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## G = C2×C9⋊C16order 288 = 25·32

### Direct product of C2 and C9⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C9⋊C16
 Chief series C1 — C3 — C9 — C18 — C36 — C72 — C9⋊C16 — C2×C9⋊C16
 Lower central C9 — C2×C9⋊C16
 Upper central C1 — C2×C8

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
(1 125)(2 126)(3 127)(4 128)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 121)(14 122)(15 123)(16 124)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 196)(34 197)(35 198)(36 199)(37 200)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 193)(47 194)(48 195)(49 108)(50 109)(51 110)(52 111)(53 112)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(81 159)(82 160)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(129 228)(130 229)(131 230)(132 231)(133 232)(134 233)(135 234)(136 235)(137 236)(138 237)(139 238)(140 239)(141 240)(142 225)(143 226)(144 227)(161 272)(162 257)(163 258)(164 259)(165 260)(166 261)(167 262)(168 263)(169 264)(170 265)(171 266)(172 267)(173 268)(174 269)(175 270)(176 271)(177 280)(178 281)(179 282)(180 283)(181 284)(182 285)(183 286)(184 287)(185 288)(186 273)(187 274)(188 275)(189 276)(190 277)(191 278)(192 279)(209 250)(210 251)(211 252)(212 253)(213 254)(214 255)(215 256)(216 241)(217 242)(218 243)(219 244)(220 245)(221 246)(222 247)(223 248)(224 249)
(1 250 42 172 277 65 58 236 145)(2 146 237 59 66 278 173 43 251)(3 252 44 174 279 67 60 238 147)(4 148 239 61 68 280 175 45 253)(5 254 46 176 281 69 62 240 149)(6 150 225 63 70 282 161 47 255)(7 256 48 162 283 71 64 226 151)(8 152 227 49 72 284 163 33 241)(9 242 34 164 285 73 50 228 153)(10 154 229 51 74 286 165 35 243)(11 244 36 166 287 75 52 230 155)(12 156 231 53 76 288 167 37 245)(13 246 38 168 273 77 54 232 157)(14 158 233 55 78 274 169 39 247)(15 248 40 170 275 79 56 234 159)(16 160 235 57 80 276 171 41 249)(17 107 143 89 115 215 195 257 180)(18 181 258 196 216 116 90 144 108)(19 109 129 91 117 217 197 259 182)(20 183 260 198 218 118 92 130 110)(21 111 131 93 119 219 199 261 184)(22 185 262 200 220 120 94 132 112)(23 97 133 95 121 221 201 263 186)(24 187 264 202 222 122 96 134 98)(25 99 135 81 123 223 203 265 188)(26 189 266 204 224 124 82 136 100)(27 101 137 83 125 209 205 267 190)(28 191 268 206 210 126 84 138 102)(29 103 139 85 127 211 207 269 192)(30 177 270 208 212 128 86 140 104)(31 105 141 87 113 213 193 271 178)(32 179 272 194 214 114 88 142 106)
(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)| (1,125)(2,126)(3,127)(4,128)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,193)(47,194)(48,195)(49,108)(50,109)(51,110)(52,111)(53,112)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(64,107)(81,159)(82,160)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(129,228)(130,229)(131,230)(132,231)(133,232)(134,233)(135,234)(136,235)(137,236)(138,237)(139,238)(140,239)(141,240)(142,225)(143,226)(144,227)(161,272)(162,257)(163,258)(164,259)(165,260)(166,261)(167,262)(168,263)(169,264)(170,265)(171,266)(172,267)(173,268)(174,269)(175,270)(176,271)(177,280)(178,281)(179,282)(180,283)(181,284)(182,285)(183,286)(184,287)(185,288)(186,273)(187,274)(188,275)(189,276)(190,277)(191,278)(192,279)(209,250)(210,251)(211,252)(212,253)(213,254)(214,255)(215,256)(216,241)(217,242)(218,243)(219,244)(220,245)(221,246)(222,247)(223,248)(224,249), (1,250,42,172,277,65,58,236,145)(2,146,237,59,66,278,173,43,251)(3,252,44,174,279,67,60,238,147)(4,148,239,61,68,280,175,45,253)(5,254,46,176,281,69,62,240,149)(6,150,225,63,70,282,161,47,255)(7,256,48,162,283,71,64,226,151)(8,152,227,49,72,284,163,33,241)(9,242,34,164,285,73,50,228,153)(10,154,229,51,74,286,165,35,243)(11,244,36,166,287,75,52,230,155)(12,156,231,53,76,288,167,37,245)(13,246,38,168,273,77,54,232,157)(14,158,233,55,78,274,169,39,247)(15,248,40,170,275,79,56,234,159)(16,160,235,57,80,276,171,41,249)(17,107,143,89,115,215,195,257,180)(18,181,258,196,216,116,90,144,108)(19,109,129,91,117,217,197,259,182)(20,183,260,198,218,118,92,130,110)(21,111,131,93,119,219,199,261,184)(22,185,262,200,220,120,94,132,112)(23,97,133,95,121,221,201,263,186)(24,187,264,202,222,122,96,134,98)(25,99,135,81,123,223,203,265,188)(26,189,266,204,224,124,82,136,100)(27,101,137,83,125,209,205,267,190)(28,191,268,206,210,126,84,138,102)(29,103,139,85,127,211,207,269,192)(30,177,270,208,212,128,86,140,104)(31,105,141,87,113,213,193,271,178)(32,179,272,194,214,114,88,142,106), (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( (1,125)(2,126)(3,127)(4,128)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,193)(47,194)(48,195)(49,108)(50,109)(51,110)(52,111)(53,112)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(64,107)(81,159)(82,160)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(129,228)(130,229)(131,230)(132,231)(133,232)(134,233)(135,234)(136,235)(137,236)(138,237)(139,238)(140,239)(141,240)(142,225)(143,226)(144,227)(161,272)(162,257)(163,258)(164,259)(165,260)(166,261)(167,262)(168,263)(169,264)(170,265)(171,266)(172,267)(173,268)(174,269)(175,270)(176,271)(177,280)(178,281)(179,282)(180,283)(181,284)(182,285)(183,286)(184,287)(185,288)(186,273)(187,274)(188,275)(189,276)(190,277)(191,278)(192,279)(209,250)(210,251)(211,252)(212,253)(213,254)(214,255)(215,256)(216,241)(217,242)(218,243)(219,244)(220,245)(221,246)(222,247)(223,248)(224,249), (1,250,42,172,277,65,58,236,145)(2,146,237,59,66,278,173,43,251)(3,252,44,174,279,67,60,238,147)(4,148,239,61,68,280,175,45,253)(5,254,46,176,281,69,62,240,149)(6,150,225,63,70,282,161,47,255)(7,256,48,162,283,71,64,226,151)(8,152,227,49,72,284,163,33,241)(9,242,34,164,285,73,50,228,153)(10,154,229,51,74,286,165,35,243)(11,244,36,166,287,75,52,230,155)(12,156,231,53,76,288,167,37,245)(13,246,38,168,273,77,54,232,157)(14,158,233,55,78,274,169,39,247)(15,248,40,170,275,79,56,234,159)(16,160,235,57,80,276,171,41,249)(17,107,143,89,115,215,195,257,180)(18,181,258,196,216,116,90,144,108)(19,109,129,91,117,217,197,259,182)(20,183,260,198,218,118,92,130,110)(21,111,131,93,119,219,199,261,184)(22,185,262,200,220,120,94,132,112)(23,97,133,95,121,221,201,263,186)(24,187,264,202,222,122,96,134,98)(25,99,135,81,123,223,203,265,188)(26,189,266,204,224,124,82,136,100)(27,101,137,83,125,209,205,267,190)(28,191,268,206,210,126,84,138,102)(29,103,139,85,127,211,207,269,192)(30,177,270,208,212,128,86,140,104)(31,105,141,87,113,213,193,271,178)(32,179,272,194,214,114,88,142,106), (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([(1,125),(2,126),(3,127),(4,128),(5,113),(6,114),(7,115),(8,116),(9,117),(10,118),(11,119),(12,120),(13,121),(14,122),(15,123),(16,124),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,196),(34,197),(35,198),(36,199),(37,200),(38,201),(39,202),(40,203),(41,204),(42,205),(43,206),(44,207),(45,208),(46,193),(47,194),(48,195),(49,108),(50,109),(51,110),(52,111),(53,112),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,103),(61,104),(62,105),(63,106),(64,107),(81,159),(82,160),(83,145),(84,146),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154),(93,155),(94,156),(95,157),(96,158),(129,228),(130,229),(131,230),(132,231),(133,232),(134,233),(135,234),(136,235),(137,236),(138,237),(139,238),(140,239),(141,240),(142,225),(143,226),(144,227),(161,272),(162,257),(163,258),(164,259),(165,260),(166,261),(167,262),(168,263),(169,264),(170,265),(171,266),(172,267),(173,268),(174,269),(175,270),(176,271),(177,280),(178,281),(179,282),(180,283),(181,284),(182,285),(183,286),(184,287),(185,288),(186,273),(187,274),(188,275),(189,276),(190,277),(191,278),(192,279),(209,250),(210,251),(211,252),(212,253),(213,254),(214,255),(215,256),(216,241),(217,242),(218,243),(219,244),(220,245),(221,246),(222,247),(223,248),(224,249)], [(1,250,42,172,277,65,58,236,145),(2,146,237,59,66,278,173,43,251),(3,252,44,174,279,67,60,238,147),(4,148,239,61,68,280,175,45,253),(5,254,46,176,281,69,62,240,149),(6,150,225,63,70,282,161,47,255),(7,256,48,162,283,71,64,226,151),(8,152,227,49,72,284,163,33,241),(9,242,34,164,285,73,50,228,153),(10,154,229,51,74,286,165,35,243),(11,244,36,166,287,75,52,230,155),(12,156,231,53,76,288,167,37,245),(13,246,38,168,273,77,54,232,157),(14,158,233,55,78,274,169,39,247),(15,248,40,170,275,79,56,234,159),(16,160,235,57,80,276,171,41,249),(17,107,143,89,115,215,195,257,180),(18,181,258,196,216,116,90,144,108),(19,109,129,91,117,217,197,259,182),(20,183,260,198,218,118,92,130,110),(21,111,131,93,119,219,199,261,184),(22,185,262,200,220,120,94,132,112),(23,97,133,95,121,221,201,263,186),(24,187,264,202,222,122,96,134,98),(25,99,135,81,123,223,203,265,188),(26,189,266,204,224,124,82,136,100),(27,101,137,83,125,209,205,267,190),(28,191,268,206,210,126,84,138,102),(29,103,139,85,127,211,207,269,192),(30,177,270,208,212,128,86,140,104),(31,105,141,87,113,213,193,271,178),(32,179,272,194,214,114,88,142,106)], [(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

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