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## G = C18.Q16order 288 = 25·32

### 2nd non-split extension by C18 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C18.Q16
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×C36 — C2×Dic18 — C18.Q16
 Lower central C9 — C18 — C36 — C18.Q16
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C18.Q16
G = < a,b,c | a18=b8=1, c2=a9b4, bab-1=a-1, ac=ca, cbc-1=a9b-1 >

Subgroups: 240 in 63 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C9, Dic3 [×2], C12 [×2], C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C18 [×3], C3⋊C8, Dic6 [×3], C2×Dic3, C2×C12, C2×C12, Q8⋊C4, Dic9 [×2], C36 [×2], C36, C2×C18, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C9⋊C8, Dic18 [×2], Dic18, C2×Dic9, C2×C36, C2×C36, C6.SD16, C2×C9⋊C8, C9×C4⋊C4, C2×Dic18, C18.Q16
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, SD16, Q16, D9, C4×S3, D12, C3⋊D4, Q8⋊C4, D18, D6⋊C4, D4.S3, C3⋊Q16, C4×D9, D36, C9⋊D4, C6.SD16, D18⋊C4, D4.D9, C9⋊Q16, C18.Q16

Smallest permutation representation of C18.Q16
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)
(1 225 44 90 98 25 245 112)(2 224 45 89 99 24 246 111)(3 223 46 88 100 23 247 110)(4 222 47 87 101 22 248 109)(5 221 48 86 102 21 249 126)(6 220 49 85 103 20 250 125)(7 219 50 84 104 19 251 124)(8 218 51 83 105 36 252 123)(9 217 52 82 106 35 235 122)(10 234 53 81 107 34 236 121)(11 233 54 80 108 33 237 120)(12 232 37 79 91 32 238 119)(13 231 38 78 92 31 239 118)(14 230 39 77 93 30 240 117)(15 229 40 76 94 29 241 116)(16 228 41 75 95 28 242 115)(17 227 42 74 96 27 243 114)(18 226 43 73 97 26 244 113)(55 277 213 137 266 184 180 145)(56 276 214 136 267 183 163 162)(57 275 215 135 268 182 164 161)(58 274 216 134 269 181 165 160)(59 273 199 133 270 198 166 159)(60 272 200 132 253 197 167 158)(61 271 201 131 254 196 168 157)(62 288 202 130 255 195 169 156)(63 287 203 129 256 194 170 155)(64 286 204 128 257 193 171 154)(65 285 205 127 258 192 172 153)(66 284 206 144 259 191 173 152)(67 283 207 143 260 190 174 151)(68 282 208 142 261 189 175 150)(69 281 209 141 262 188 176 149)(70 280 210 140 263 187 177 148)(71 279 211 139 264 186 178 147)(72 278 212 138 265 185 179 146)
(1 263 107 61)(2 264 108 62)(3 265 91 63)(4 266 92 64)(5 267 93 65)(6 268 94 66)(7 269 95 67)(8 270 96 68)(9 253 97 69)(10 254 98 70)(11 255 99 71)(12 256 100 72)(13 257 101 55)(14 258 102 56)(15 259 103 57)(16 260 104 58)(17 261 105 59)(18 262 106 60)(19 151 228 134)(20 152 229 135)(21 153 230 136)(22 154 231 137)(23 155 232 138)(24 156 233 139)(25 157 234 140)(26 158 217 141)(27 159 218 142)(28 160 219 143)(29 161 220 144)(30 162 221 127)(31 145 222 128)(32 146 223 129)(33 147 224 130)(34 148 225 131)(35 149 226 132)(36 150 227 133)(37 203 247 179)(38 204 248 180)(39 205 249 163)(40 206 250 164)(41 207 251 165)(42 208 252 166)(43 209 235 167)(44 210 236 168)(45 211 237 169)(46 212 238 170)(47 213 239 171)(48 214 240 172)(49 215 241 173)(50 216 242 174)(51 199 243 175)(52 200 244 176)(53 201 245 177)(54 202 246 178)(73 272 122 188)(74 273 123 189)(75 274 124 190)(76 275 125 191)(77 276 126 192)(78 277 109 193)(79 278 110 194)(80 279 111 195)(81 280 112 196)(82 281 113 197)(83 282 114 198)(84 283 115 181)(85 284 116 182)(86 285 117 183)(87 286 118 184)(88 287 119 185)(89 288 120 186)(90 271 121 187)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 4 4 36 36 2 2 2 18 18 18 18 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + + - - - - image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 Q16 D9 C4×S3 D12 C3⋊D4 D18 C4×D9 D36 C9⋊D4 D4.S3 C3⋊Q16 D4.D9 C9⋊Q16 kernel C18.Q16 C2×C9⋊C8 C9×C4⋊C4 C2×Dic18 Dic18 C3×C4⋊C4 C36 C2×C18 C2×C12 C18 C18 C4⋊C4 C12 C12 C2×C6 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 3 2 2 2 3 6 6 6 1 1 3 3

Matrix representation of C18.Q16 in GL4(𝔽73) generated by

 28 31 0 0 42 70 0 0 0 0 72 0 0 0 0 72
,
 12 63 0 0 51 61 0 0 0 0 61 61 0 0 6 0
,
 27 0 0 0 0 27 0 0 0 0 35 58 0 0 67 38
`G:=sub<GL(4,GF(73))| [28,42,0,0,31,70,0,0,0,0,72,0,0,0,0,72],[12,51,0,0,63,61,0,0,0,0,61,6,0,0,61,0],[27,0,0,0,0,27,0,0,0,0,35,67,0,0,58,38] >;`

C18.Q16 in GAP, Magma, Sage, TeX

`C_{18}.Q_{16}`
`% in TeX`

`G:=Group("C18.Q16");`
`// GroupNames label`

`G:=SmallGroup(288,16);`
`// by ID`

`G=gap.SmallGroup(288,16);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,675,346,80,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^18=b^8=1,c^2=a^9*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

׿
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