metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.9D36, C36.1D4, C12.1D12, C18.4Q16, Dic18⋊3C4, C18.5SD16, C4⋊C4.3D9, C4.1(C4×D9), C12.3(C4×S3), C36.3(C2×C4), C9⋊1(Q8⋊C4), (C2×C12).39D6, (C2×C18).31D4, (C2×C4).37D18, C6.10(D6⋊C4), C2.5(D18⋊C4), C6.8(D4.S3), C2.2(D4.D9), C3.(C6.SD16), C6.7(C3⋊Q16), C2.2(C9⋊Q16), C18.3(C22⋊C4), (C2×C36).17C22, (C2×Dic18).5C2, C22.14(C9⋊D4), (C2×C9⋊C8).3C2, (C9×C4⋊C4).3C2, (C3×C4⋊C4).3S3, (C2×C6).69(C3⋊D4), SmallGroup(288,16)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C9, Dic3, C12, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C18, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, Q8⋊C4, Dic9, C36, C36, C2×C18, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C9⋊C8, Dic18, Dic18, C2×Dic9, C2×C36, C2×C36, C6.SD16, C2×C9⋊C8, C9×C4⋊C4, C2×Dic18, C18.Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, 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
►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)
(1 271 148 248 52 88 118 261)(2 288 149 247 53 87 119 260)(3 287 150 246 54 86 120 259)(4 286 151 245 37 85 121 258)(5 285 152 244 38 84 122 257)(6 284 153 243 39 83 123 256)(7 283 154 242 40 82 124 255)(8 282 155 241 41 81 125 254)(9 281 156 240 42 80 126 253)(10 280 157 239 43 79 109 270)(11 279 158 238 44 78 110 269)(12 278 159 237 45 77 111 268)(13 277 160 236 46 76 112 267)(14 276 161 235 47 75 113 266)(15 275 162 252 48 74 114 265)(16 274 145 251 49 73 115 264)(17 273 146 250 50 90 116 263)(18 272 147 249 51 89 117 262)(19 98 142 65 183 208 178 233)(20 97 143 64 184 207 179 232)(21 96 144 63 185 206 180 231)(22 95 127 62 186 205 163 230)(23 94 128 61 187 204 164 229)(24 93 129 60 188 203 165 228)(25 92 130 59 189 202 166 227)(26 91 131 58 190 201 167 226)(27 108 132 57 191 200 168 225)(28 107 133 56 192 199 169 224)(29 106 134 55 193 216 170 223)(30 105 135 72 194 215 171 222)(31 104 136 71 195 214 172 221)(32 103 137 70 196 213 173 220)(33 102 138 69 197 212 174 219)(34 101 139 68 198 211 175 218)(35 100 140 67 181 210 176 217)(36 99 141 66 182 209 177 234)
(1 19 43 192)(2 20 44 193)(3 21 45 194)(4 22 46 195)(5 23 47 196)(6 24 48 197)(7 25 49 198)(8 26 50 181)(9 27 51 182)(10 28 52 183)(11 29 53 184)(12 30 54 185)(13 31 37 186)(14 32 38 187)(15 33 39 188)(16 34 40 189)(17 35 41 190)(18 36 42 191)(55 279 232 87)(56 280 233 88)(57 281 234 89)(58 282 217 90)(59 283 218 73)(60 284 219 74)(61 285 220 75)(62 286 221 76)(63 287 222 77)(64 288 223 78)(65 271 224 79)(66 272 225 80)(67 273 226 81)(68 274 227 82)(69 275 228 83)(70 276 229 84)(71 277 230 85)(72 278 231 86)(91 241 210 263)(92 242 211 264)(93 243 212 265)(94 244 213 266)(95 245 214 267)(96 246 215 268)(97 247 216 269)(98 248 199 270)(99 249 200 253)(100 250 201 254)(101 251 202 255)(102 252 203 256)(103 235 204 257)(104 236 205 258)(105 237 206 259)(106 238 207 260)(107 239 208 261)(108 240 209 262)(109 133 148 178)(110 134 149 179)(111 135 150 180)(112 136 151 163)(113 137 152 164)(114 138 153 165)(115 139 154 166)(116 140 155 167)(117 141 156 168)(118 142 157 169)(119 143 158 170)(120 144 159 171)(121 127 160 172)(122 128 161 173)(123 129 162 174)(124 130 145 175)(125 131 146 176)(126 132 147 177)
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), (1,271,148,248,52,88,118,261)(2,288,149,247,53,87,119,260)(3,287,150,246,54,86,120,259)(4,286,151,245,37,85,121,258)(5,285,152,244,38,84,122,257)(6,284,153,243,39,83,123,256)(7,283,154,242,40,82,124,255)(8,282,155,241,41,81,125,254)(9,281,156,240,42,80,126,253)(10,280,157,239,43,79,109,270)(11,279,158,238,44,78,110,269)(12,278,159,237,45,77,111,268)(13,277,160,236,46,76,112,267)(14,276,161,235,47,75,113,266)(15,275,162,252,48,74,114,265)(16,274,145,251,49,73,115,264)(17,273,146,250,50,90,116,263)(18,272,147,249,51,89,117,262)(19,98,142,65,183,208,178,233)(20,97,143,64,184,207,179,232)(21,96,144,63,185,206,180,231)(22,95,127,62,186,205,163,230)(23,94,128,61,187,204,164,229)(24,93,129,60,188,203,165,228)(25,92,130,59,189,202,166,227)(26,91,131,58,190,201,167,226)(27,108,132,57,191,200,168,225)(28,107,133,56,192,199,169,224)(29,106,134,55,193,216,170,223)(30,105,135,72,194,215,171,222)(31,104,136,71,195,214,172,221)(32,103,137,70,196,213,173,220)(33,102,138,69,197,212,174,219)(34,101,139,68,198,211,175,218)(35,100,140,67,181,210,176,217)(36,99,141,66,182,209,177,234), (1,19,43,192)(2,20,44,193)(3,21,45,194)(4,22,46,195)(5,23,47,196)(6,24,48,197)(7,25,49,198)(8,26,50,181)(9,27,51,182)(10,28,52,183)(11,29,53,184)(12,30,54,185)(13,31,37,186)(14,32,38,187)(15,33,39,188)(16,34,40,189)(17,35,41,190)(18,36,42,191)(55,279,232,87)(56,280,233,88)(57,281,234,89)(58,282,217,90)(59,283,218,73)(60,284,219,74)(61,285,220,75)(62,286,221,76)(63,287,222,77)(64,288,223,78)(65,271,224,79)(66,272,225,80)(67,273,226,81)(68,274,227,82)(69,275,228,83)(70,276,229,84)(71,277,230,85)(72,278,231,86)(91,241,210,263)(92,242,211,264)(93,243,212,265)(94,244,213,266)(95,245,214,267)(96,246,215,268)(97,247,216,269)(98,248,199,270)(99,249,200,253)(100,250,201,254)(101,251,202,255)(102,252,203,256)(103,235,204,257)(104,236,205,258)(105,237,206,259)(106,238,207,260)(107,239,208,261)(108,240,209,262)(109,133,148,178)(110,134,149,179)(111,135,150,180)(112,136,151,163)(113,137,152,164)(114,138,153,165)(115,139,154,166)(116,140,155,167)(117,141,156,168)(118,142,157,169)(119,143,158,170)(120,144,159,171)(121,127,160,172)(122,128,161,173)(123,129,162,174)(124,130,145,175)(125,131,146,176)(126,132,147,177)>;
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), (1,271,148,248,52,88,118,261)(2,288,149,247,53,87,119,260)(3,287,150,246,54,86,120,259)(4,286,151,245,37,85,121,258)(5,285,152,244,38,84,122,257)(6,284,153,243,39,83,123,256)(7,283,154,242,40,82,124,255)(8,282,155,241,41,81,125,254)(9,281,156,240,42,80,126,253)(10,280,157,239,43,79,109,270)(11,279,158,238,44,78,110,269)(12,278,159,237,45,77,111,268)(13,277,160,236,46,76,112,267)(14,276,161,235,47,75,113,266)(15,275,162,252,48,74,114,265)(16,274,145,251,49,73,115,264)(17,273,146,250,50,90,116,263)(18,272,147,249,51,89,117,262)(19,98,142,65,183,208,178,233)(20,97,143,64,184,207,179,232)(21,96,144,63,185,206,180,231)(22,95,127,62,186,205,163,230)(23,94,128,61,187,204,164,229)(24,93,129,60,188,203,165,228)(25,92,130,59,189,202,166,227)(26,91,131,58,190,201,167,226)(27,108,132,57,191,200,168,225)(28,107,133,56,192,199,169,224)(29,106,134,55,193,216,170,223)(30,105,135,72,194,215,171,222)(31,104,136,71,195,214,172,221)(32,103,137,70,196,213,173,220)(33,102,138,69,197,212,174,219)(34,101,139,68,198,211,175,218)(35,100,140,67,181,210,176,217)(36,99,141,66,182,209,177,234), (1,19,43,192)(2,20,44,193)(3,21,45,194)(4,22,46,195)(5,23,47,196)(6,24,48,197)(7,25,49,198)(8,26,50,181)(9,27,51,182)(10,28,52,183)(11,29,53,184)(12,30,54,185)(13,31,37,186)(14,32,38,187)(15,33,39,188)(16,34,40,189)(17,35,41,190)(18,36,42,191)(55,279,232,87)(56,280,233,88)(57,281,234,89)(58,282,217,90)(59,283,218,73)(60,284,219,74)(61,285,220,75)(62,286,221,76)(63,287,222,77)(64,288,223,78)(65,271,224,79)(66,272,225,80)(67,273,226,81)(68,274,227,82)(69,275,228,83)(70,276,229,84)(71,277,230,85)(72,278,231,86)(91,241,210,263)(92,242,211,264)(93,243,212,265)(94,244,213,266)(95,245,214,267)(96,246,215,268)(97,247,216,269)(98,248,199,270)(99,249,200,253)(100,250,201,254)(101,251,202,255)(102,252,203,256)(103,235,204,257)(104,236,205,258)(105,237,206,259)(106,238,207,260)(107,239,208,261)(108,240,209,262)(109,133,148,178)(110,134,149,179)(111,135,150,180)(112,136,151,163)(113,137,152,164)(114,138,153,165)(115,139,154,166)(116,140,155,167)(117,141,156,168)(118,142,157,169)(119,143,158,170)(120,144,159,171)(121,127,160,172)(122,128,161,173)(123,129,162,174)(124,130,145,175)(125,131,146,176)(126,132,147,177) );
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)], [(1,271,148,248,52,88,118,261),(2,288,149,247,53,87,119,260),(3,287,150,246,54,86,120,259),(4,286,151,245,37,85,121,258),(5,285,152,244,38,84,122,257),(6,284,153,243,39,83,123,256),(7,283,154,242,40,82,124,255),(8,282,155,241,41,81,125,254),(9,281,156,240,42,80,126,253),(10,280,157,239,43,79,109,270),(11,279,158,238,44,78,110,269),(12,278,159,237,45,77,111,268),(13,277,160,236,46,76,112,267),(14,276,161,235,47,75,113,266),(15,275,162,252,48,74,114,265),(16,274,145,251,49,73,115,264),(17,273,146,250,50,90,116,263),(18,272,147,249,51,89,117,262),(19,98,142,65,183,208,178,233),(20,97,143,64,184,207,179,232),(21,96,144,63,185,206,180,231),(22,95,127,62,186,205,163,230),(23,94,128,61,187,204,164,229),(24,93,129,60,188,203,165,228),(25,92,130,59,189,202,166,227),(26,91,131,58,190,201,167,226),(27,108,132,57,191,200,168,225),(28,107,133,56,192,199,169,224),(29,106,134,55,193,216,170,223),(30,105,135,72,194,215,171,222),(31,104,136,71,195,214,172,221),(32,103,137,70,196,213,173,220),(33,102,138,69,197,212,174,219),(34,101,139,68,198,211,175,218),(35,100,140,67,181,210,176,217),(36,99,141,66,182,209,177,234)], [(1,19,43,192),(2,20,44,193),(3,21,45,194),(4,22,46,195),(5,23,47,196),(6,24,48,197),(7,25,49,198),(8,26,50,181),(9,27,51,182),(10,28,52,183),(11,29,53,184),(12,30,54,185),(13,31,37,186),(14,32,38,187),(15,33,39,188),(16,34,40,189),(17,35,41,190),(18,36,42,191),(55,279,232,87),(56,280,233,88),(57,281,234,89),(58,282,217,90),(59,283,218,73),(60,284,219,74),(61,285,220,75),(62,286,221,76),(63,287,222,77),(64,288,223,78),(65,271,224,79),(66,272,225,80),(67,273,226,81),(68,274,227,82),(69,275,228,83),(70,276,229,84),(71,277,230,85),(72,278,231,86),(91,241,210,263),(92,242,211,264),(93,243,212,265),(94,244,213,266),(95,245,214,267),(96,246,215,268),(97,247,216,269),(98,248,199,270),(99,249,200,253),(100,250,201,254),(101,251,202,255),(102,252,203,256),(103,235,204,257),(104,236,205,258),(105,237,206,259),(106,238,207,260),(107,239,208,261),(108,240,209,262),(109,133,148,178),(110,134,149,179),(111,135,150,180),(112,136,151,163),(113,137,152,164),(114,138,153,165),(115,139,154,166),(116,140,155,167),(117,141,156,168),(118,142,157,169),(119,143,158,170),(120,144,159,171),(121,127,160,172),(122,128,161,173),(123,129,162,174),(124,130,145,175),(125,131,146,176),(126,132,147,177)]])
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