direct product, abelian, monomial
Aliases: C12×C24, SmallGroup(288,314)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C12×C24 |
| C1 — C12×C24 |
| C1 — C12×C24 |
Generators and relations for C12×C24
G = < a,b | a12=b24=1, ab=ba >
Subgroups: 132, all normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C8, C2×C4, C2×C4, C32, C12, C2×C6, C42, C2×C8, C3×C6, C3×C6, C24, C2×C12, C4×C8, C3×C12, C62, C4×C12, C2×C24, C3×C24, C6×C12, C6×C12, C4×C24, C122, C6×C24, C12×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C42, C2×C8, C3×C6, C24, C2×C12, C4×C8, C3×C12, C62, C4×C12, C2×C24, C3×C24, C6×C12, C4×C24, C122, C6×C24, C12×C24
►Regular action on 288 points
Generators in S288
(1 121 201 226 96 29 153 274 108 187 54 263)(2 122 202 227 73 30 154 275 109 188 55 264)(3 123 203 228 74 31 155 276 110 189 56 241)(4 124 204 229 75 32 156 277 111 190 57 242)(5 125 205 230 76 33 157 278 112 191 58 243)(6 126 206 231 77 34 158 279 113 192 59 244)(7 127 207 232 78 35 159 280 114 169 60 245)(8 128 208 233 79 36 160 281 115 170 61 246)(9 129 209 234 80 37 161 282 116 171 62 247)(10 130 210 235 81 38 162 283 117 172 63 248)(11 131 211 236 82 39 163 284 118 173 64 249)(12 132 212 237 83 40 164 285 119 174 65 250)(13 133 213 238 84 41 165 286 120 175 66 251)(14 134 214 239 85 42 166 287 97 176 67 252)(15 135 215 240 86 43 167 288 98 177 68 253)(16 136 216 217 87 44 168 265 99 178 69 254)(17 137 193 218 88 45 145 266 100 179 70 255)(18 138 194 219 89 46 146 267 101 180 71 256)(19 139 195 220 90 47 147 268 102 181 72 257)(20 140 196 221 91 48 148 269 103 182 49 258)(21 141 197 222 92 25 149 270 104 183 50 259)(22 142 198 223 93 26 150 271 105 184 51 260)(23 143 199 224 94 27 151 272 106 185 52 261)(24 144 200 225 95 28 152 273 107 186 53 262)
(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)| (1,121,201,226,96,29,153,274,108,187,54,263)(2,122,202,227,73,30,154,275,109,188,55,264)(3,123,203,228,74,31,155,276,110,189,56,241)(4,124,204,229,75,32,156,277,111,190,57,242)(5,125,205,230,76,33,157,278,112,191,58,243)(6,126,206,231,77,34,158,279,113,192,59,244)(7,127,207,232,78,35,159,280,114,169,60,245)(8,128,208,233,79,36,160,281,115,170,61,246)(9,129,209,234,80,37,161,282,116,171,62,247)(10,130,210,235,81,38,162,283,117,172,63,248)(11,131,211,236,82,39,163,284,118,173,64,249)(12,132,212,237,83,40,164,285,119,174,65,250)(13,133,213,238,84,41,165,286,120,175,66,251)(14,134,214,239,85,42,166,287,97,176,67,252)(15,135,215,240,86,43,167,288,98,177,68,253)(16,136,216,217,87,44,168,265,99,178,69,254)(17,137,193,218,88,45,145,266,100,179,70,255)(18,138,194,219,89,46,146,267,101,180,71,256)(19,139,195,220,90,47,147,268,102,181,72,257)(20,140,196,221,91,48,148,269,103,182,49,258)(21,141,197,222,92,25,149,270,104,183,50,259)(22,142,198,223,93,26,150,271,105,184,51,260)(23,143,199,224,94,27,151,272,106,185,52,261)(24,144,200,225,95,28,152,273,107,186,53,262), (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( (1,121,201,226,96,29,153,274,108,187,54,263)(2,122,202,227,73,30,154,275,109,188,55,264)(3,123,203,228,74,31,155,276,110,189,56,241)(4,124,204,229,75,32,156,277,111,190,57,242)(5,125,205,230,76,33,157,278,112,191,58,243)(6,126,206,231,77,34,158,279,113,192,59,244)(7,127,207,232,78,35,159,280,114,169,60,245)(8,128,208,233,79,36,160,281,115,170,61,246)(9,129,209,234,80,37,161,282,116,171,62,247)(10,130,210,235,81,38,162,283,117,172,63,248)(11,131,211,236,82,39,163,284,118,173,64,249)(12,132,212,237,83,40,164,285,119,174,65,250)(13,133,213,238,84,41,165,286,120,175,66,251)(14,134,214,239,85,42,166,287,97,176,67,252)(15,135,215,240,86,43,167,288,98,177,68,253)(16,136,216,217,87,44,168,265,99,178,69,254)(17,137,193,218,88,45,145,266,100,179,70,255)(18,138,194,219,89,46,146,267,101,180,71,256)(19,139,195,220,90,47,147,268,102,181,72,257)(20,140,196,221,91,48,148,269,103,182,49,258)(21,141,197,222,92,25,149,270,104,183,50,259)(22,142,198,223,93,26,150,271,105,184,51,260)(23,143,199,224,94,27,151,272,106,185,52,261)(24,144,200,225,95,28,152,273,107,186,53,262), (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([[(1,121,201,226,96,29,153,274,108,187,54,263),(2,122,202,227,73,30,154,275,109,188,55,264),(3,123,203,228,74,31,155,276,110,189,56,241),(4,124,204,229,75,32,156,277,111,190,57,242),(5,125,205,230,76,33,157,278,112,191,58,243),(6,126,206,231,77,34,158,279,113,192,59,244),(7,127,207,232,78,35,159,280,114,169,60,245),(8,128,208,233,79,36,160,281,115,170,61,246),(9,129,209,234,80,37,161,282,116,171,62,247),(10,130,210,235,81,38,162,283,117,172,63,248),(11,131,211,236,82,39,163,284,118,173,64,249),(12,132,212,237,83,40,164,285,119,174,65,250),(13,133,213,238,84,41,165,286,120,175,66,251),(14,134,214,239,85,42,166,287,97,176,67,252),(15,135,215,240,86,43,167,288,98,177,68,253),(16,136,216,217,87,44,168,265,99,178,69,254),(17,137,193,218,88,45,145,266,100,179,70,255),(18,138,194,219,89,46,146,267,101,180,71,256),(19,139,195,220,90,47,147,268,102,181,72,257),(20,140,196,221,91,48,148,269,103,182,49,258),(21,141,197,222,92,25,149,270,104,183,50,259),(22,142,198,223,93,26,150,271,105,184,51,260),(23,143,199,224,94,27,151,272,106,185,52,261),(24,144,200,225,95,28,152,273,107,186,53,262)], [(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 | 2B | 2C | 3A | ··· | 3H | 4A | ··· | 4L | 6A | ··· | 6X | 8A | ··· | 8P | 12A | ··· | 12CR | 24A | ··· | 24DX |
| order | 1 | 2 | 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 | ··· | 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 | C12×C24 | C122 | C6×C24 | C4×C24 | C3×C24 | C6×C12 | C4×C12 | C2×C24 | C3×C12 | C24 | C2×C12 | C12 |
| # reps | 1 | 1 | 2 | 8 | 8 | 4 | 8 | 16 | 16 | 64 | 32 | 128 |
Matrix representation of C12×C24 ►in GL3(𝔽73) generated by
| 49 | 0 | 0 |
| 0 | 65 | 0 |
| 0 | 0 | 46 |
| 17 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [49,0,0,0,65,0,0,0,46],[17,0,0,0,46,0,0,0,72] >;
C12×C24 in GAP, Magma, Sage, TeX
C_{12}\times C_{24} % in TeX
G:=Group("C12xC24"); // GroupNames label
G:=SmallGroup(288,314);
// by ID
G=gap.SmallGroup(288,314);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,252,512,172]);
// Polycyclic
G:=Group<a,b|a^12=b^24=1,a*b=b*a>;
// generators/relations