direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C18, C36.44D4, C36.46C23, C72.27C22, C3.(C6×Q16), C4.8(D4×C9), (C6×Q16).C3, C8.5(C2×C18), (C2×C8).4C18, C6.76(C6×D4), C6.8(C3×Q16), (C2×C24).15C6, (C2×C72).14C2, C24.25(C2×C6), C2.13(D4×C18), (C2×C18).54D4, C12.44(C3×D4), C18.76(C2×D4), (C6×Q8).20C6, Q8.4(C2×C18), (Q8×C18).9C2, (C2×Q8).6C18, (C3×Q16).6C6, C4.3(C22×C18), C22.16(D4×C9), C12.46(C22×C6), (Q8×C9).12C22, (C2×C36).129C22, (C2×C6).63(C3×D4), (C2×C4).28(C2×C18), (C3×Q8).25(C2×C6), (C2×C12).146(C2×C6), SmallGroup(288,184)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C9, C12, C12, C2×C6, C2×C8, Q16, C2×Q8, C18, C18, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C2×Q16, C36, C36, C2×C18, C2×C24, C3×Q16, C6×Q8, C72, C2×C36, C2×C36, Q8×C9, Q8×C9, C6×Q16, C2×C72, C9×Q16, Q8×C18, Q16×C18
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, Q16, C2×D4, C18, C3×D4, C22×C6, C2×Q16, C2×C18, C3×Q16, C6×D4, D4×C9, C22×C18, C6×Q16, C9×Q16, D4×C18, 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)
(1 214 185 113 21 67 41 157)(2 215 186 114 22 68 42 158)(3 216 187 115 23 69 43 159)(4 199 188 116 24 70 44 160)(5 200 189 117 25 71 45 161)(6 201 190 118 26 72 46 162)(7 202 191 119 27 55 47 145)(8 203 192 120 28 56 48 146)(9 204 193 121 29 57 49 147)(10 205 194 122 30 58 50 148)(11 206 195 123 31 59 51 149)(12 207 196 124 32 60 52 150)(13 208 197 125 33 61 53 151)(14 209 198 126 34 62 54 152)(15 210 181 109 35 63 37 153)(16 211 182 110 36 64 38 154)(17 212 183 111 19 65 39 155)(18 213 184 112 20 66 40 156)(73 288 94 166 261 239 231 141)(74 271 95 167 262 240 232 142)(75 272 96 168 263 241 233 143)(76 273 97 169 264 242 234 144)(77 274 98 170 265 243 217 127)(78 275 99 171 266 244 218 128)(79 276 100 172 267 245 219 129)(80 277 101 173 268 246 220 130)(81 278 102 174 269 247 221 131)(82 279 103 175 270 248 222 132)(83 280 104 176 253 249 223 133)(84 281 105 177 254 250 224 134)(85 282 106 178 255 251 225 135)(86 283 107 179 256 252 226 136)(87 284 108 180 257 235 227 137)(88 285 91 163 258 236 228 138)(89 286 92 164 259 237 229 139)(90 287 93 165 260 238 230 140)
(1 232 21 95)(2 233 22 96)(3 234 23 97)(4 217 24 98)(5 218 25 99)(6 219 26 100)(7 220 27 101)(8 221 28 102)(9 222 29 103)(10 223 30 104)(11 224 31 105)(12 225 32 106)(13 226 33 107)(14 227 34 108)(15 228 35 91)(16 229 36 92)(17 230 19 93)(18 231 20 94)(37 88 181 258)(38 89 182 259)(39 90 183 260)(40 73 184 261)(41 74 185 262)(42 75 186 263)(43 76 187 264)(44 77 188 265)(45 78 189 266)(46 79 190 267)(47 80 191 268)(48 81 192 269)(49 82 193 270)(50 83 194 253)(51 84 195 254)(52 85 196 255)(53 86 197 256)(54 87 198 257)(55 277 202 246)(56 278 203 247)(57 279 204 248)(58 280 205 249)(59 281 206 250)(60 282 207 251)(61 283 208 252)(62 284 209 235)(63 285 210 236)(64 286 211 237)(65 287 212 238)(66 288 213 239)(67 271 214 240)(68 272 215 241)(69 273 216 242)(70 274 199 243)(71 275 200 244)(72 276 201 245)(109 163 153 138)(110 164 154 139)(111 165 155 140)(112 166 156 141)(113 167 157 142)(114 168 158 143)(115 169 159 144)(116 170 160 127)(117 171 161 128)(118 172 162 129)(119 173 145 130)(120 174 146 131)(121 175 147 132)(122 176 148 133)(123 177 149 134)(124 178 150 135)(125 179 151 136)(126 180 152 137)
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), (1,214,185,113,21,67,41,157)(2,215,186,114,22,68,42,158)(3,216,187,115,23,69,43,159)(4,199,188,116,24,70,44,160)(5,200,189,117,25,71,45,161)(6,201,190,118,26,72,46,162)(7,202,191,119,27,55,47,145)(8,203,192,120,28,56,48,146)(9,204,193,121,29,57,49,147)(10,205,194,122,30,58,50,148)(11,206,195,123,31,59,51,149)(12,207,196,124,32,60,52,150)(13,208,197,125,33,61,53,151)(14,209,198,126,34,62,54,152)(15,210,181,109,35,63,37,153)(16,211,182,110,36,64,38,154)(17,212,183,111,19,65,39,155)(18,213,184,112,20,66,40,156)(73,288,94,166,261,239,231,141)(74,271,95,167,262,240,232,142)(75,272,96,168,263,241,233,143)(76,273,97,169,264,242,234,144)(77,274,98,170,265,243,217,127)(78,275,99,171,266,244,218,128)(79,276,100,172,267,245,219,129)(80,277,101,173,268,246,220,130)(81,278,102,174,269,247,221,131)(82,279,103,175,270,248,222,132)(83,280,104,176,253,249,223,133)(84,281,105,177,254,250,224,134)(85,282,106,178,255,251,225,135)(86,283,107,179,256,252,226,136)(87,284,108,180,257,235,227,137)(88,285,91,163,258,236,228,138)(89,286,92,164,259,237,229,139)(90,287,93,165,260,238,230,140), (1,232,21,95)(2,233,22,96)(3,234,23,97)(4,217,24,98)(5,218,25,99)(6,219,26,100)(7,220,27,101)(8,221,28,102)(9,222,29,103)(10,223,30,104)(11,224,31,105)(12,225,32,106)(13,226,33,107)(14,227,34,108)(15,228,35,91)(16,229,36,92)(17,230,19,93)(18,231,20,94)(37,88,181,258)(38,89,182,259)(39,90,183,260)(40,73,184,261)(41,74,185,262)(42,75,186,263)(43,76,187,264)(44,77,188,265)(45,78,189,266)(46,79,190,267)(47,80,191,268)(48,81,192,269)(49,82,193,270)(50,83,194,253)(51,84,195,254)(52,85,196,255)(53,86,197,256)(54,87,198,257)(55,277,202,246)(56,278,203,247)(57,279,204,248)(58,280,205,249)(59,281,206,250)(60,282,207,251)(61,283,208,252)(62,284,209,235)(63,285,210,236)(64,286,211,237)(65,287,212,238)(66,288,213,239)(67,271,214,240)(68,272,215,241)(69,273,216,242)(70,274,199,243)(71,275,200,244)(72,276,201,245)(109,163,153,138)(110,164,154,139)(111,165,155,140)(112,166,156,141)(113,167,157,142)(114,168,158,143)(115,169,159,144)(116,170,160,127)(117,171,161,128)(118,172,162,129)(119,173,145,130)(120,174,146,131)(121,175,147,132)(122,176,148,133)(123,177,149,134)(124,178,150,135)(125,179,151,136)(126,180,152,137)>;
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), (1,214,185,113,21,67,41,157)(2,215,186,114,22,68,42,158)(3,216,187,115,23,69,43,159)(4,199,188,116,24,70,44,160)(5,200,189,117,25,71,45,161)(6,201,190,118,26,72,46,162)(7,202,191,119,27,55,47,145)(8,203,192,120,28,56,48,146)(9,204,193,121,29,57,49,147)(10,205,194,122,30,58,50,148)(11,206,195,123,31,59,51,149)(12,207,196,124,32,60,52,150)(13,208,197,125,33,61,53,151)(14,209,198,126,34,62,54,152)(15,210,181,109,35,63,37,153)(16,211,182,110,36,64,38,154)(17,212,183,111,19,65,39,155)(18,213,184,112,20,66,40,156)(73,288,94,166,261,239,231,141)(74,271,95,167,262,240,232,142)(75,272,96,168,263,241,233,143)(76,273,97,169,264,242,234,144)(77,274,98,170,265,243,217,127)(78,275,99,171,266,244,218,128)(79,276,100,172,267,245,219,129)(80,277,101,173,268,246,220,130)(81,278,102,174,269,247,221,131)(82,279,103,175,270,248,222,132)(83,280,104,176,253,249,223,133)(84,281,105,177,254,250,224,134)(85,282,106,178,255,251,225,135)(86,283,107,179,256,252,226,136)(87,284,108,180,257,235,227,137)(88,285,91,163,258,236,228,138)(89,286,92,164,259,237,229,139)(90,287,93,165,260,238,230,140), (1,232,21,95)(2,233,22,96)(3,234,23,97)(4,217,24,98)(5,218,25,99)(6,219,26,100)(7,220,27,101)(8,221,28,102)(9,222,29,103)(10,223,30,104)(11,224,31,105)(12,225,32,106)(13,226,33,107)(14,227,34,108)(15,228,35,91)(16,229,36,92)(17,230,19,93)(18,231,20,94)(37,88,181,258)(38,89,182,259)(39,90,183,260)(40,73,184,261)(41,74,185,262)(42,75,186,263)(43,76,187,264)(44,77,188,265)(45,78,189,266)(46,79,190,267)(47,80,191,268)(48,81,192,269)(49,82,193,270)(50,83,194,253)(51,84,195,254)(52,85,196,255)(53,86,197,256)(54,87,198,257)(55,277,202,246)(56,278,203,247)(57,279,204,248)(58,280,205,249)(59,281,206,250)(60,282,207,251)(61,283,208,252)(62,284,209,235)(63,285,210,236)(64,286,211,237)(65,287,212,238)(66,288,213,239)(67,271,214,240)(68,272,215,241)(69,273,216,242)(70,274,199,243)(71,275,200,244)(72,276,201,245)(109,163,153,138)(110,164,154,139)(111,165,155,140)(112,166,156,141)(113,167,157,142)(114,168,158,143)(115,169,159,144)(116,170,160,127)(117,171,161,128)(118,172,162,129)(119,173,145,130)(120,174,146,131)(121,175,147,132)(122,176,148,133)(123,177,149,134)(124,178,150,135)(125,179,151,136)(126,180,152,137) );
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)], [(1,214,185,113,21,67,41,157),(2,215,186,114,22,68,42,158),(3,216,187,115,23,69,43,159),(4,199,188,116,24,70,44,160),(5,200,189,117,25,71,45,161),(6,201,190,118,26,72,46,162),(7,202,191,119,27,55,47,145),(8,203,192,120,28,56,48,146),(9,204,193,121,29,57,49,147),(10,205,194,122,30,58,50,148),(11,206,195,123,31,59,51,149),(12,207,196,124,32,60,52,150),(13,208,197,125,33,61,53,151),(14,209,198,126,34,62,54,152),(15,210,181,109,35,63,37,153),(16,211,182,110,36,64,38,154),(17,212,183,111,19,65,39,155),(18,213,184,112,20,66,40,156),(73,288,94,166,261,239,231,141),(74,271,95,167,262,240,232,142),(75,272,96,168,263,241,233,143),(76,273,97,169,264,242,234,144),(77,274,98,170,265,243,217,127),(78,275,99,171,266,244,218,128),(79,276,100,172,267,245,219,129),(80,277,101,173,268,246,220,130),(81,278,102,174,269,247,221,131),(82,279,103,175,270,248,222,132),(83,280,104,176,253,249,223,133),(84,281,105,177,254,250,224,134),(85,282,106,178,255,251,225,135),(86,283,107,179,256,252,226,136),(87,284,108,180,257,235,227,137),(88,285,91,163,258,236,228,138),(89,286,92,164,259,237,229,139),(90,287,93,165,260,238,230,140)], [(1,232,21,95),(2,233,22,96),(3,234,23,97),(4,217,24,98),(5,218,25,99),(6,219,26,100),(7,220,27,101),(8,221,28,102),(9,222,29,103),(10,223,30,104),(11,224,31,105),(12,225,32,106),(13,226,33,107),(14,227,34,108),(15,228,35,91),(16,229,36,92),(17,230,19,93),(18,231,20,94),(37,88,181,258),(38,89,182,259),(39,90,183,260),(40,73,184,261),(41,74,185,262),(42,75,186,263),(43,76,187,264),(44,77,188,265),(45,78,189,266),(46,79,190,267),(47,80,191,268),(48,81,192,269),(49,82,193,270),(50,83,194,253),(51,84,195,254),(52,85,196,255),(53,86,197,256),(54,87,198,257),(55,277,202,246),(56,278,203,247),(57,279,204,248),(58,280,205,249),(59,281,206,250),(60,282,207,251),(61,283,208,252),(62,284,209,235),(63,285,210,236),(64,286,211,237),(65,287,212,238),(66,288,213,239),(67,271,214,240),(68,272,215,241),(69,273,216,242),(70,274,199,243),(71,275,200,244),(72,276,201,245),(109,163,153,138),(110,164,154,139),(111,165,155,140),(112,166,156,141),(113,167,157,142),(114,168,158,143),(115,169,159,144),(116,170,160,127),(117,171,161,128),(118,172,162,129),(119,173,145,130),(120,174,146,131),(121,175,147,132),(122,176,148,133),(123,177,149,134),(124,178,150,135),(125,179,151,136),(126,180,152,137)]])
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