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## G = Q16×C18order 288 = 25·32

### Direct product of C18 and Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — Q16×C18
 Chief series C1 — C2 — C6 — C12 — C36 — Q8×C9 — C9×Q16 — Q16×C18
 Lower central C1 — C2 — C4 — Q16×C18
 Upper central C1 — C2×C18 — C2×C36 — Q16×C18

Generators and relations for Q16×C18
G = < a,b,c | a18=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 114 in 90 conjugacy classes, 66 normal (24 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×4], Q8 [×2], C9, C12 [×2], C12 [×4], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C18, C18 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C3×Q8 [×2], C2×Q16, C36 [×2], C36 [×4], C2×C18, C2×C24, C3×Q16 [×4], C6×Q8 [×2], C72 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×4], Q8×C9 [×2], C6×Q16, C2×C72, C9×Q16 [×4], Q8×C18 [×2], Q16×C18
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], Q16 [×2], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C2×Q16, C2×C18 [×7], C3×Q16 [×2], C6×D4, D4×C9 [×2], C22×C18, C6×Q16, C9×Q16 [×2], D4×C18, Q16×C18

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

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 12E ··· 12L 18A ··· 18R 24A ··· 24H 36A ··· 36L 36M ··· 36AJ 72A ··· 72X order 1 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 1 1 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 D4 D4 Q16 C3×D4 C3×D4 C3×Q16 D4×C9 D4×C9 C9×Q16 kernel Q16×C18 C2×C72 C9×Q16 Q8×C18 C6×Q16 C2×C24 C3×Q16 C6×Q8 C2×Q16 C2×C8 Q16 C2×Q8 C36 C2×C18 C18 C12 C2×C6 C6 C4 C22 C2 # reps 1 1 4 2 2 2 8 4 6 6 24 12 1 1 4 2 2 8 6 6 24

Matrix representation of Q16×C18 in GL3(𝔽73) generated by

 72 0 0 0 37 0 0 0 37
,
 72 0 0 0 41 41 0 16 0
,
 72 0 0 0 5 56 0 23 68
G:=sub<GL(3,GF(73))| [72,0,0,0,37,0,0,0,37],[72,0,0,0,41,16,0,41,0],[72,0,0,0,5,23,0,56,68] >;

Q16×C18 in GAP, Magma, Sage, TeX

Q_{16}\times C_{18}
% in TeX

G:=Group("Q16xC18");
// GroupNames label

G:=SmallGroup(288,184);
// by ID

G=gap.SmallGroup(288,184);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,1008,365,1016,192,5884,2951,242]);
// Polycyclic

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

׿
×
𝔽