direct product, abelian, monomial
Aliases: C2×C6×C24, SmallGroup(288,826)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C6×C24 |
| C1 — C2×C6×C24 |
| C1 — C2×C6×C24 |
Generators and relations for C2×C6×C24
G = < a,b,c | a2=b6=c24=1, ab=ba, ac=ca, bc=cb >
Subgroups: 228, all normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C2×C4, C23, C32, C12, C2×C6, C2×C8, C22×C4, C3×C6, C3×C6, C24, C2×C12, C22×C6, C22×C8, C3×C12, C3×C12, C62, C2×C24, C22×C12, C3×C24, C6×C12, C2×C62, C22×C24, C6×C24, C2×C6×C12, C2×C6×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C32, C12, C2×C6, C2×C8, C22×C4, C3×C6, C24, C2×C12, C22×C6, C22×C8, C3×C12, C62, C2×C24, C22×C12, C3×C24, C6×C12, C2×C62, C22×C24, C6×C24, C2×C6×C12, C2×C6×C24
►Regular action on 288 points
Generators in S288
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 215)(26 216)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 209)(44 210)(45 211)(46 212)(47 213)(48 214)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 275)(74 276)(75 277)(76 278)(77 279)(78 280)(79 281)(80 282)(81 283)(82 284)(83 285)(84 286)(85 287)(86 288)(87 265)(88 266)(89 267)(90 268)(91 269)(92 270)(93 271)(94 272)(95 273)(96 274)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(133 175)(134 176)(135 177)(136 178)(137 179)(138 180)(139 181)(140 182)(141 183)(142 184)(143 185)(144 186)(217 257)(218 258)(219 259)(220 260)(221 261)(222 262)(223 263)(224 264)(225 241)(226 242)(227 243)(228 244)(229 245)(230 246)(231 247)(232 248)(233 249)(234 250)(235 251)(236 252)(237 253)(238 254)(239 255)(240 256)
(1 100 38 126 241 279)(2 101 39 127 242 280)(3 102 40 128 243 281)(4 103 41 129 244 282)(5 104 42 130 245 283)(6 105 43 131 246 284)(7 106 44 132 247 285)(8 107 45 133 248 286)(9 108 46 134 249 287)(10 109 47 135 250 288)(11 110 48 136 251 265)(12 111 25 137 252 266)(13 112 26 138 253 267)(14 113 27 139 254 268)(15 114 28 140 255 269)(16 115 29 141 256 270)(17 116 30 142 257 271)(18 117 31 143 258 272)(19 118 32 144 259 273)(20 119 33 121 260 274)(21 120 34 122 261 275)(22 97 35 123 262 276)(23 98 36 124 263 277)(24 99 37 125 264 278)(49 194 182 239 91 166)(50 195 183 240 92 167)(51 196 184 217 93 168)(52 197 185 218 94 145)(53 198 186 219 95 146)(54 199 187 220 96 147)(55 200 188 221 73 148)(56 201 189 222 74 149)(57 202 190 223 75 150)(58 203 191 224 76 151)(59 204 192 225 77 152)(60 205 169 226 78 153)(61 206 170 227 79 154)(62 207 171 228 80 155)(63 208 172 229 81 156)(64 209 173 230 82 157)(65 210 174 231 83 158)(66 211 175 232 84 159)(67 212 176 233 85 160)(68 213 177 234 86 161)(69 214 178 235 87 162)(70 215 179 236 88 163)(71 216 180 237 89 164)(72 193 181 238 90 165)
(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)
G:=sub<Sym(288)| (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,215)(26,216)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,281)(80,282)(81,283)(82,284)(83,285)(84,286)(85,287)(86,288)(87,265)(88,266)(89,267)(90,268)(91,269)(92,270)(93,271)(94,272)(95,273)(96,274)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186)(217,257)(218,258)(219,259)(220,260)(221,261)(222,262)(223,263)(224,264)(225,241)(226,242)(227,243)(228,244)(229,245)(230,246)(231,247)(232,248)(233,249)(234,250)(235,251)(236,252)(237,253)(238,254)(239,255)(240,256), (1,100,38,126,241,279)(2,101,39,127,242,280)(3,102,40,128,243,281)(4,103,41,129,244,282)(5,104,42,130,245,283)(6,105,43,131,246,284)(7,106,44,132,247,285)(8,107,45,133,248,286)(9,108,46,134,249,287)(10,109,47,135,250,288)(11,110,48,136,251,265)(12,111,25,137,252,266)(13,112,26,138,253,267)(14,113,27,139,254,268)(15,114,28,140,255,269)(16,115,29,141,256,270)(17,116,30,142,257,271)(18,117,31,143,258,272)(19,118,32,144,259,273)(20,119,33,121,260,274)(21,120,34,122,261,275)(22,97,35,123,262,276)(23,98,36,124,263,277)(24,99,37,125,264,278)(49,194,182,239,91,166)(50,195,183,240,92,167)(51,196,184,217,93,168)(52,197,185,218,94,145)(53,198,186,219,95,146)(54,199,187,220,96,147)(55,200,188,221,73,148)(56,201,189,222,74,149)(57,202,190,223,75,150)(58,203,191,224,76,151)(59,204,192,225,77,152)(60,205,169,226,78,153)(61,206,170,227,79,154)(62,207,171,228,80,155)(63,208,172,229,81,156)(64,209,173,230,82,157)(65,210,174,231,83,158)(66,211,175,232,84,159)(67,212,176,233,85,160)(68,213,177,234,86,161)(69,214,178,235,87,162)(70,215,179,236,88,163)(71,216,180,237,89,164)(72,193,181,238,90,165), (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)>;
G:=Group( (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,215)(26,216)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,281)(80,282)(81,283)(82,284)(83,285)(84,286)(85,287)(86,288)(87,265)(88,266)(89,267)(90,268)(91,269)(92,270)(93,271)(94,272)(95,273)(96,274)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186)(217,257)(218,258)(219,259)(220,260)(221,261)(222,262)(223,263)(224,264)(225,241)(226,242)(227,243)(228,244)(229,245)(230,246)(231,247)(232,248)(233,249)(234,250)(235,251)(236,252)(237,253)(238,254)(239,255)(240,256), (1,100,38,126,241,279)(2,101,39,127,242,280)(3,102,40,128,243,281)(4,103,41,129,244,282)(5,104,42,130,245,283)(6,105,43,131,246,284)(7,106,44,132,247,285)(8,107,45,133,248,286)(9,108,46,134,249,287)(10,109,47,135,250,288)(11,110,48,136,251,265)(12,111,25,137,252,266)(13,112,26,138,253,267)(14,113,27,139,254,268)(15,114,28,140,255,269)(16,115,29,141,256,270)(17,116,30,142,257,271)(18,117,31,143,258,272)(19,118,32,144,259,273)(20,119,33,121,260,274)(21,120,34,122,261,275)(22,97,35,123,262,276)(23,98,36,124,263,277)(24,99,37,125,264,278)(49,194,182,239,91,166)(50,195,183,240,92,167)(51,196,184,217,93,168)(52,197,185,218,94,145)(53,198,186,219,95,146)(54,199,187,220,96,147)(55,200,188,221,73,148)(56,201,189,222,74,149)(57,202,190,223,75,150)(58,203,191,224,76,151)(59,204,192,225,77,152)(60,205,169,226,78,153)(61,206,170,227,79,154)(62,207,171,228,80,155)(63,208,172,229,81,156)(64,209,173,230,82,157)(65,210,174,231,83,158)(66,211,175,232,84,159)(67,212,176,233,85,160)(68,213,177,234,86,161)(69,214,178,235,87,162)(70,215,179,236,88,163)(71,216,180,237,89,164)(72,193,181,238,90,165), (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) );
G=PermutationGroup([[(1,152),(2,153),(3,154),(4,155),(5,156),(6,157),(7,158),(8,159),(9,160),(10,161),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,168),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(25,215),(26,216),(27,193),(28,194),(29,195),(30,196),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,209),(44,210),(45,211),(46,212),(47,213),(48,214),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(73,275),(74,276),(75,277),(76,278),(77,279),(78,280),(79,281),(80,282),(81,283),(82,284),(83,285),(84,286),(85,287),(86,288),(87,265),(88,266),(89,267),(90,268),(91,269),(92,270),(93,271),(94,272),(95,273),(96,274),(121,187),(122,188),(123,189),(124,190),(125,191),(126,192),(127,169),(128,170),(129,171),(130,172),(131,173),(132,174),(133,175),(134,176),(135,177),(136,178),(137,179),(138,180),(139,181),(140,182),(141,183),(142,184),(143,185),(144,186),(217,257),(218,258),(219,259),(220,260),(221,261),(222,262),(223,263),(224,264),(225,241),(226,242),(227,243),(228,244),(229,245),(230,246),(231,247),(232,248),(233,249),(234,250),(235,251),(236,252),(237,253),(238,254),(239,255),(240,256)], [(1,100,38,126,241,279),(2,101,39,127,242,280),(3,102,40,128,243,281),(4,103,41,129,244,282),(5,104,42,130,245,283),(6,105,43,131,246,284),(7,106,44,132,247,285),(8,107,45,133,248,286),(9,108,46,134,249,287),(10,109,47,135,250,288),(11,110,48,136,251,265),(12,111,25,137,252,266),(13,112,26,138,253,267),(14,113,27,139,254,268),(15,114,28,140,255,269),(16,115,29,141,256,270),(17,116,30,142,257,271),(18,117,31,143,258,272),(19,118,32,144,259,273),(20,119,33,121,260,274),(21,120,34,122,261,275),(22,97,35,123,262,276),(23,98,36,124,263,277),(24,99,37,125,264,278),(49,194,182,239,91,166),(50,195,183,240,92,167),(51,196,184,217,93,168),(52,197,185,218,94,145),(53,198,186,219,95,146),(54,199,187,220,96,147),(55,200,188,221,73,148),(56,201,189,222,74,149),(57,202,190,223,75,150),(58,203,191,224,76,151),(59,204,192,225,77,152),(60,205,169,226,78,153),(61,206,170,227,79,154),(62,207,171,228,80,155),(63,208,172,229,81,156),(64,209,173,230,82,157),(65,210,174,231,83,158),(66,211,175,232,84,159),(67,212,176,233,85,160),(68,213,177,234,86,161),(69,214,178,235,87,162),(70,215,179,236,88,163),(71,216,180,237,89,164),(72,193,181,238,90,165)], [(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)]])
288 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | ··· | 3H | 4A | ··· | 4H | 6A | ··· | 6BD | 8A | ··· | 8P | 12A | ··· | 12BL | 24A | ··· | 24DX |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
288 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 |
| kernel | C2×C6×C24 | C6×C24 | C2×C6×C12 | C22×C24 | C6×C12 | C2×C62 | C2×C24 | C22×C12 | C62 | C2×C12 | C22×C6 | C2×C6 |
| # reps | 1 | 6 | 1 | 8 | 6 | 2 | 48 | 8 | 16 | 48 | 16 | 128 |
Matrix representation of C2×C6×C24 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 65 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 65 |
| 24 | 0 | 0 |
| 0 | 43 | 0 |
| 0 | 0 | 24 |
G:=sub<GL(3,GF(73))| [1,0,0,0,72,0,0,0,72],[65,0,0,0,1,0,0,0,65],[24,0,0,0,43,0,0,0,24] >;
C2×C6×C24 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_{24} % in TeX
G:=Group("C2xC6xC24"); // GroupNames label
G:=SmallGroup(288,826);
// by ID
G=gap.SmallGroup(288,826);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^6=c^24=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations