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## G = C18.C42order 288 = 25·32

### 5th non-split extension by C18 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C18.C42
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×C18 — C22×Dic9 — C18.C42
 Lower central C9 — C18 — C18.C42
 Upper central C1 — C23 — C22×C4

Generators and relations for C18.C42
G = < a,b,c | a18=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a9b >

Subgroups: 380 in 114 conjugacy classes, 64 normal (28 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, C2×C6, C2×C6, C22×C4, C22×C4, C18, C18, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, Dic9, C36, C2×C18, C2×C18, C22×Dic3, C22×C12, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C22×C18, C6.C42, C22×Dic9, C22×C36, C18.C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, D9, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, Dic9, D18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C6.C42, C4×Dic9, Dic9⋊C4, C4⋊Dic9, D18⋊C4, C18.D4, C18.C42

Smallest permutation representation of C18.C42
Regular action on 288 points
Generators in S288
```(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)
(1 190 50 149)(2 189 51 148)(3 188 52 147)(4 187 53 146)(5 186 54 145)(6 185 37 162)(7 184 38 161)(8 183 39 160)(9 182 40 159)(10 181 41 158)(11 198 42 157)(12 197 43 156)(13 196 44 155)(14 195 45 154)(15 194 46 153)(16 193 47 152)(17 192 48 151)(18 191 49 150)(19 210 57 163)(20 209 58 180)(21 208 59 179)(22 207 60 178)(23 206 61 177)(24 205 62 176)(25 204 63 175)(26 203 64 174)(27 202 65 173)(28 201 66 172)(29 200 67 171)(30 199 68 170)(31 216 69 169)(32 215 70 168)(33 214 71 167)(34 213 72 166)(35 212 55 165)(36 211 56 164)(73 266 114 220)(74 265 115 219)(75 264 116 218)(76 263 117 217)(77 262 118 234)(78 261 119 233)(79 260 120 232)(80 259 121 231)(81 258 122 230)(82 257 123 229)(83 256 124 228)(84 255 125 227)(85 254 126 226)(86 253 109 225)(87 270 110 224)(88 269 111 223)(89 268 112 222)(90 267 113 221)(91 282 135 242)(92 281 136 241)(93 280 137 240)(94 279 138 239)(95 278 139 238)(96 277 140 237)(97 276 141 236)(98 275 142 235)(99 274 143 252)(100 273 144 251)(101 272 127 250)(102 271 128 249)(103 288 129 248)(104 287 130 247)(105 286 131 246)(106 285 132 245)(107 284 133 244)(108 283 134 243)
(1 102 30 90)(2 103 31 73)(3 104 32 74)(4 105 33 75)(5 106 34 76)(6 107 35 77)(7 108 36 78)(8 91 19 79)(9 92 20 80)(10 93 21 81)(11 94 22 82)(12 95 23 83)(13 96 24 84)(14 97 25 85)(15 98 26 86)(16 99 27 87)(17 100 28 88)(18 101 29 89)(37 133 55 118)(38 134 56 119)(39 135 57 120)(40 136 58 121)(41 137 59 122)(42 138 60 123)(43 139 61 124)(44 140 62 125)(45 141 63 126)(46 142 64 109)(47 143 65 110)(48 144 66 111)(49 127 67 112)(50 128 68 113)(51 129 69 114)(52 130 70 115)(53 131 71 116)(54 132 72 117)(145 236 166 226)(146 237 167 227)(147 238 168 228)(148 239 169 229)(149 240 170 230)(150 241 171 231)(151 242 172 232)(152 243 173 233)(153 244 174 234)(154 245 175 217)(155 246 176 218)(156 247 177 219)(157 248 178 220)(158 249 179 221)(159 250 180 222)(160 251 163 223)(161 252 164 224)(162 235 165 225)(181 271 208 267)(182 272 209 268)(183 273 210 269)(184 274 211 270)(185 275 212 253)(186 276 213 254)(187 277 214 255)(188 278 215 256)(189 279 216 257)(190 280 199 258)(191 281 200 259)(192 282 201 260)(193 283 202 261)(194 284 203 262)(195 285 204 263)(196 286 205 264)(197 287 206 265)(198 288 207 266)```

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 2 2 2 2 2 18 ··· 18 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + + - + - + - + image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 D9 Dic6 C4×S3 D12 C3⋊D4 Dic9 D18 Dic18 C4×D9 D36 C9⋊D4 kernel C18.C42 C22×Dic9 C22×C36 C2×Dic9 C2×C36 C22×C12 C2×C18 C2×C18 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 C22 # reps 1 2 1 8 4 1 3 1 2 1 3 2 4 2 4 6 3 6 12 6 12

Matrix representation of C18.C42 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 17 6 0 0 31 11
,
 6 0 0 0 0 1 0 0 0 0 25 33 0 0 8 12
,
 6 0 0 0 0 31 0 0 0 0 5 10 0 0 27 32
`G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,17,31,0,0,6,11],[6,0,0,0,0,1,0,0,0,0,25,8,0,0,33,12],[6,0,0,0,0,31,0,0,0,0,5,27,0,0,10,32] >;`

C18.C42 in GAP, Magma, Sage, TeX

`C_{18}.C_4^2`
`% in TeX`

`G:=Group("C18.C4^2");`
`// GroupNames label`

`G:=SmallGroup(288,38);`
`// by ID`

`G=gap.SmallGroup(288,38);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^18=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;`
`// generators/relations`

׿
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