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G = C12×C24order 288 = 25·32

Abelian group of type [12,24]

Aliases: C12×C24, SmallGroup(288,314)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12×C24
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×C12 — C6×C24 — C12×C24
 Lower central C1 — C12×C24
 Upper central 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 [×2], C3 [×4], C4 [×6], C22, C6 [×12], C8 [×4], C2×C4, C2×C4 [×2], C32, C12 [×24], C2×C6 [×4], C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C24 [×16], C2×C12 [×12], C4×C8, C3×C12 [×6], C62, C4×C12 [×4], C2×C24 [×8], C3×C24 [×4], C6×C12, C6×C12 [×2], C4×C24 [×4], C122, C6×C24 [×2], C12×C24
Quotients: C1, C2 [×3], C3 [×4], C4 [×6], C22, C6 [×12], C8 [×4], C2×C4 [×3], C32, C12 [×24], C2×C6 [×4], C42, C2×C8 [×2], C3×C6 [×3], C24 [×16], C2×C12 [×12], C4×C8, C3×C12 [×6], C62, C4×C12 [×4], C2×C24 [×8], C3×C24 [×4], C6×C12 [×3], C4×C24 [×4], C122, C6×C24 [×2], C12×C24

Smallest permutation representation of C12×C24
Regular action on 288 points
Generators in S288
(1 161 26 203 129 249 64 276 222 178 108 83)(2 162 27 204 130 250 65 277 223 179 109 84)(3 163 28 205 131 251 66 278 224 180 110 85)(4 164 29 206 132 252 67 279 225 181 111 86)(5 165 30 207 133 253 68 280 226 182 112 87)(6 166 31 208 134 254 69 281 227 183 113 88)(7 167 32 209 135 255 70 282 228 184 114 89)(8 168 33 210 136 256 71 283 229 185 115 90)(9 145 34 211 137 257 72 284 230 186 116 91)(10 146 35 212 138 258 49 285 231 187 117 92)(11 147 36 213 139 259 50 286 232 188 118 93)(12 148 37 214 140 260 51 287 233 189 119 94)(13 149 38 215 141 261 52 288 234 190 120 95)(14 150 39 216 142 262 53 265 235 191 97 96)(15 151 40 193 143 263 54 266 236 192 98 73)(16 152 41 194 144 264 55 267 237 169 99 74)(17 153 42 195 121 241 56 268 238 170 100 75)(18 154 43 196 122 242 57 269 239 171 101 76)(19 155 44 197 123 243 58 270 240 172 102 77)(20 156 45 198 124 244 59 271 217 173 103 78)(21 157 46 199 125 245 60 272 218 174 104 79)(22 158 47 200 126 246 61 273 219 175 105 80)(23 159 48 201 127 247 62 274 220 176 106 81)(24 160 25 202 128 248 63 275 221 177 107 82)
(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)| (1,161,26,203,129,249,64,276,222,178,108,83)(2,162,27,204,130,250,65,277,223,179,109,84)(3,163,28,205,131,251,66,278,224,180,110,85)(4,164,29,206,132,252,67,279,225,181,111,86)(5,165,30,207,133,253,68,280,226,182,112,87)(6,166,31,208,134,254,69,281,227,183,113,88)(7,167,32,209,135,255,70,282,228,184,114,89)(8,168,33,210,136,256,71,283,229,185,115,90)(9,145,34,211,137,257,72,284,230,186,116,91)(10,146,35,212,138,258,49,285,231,187,117,92)(11,147,36,213,139,259,50,286,232,188,118,93)(12,148,37,214,140,260,51,287,233,189,119,94)(13,149,38,215,141,261,52,288,234,190,120,95)(14,150,39,216,142,262,53,265,235,191,97,96)(15,151,40,193,143,263,54,266,236,192,98,73)(16,152,41,194,144,264,55,267,237,169,99,74)(17,153,42,195,121,241,56,268,238,170,100,75)(18,154,43,196,122,242,57,269,239,171,101,76)(19,155,44,197,123,243,58,270,240,172,102,77)(20,156,45,198,124,244,59,271,217,173,103,78)(21,157,46,199,125,245,60,272,218,174,104,79)(22,158,47,200,126,246,61,273,219,175,105,80)(23,159,48,201,127,247,62,274,220,176,106,81)(24,160,25,202,128,248,63,275,221,177,107,82), (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( (1,161,26,203,129,249,64,276,222,178,108,83)(2,162,27,204,130,250,65,277,223,179,109,84)(3,163,28,205,131,251,66,278,224,180,110,85)(4,164,29,206,132,252,67,279,225,181,111,86)(5,165,30,207,133,253,68,280,226,182,112,87)(6,166,31,208,134,254,69,281,227,183,113,88)(7,167,32,209,135,255,70,282,228,184,114,89)(8,168,33,210,136,256,71,283,229,185,115,90)(9,145,34,211,137,257,72,284,230,186,116,91)(10,146,35,212,138,258,49,285,231,187,117,92)(11,147,36,213,139,259,50,286,232,188,118,93)(12,148,37,214,140,260,51,287,233,189,119,94)(13,149,38,215,141,261,52,288,234,190,120,95)(14,150,39,216,142,262,53,265,235,191,97,96)(15,151,40,193,143,263,54,266,236,192,98,73)(16,152,41,194,144,264,55,267,237,169,99,74)(17,153,42,195,121,241,56,268,238,170,100,75)(18,154,43,196,122,242,57,269,239,171,101,76)(19,155,44,197,123,243,58,270,240,172,102,77)(20,156,45,198,124,244,59,271,217,173,103,78)(21,157,46,199,125,245,60,272,218,174,104,79)(22,158,47,200,126,246,61,273,219,175,105,80)(23,159,48,201,127,247,62,274,220,176,106,81)(24,160,25,202,128,248,63,275,221,177,107,82), (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([(1,161,26,203,129,249,64,276,222,178,108,83),(2,162,27,204,130,250,65,277,223,179,109,84),(3,163,28,205,131,251,66,278,224,180,110,85),(4,164,29,206,132,252,67,279,225,181,111,86),(5,165,30,207,133,253,68,280,226,182,112,87),(6,166,31,208,134,254,69,281,227,183,113,88),(7,167,32,209,135,255,70,282,228,184,114,89),(8,168,33,210,136,256,71,283,229,185,115,90),(9,145,34,211,137,257,72,284,230,186,116,91),(10,146,35,212,138,258,49,285,231,187,117,92),(11,147,36,213,139,259,50,286,232,188,118,93),(12,148,37,214,140,260,51,287,233,189,119,94),(13,149,38,215,141,261,52,288,234,190,120,95),(14,150,39,216,142,262,53,265,235,191,97,96),(15,151,40,193,143,263,54,266,236,192,98,73),(16,152,41,194,144,264,55,267,237,169,99,74),(17,153,42,195,121,241,56,268,238,170,100,75),(18,154,43,196,122,242,57,269,239,171,101,76),(19,155,44,197,123,243,58,270,240,172,102,77),(20,156,45,198,124,244,59,271,217,173,103,78),(21,157,46,199,125,245,60,272,218,174,104,79),(22,158,47,200,126,246,61,273,219,175,105,80),(23,159,48,201,127,247,62,274,220,176,106,81),(24,160,25,202,128,248,63,275,221,177,107,82)], [(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

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