,129),(48,130),(49,191),(50,192),(51,177),(52,178),(53,179),(54,180),(55,181),(56,182),(57,183),(58,184),(59,185),(60,186),(61,187),(62,188),(63,189),(64,190),(65,265),(66,266),(67,267),(68,268),(69,269),(70,270),(71,271),(72,272),(73,257),(74,258),(75,259),(76,260),(77,261),(78,262),(79,263),(80,264),(81,284),(82,285),(83,286),(84,287),(85,288),(86,273),(87,274),(88,275),(89,276),(90,277),(91,278),(92,279),(93,280),(94,281),(95,282),(96,283),(97,200),(98,201),(99,202),(100,203),(101,204),(102,205),(103,206),(104,207),(105,208),(106,193),(107,194),(108,195),(109,196),(110,197),(111,198),(112,199),(161,218),(162,219),(163,220),(164,221),(165,222),(166,223),(167,224),(168,209),(169,210),(170,211),(171,212),(172,213),(173,214),(174,215),(175,216),(176,217),(225,253),(226,254),(227,255),(228,256),(229,241),(230,242),(231,243),(232,244),(233,245),(234,246),(235,247),(236,248),(237,249),(238,250),(239,251),(240,252)], [(1,249,114,221,192,80,36,97,93),(2,94,98,37,65,177,222,115,250),(3,251,116,223,178,66,38,99,95),(4,96,100,39,67,179,224,117,252),(5,253,118,209,180,68,40,101,81),(6,82,102,41,69,181,210,119,254),(7,255,120,211,182,70,42,103,83),(8,84,104,43,71,183,212,121,256),(9,241,122,213,184,72,44,105,85),(10,86,106,45,73,185,214,123,242),(11,243,124,215,186,74,46,107,87),(12,88,108,47,75,187,216,125,244),(13,245,126,217,188,76,48,109,89),(14,90,110,33,77,189,218,127,246),(15,247,128,219,190,78,34,111,91),(16,92,112,35,79,191,220,113,248),(17,230,152,273,193,143,257,59,173),(18,174,60,258,144,194,274,153,231),(19,232,154,275,195,129,259,61,175),(20,176,62,260,130,196,276,155,233),(21,234,156,277,197,131,261,63,161),(22,162,64,262,132,198,278,157,235),(23,236,158,279,199,133,263,49,163),(24,164,50,264,134,200,280,159,237),(25,238,160,281,201,135,265,51,165),(26,166,52,266,136,202,282,145,239),(27,240,146,283,203,137,267,53,167),(28,168,54,268,138,204,284,147,225),(29,226,148,285,205,139,269,55,169),(30,170,56,270,140,206,286,149,227),(31,228,150,287,207,141,271,57,171),(32,172,58,272,142,208,288,151,229)], [(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 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | ··· | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 16A | ··· | 16P | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | Dic3 | D6 | Dic3 | D9 | C3⋊C8 | C3⋊C8 | Dic9 | D18 | Dic9 | C3⋊C16 | C9⋊C8 | C9⋊C8 | C9⋊C16 |
| kernel | C2×C9⋊C16 | C9⋊C16 | C2×C72 | C72 | C2×C36 | C36 | C2×C18 | C18 | C2×C24 | C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C2×C9⋊C16 ►in GL4(𝔽433) generated by
| 432 | 0 | 0 | 0 |
| 0 | 432 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 350 | 397 |
| 0 | 0 | 36 | 386 |
| 250 | 0 | 0 | 0 |
| 0 | 432 | 0 | 0 |
| 0 | 0 | 316 | 346 |
| 0 | 0 | 30 | 117 |
G:=sub<GL(4,GF(433))| [432,0,0,0,0,432,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,350,36,0,0,397,386],[250,0,0,0,0,432,0,0,0,0,316,30,0,0,346,117] >;
C2×C9⋊C16 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes C_{16} % in TeX
G:=Group("C2xC9:C16"); // GroupNames label
G:=SmallGroup(288,18);
// by ID
G=gap.SmallGroup(288,18);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^2=b^9=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export