metabelian, supersoluble, monomial
Aliases: C12⋊6Dic6, C12.46D12, C122.2C2, C62.213C23, (C3×C12)⋊8Q8, (C4×C12).8S3, C6.48(C2×D12), C32⋊10(C4⋊Q8), C3⋊2(C12⋊2Q8), (C2×C12).382D6, (C3×C12).121D4, C42.4(C3⋊S3), C6.39(C2×Dic6), C4⋊2(C32⋊4Q8), C4.4(C12⋊S3), (C6×C12).297C22, C12⋊Dic3.5C2, (C3×C6).53(C2×Q8), C2.4(C2×C12⋊S3), (C3×C6).188(C2×D4), C2.4(C2×C32⋊4Q8), (C2×C32⋊4Q8).4C2, (C2×C6).230(C22×S3), C22.34(C22×C3⋊S3), (C2×C3⋊Dic3).74C22, (C2×C4).76(C2×C3⋊S3), SmallGroup(288,726)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C2×C32⋊4Q8 — C12⋊6Dic6 |
Generators and relations for C12⋊6Dic6
G = < a,b,c | a12=b12=1, c2=b6, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 684 in 204 conjugacy classes, 101 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C4⋊Q8, C3⋊Dic3, C3×C12, C62, C4⋊Dic3, C4×C12, C2×Dic6, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C12⋊2Q8, C12⋊Dic3, C122, C2×C32⋊4Q8, C12⋊6Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C3⋊S3, Dic6, D12, C22×S3, C4⋊Q8, C2×C3⋊S3, C2×Dic6, C2×D12, C32⋊4Q8, C12⋊S3, C22×C3⋊S3, C12⋊2Q8, C2×C32⋊4Q8, C2×C12⋊S3, C12⋊6Dic6
►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 32 123 48 234 136 108 111 62 222 282 259)(2 33 124 37 235 137 97 112 63 223 283 260)(3 34 125 38 236 138 98 113 64 224 284 261)(4 35 126 39 237 139 99 114 65 225 285 262)(5 36 127 40 238 140 100 115 66 226 286 263)(6 25 128 41 239 141 101 116 67 227 287 264)(7 26 129 42 240 142 102 117 68 228 288 253)(8 27 130 43 229 143 103 118 69 217 277 254)(9 28 131 44 230 144 104 119 70 218 278 255)(10 29 132 45 231 133 105 120 71 219 279 256)(11 30 121 46 232 134 106 109 72 220 280 257)(12 31 122 47 233 135 107 110 61 221 281 258)(13 211 56 179 161 272 249 151 91 195 183 78)(14 212 57 180 162 273 250 152 92 196 184 79)(15 213 58 169 163 274 251 153 93 197 185 80)(16 214 59 170 164 275 252 154 94 198 186 81)(17 215 60 171 165 276 241 155 95 199 187 82)(18 216 49 172 166 265 242 156 96 200 188 83)(19 205 50 173 167 266 243 145 85 201 189 84)(20 206 51 174 168 267 244 146 86 202 190 73)(21 207 52 175 157 268 245 147 87 203 191 74)(22 208 53 176 158 269 246 148 88 204 192 75)(23 209 54 177 159 270 247 149 89 193 181 76)(24 210 55 178 160 271 248 150 90 194 182 77)
(1 182 108 160)(2 181 97 159)(3 192 98 158)(4 191 99 157)(5 190 100 168)(6 189 101 167)(7 188 102 166)(8 187 103 165)(9 186 104 164)(10 185 105 163)(11 184 106 162)(12 183 107 161)(13 233 249 281)(14 232 250 280)(15 231 251 279)(16 230 252 278)(17 229 241 277)(18 240 242 288)(19 239 243 287)(20 238 244 286)(21 237 245 285)(22 236 246 284)(23 235 247 283)(24 234 248 282)(25 201 116 173)(26 200 117 172)(27 199 118 171)(28 198 119 170)(29 197 120 169)(30 196 109 180)(31 195 110 179)(32 194 111 178)(33 193 112 177)(34 204 113 176)(35 203 114 175)(36 202 115 174)(37 149 223 209)(38 148 224 208)(39 147 225 207)(40 146 226 206)(41 145 227 205)(42 156 228 216)(43 155 217 215)(44 154 218 214)(45 153 219 213)(46 152 220 212)(47 151 221 211)(48 150 222 210)(49 129 96 68)(50 128 85 67)(51 127 86 66)(52 126 87 65)(53 125 88 64)(54 124 89 63)(55 123 90 62)(56 122 91 61)(57 121 92 72)(58 132 93 71)(59 131 94 70)(60 130 95 69)(73 140 267 263)(74 139 268 262)(75 138 269 261)(76 137 270 260)(77 136 271 259)(78 135 272 258)(79 134 273 257)(80 133 274 256)(81 144 275 255)(82 143 276 254)(83 142 265 253)(84 141 266 264)
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), (1,32,123,48,234,136,108,111,62,222,282,259)(2,33,124,37,235,137,97,112,63,223,283,260)(3,34,125,38,236,138,98,113,64,224,284,261)(4,35,126,39,237,139,99,114,65,225,285,262)(5,36,127,40,238,140,100,115,66,226,286,263)(6,25,128,41,239,141,101,116,67,227,287,264)(7,26,129,42,240,142,102,117,68,228,288,253)(8,27,130,43,229,143,103,118,69,217,277,254)(9,28,131,44,230,144,104,119,70,218,278,255)(10,29,132,45,231,133,105,120,71,219,279,256)(11,30,121,46,232,134,106,109,72,220,280,257)(12,31,122,47,233,135,107,110,61,221,281,258)(13,211,56,179,161,272,249,151,91,195,183,78)(14,212,57,180,162,273,250,152,92,196,184,79)(15,213,58,169,163,274,251,153,93,197,185,80)(16,214,59,170,164,275,252,154,94,198,186,81)(17,215,60,171,165,276,241,155,95,199,187,82)(18,216,49,172,166,265,242,156,96,200,188,83)(19,205,50,173,167,266,243,145,85,201,189,84)(20,206,51,174,168,267,244,146,86,202,190,73)(21,207,52,175,157,268,245,147,87,203,191,74)(22,208,53,176,158,269,246,148,88,204,192,75)(23,209,54,177,159,270,247,149,89,193,181,76)(24,210,55,178,160,271,248,150,90,194,182,77), (1,182,108,160)(2,181,97,159)(3,192,98,158)(4,191,99,157)(5,190,100,168)(6,189,101,167)(7,188,102,166)(8,187,103,165)(9,186,104,164)(10,185,105,163)(11,184,106,162)(12,183,107,161)(13,233,249,281)(14,232,250,280)(15,231,251,279)(16,230,252,278)(17,229,241,277)(18,240,242,288)(19,239,243,287)(20,238,244,286)(21,237,245,285)(22,236,246,284)(23,235,247,283)(24,234,248,282)(25,201,116,173)(26,200,117,172)(27,199,118,171)(28,198,119,170)(29,197,120,169)(30,196,109,180)(31,195,110,179)(32,194,111,178)(33,193,112,177)(34,204,113,176)(35,203,114,175)(36,202,115,174)(37,149,223,209)(38,148,224,208)(39,147,225,207)(40,146,226,206)(41,145,227,205)(42,156,228,216)(43,155,217,215)(44,154,218,214)(45,153,219,213)(46,152,220,212)(47,151,221,211)(48,150,222,210)(49,129,96,68)(50,128,85,67)(51,127,86,66)(52,126,87,65)(53,125,88,64)(54,124,89,63)(55,123,90,62)(56,122,91,61)(57,121,92,72)(58,132,93,71)(59,131,94,70)(60,130,95,69)(73,140,267,263)(74,139,268,262)(75,138,269,261)(76,137,270,260)(77,136,271,259)(78,135,272,258)(79,134,273,257)(80,133,274,256)(81,144,275,255)(82,143,276,254)(83,142,265,253)(84,141,266,264)>;
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), (1,32,123,48,234,136,108,111,62,222,282,259)(2,33,124,37,235,137,97,112,63,223,283,260)(3,34,125,38,236,138,98,113,64,224,284,261)(4,35,126,39,237,139,99,114,65,225,285,262)(5,36,127,40,238,140,100,115,66,226,286,263)(6,25,128,41,239,141,101,116,67,227,287,264)(7,26,129,42,240,142,102,117,68,228,288,253)(8,27,130,43,229,143,103,118,69,217,277,254)(9,28,131,44,230,144,104,119,70,218,278,255)(10,29,132,45,231,133,105,120,71,219,279,256)(11,30,121,46,232,134,106,109,72,220,280,257)(12,31,122,47,233,135,107,110,61,221,281,258)(13,211,56,179,161,272,249,151,91,195,183,78)(14,212,57,180,162,273,250,152,92,196,184,79)(15,213,58,169,163,274,251,153,93,197,185,80)(16,214,59,170,164,275,252,154,94,198,186,81)(17,215,60,171,165,276,241,155,95,199,187,82)(18,216,49,172,166,265,242,156,96,200,188,83)(19,205,50,173,167,266,243,145,85,201,189,84)(20,206,51,174,168,267,244,146,86,202,190,73)(21,207,52,175,157,268,245,147,87,203,191,74)(22,208,53,176,158,269,246,148,88,204,192,75)(23,209,54,177,159,270,247,149,89,193,181,76)(24,210,55,178,160,271,248,150,90,194,182,77), (1,182,108,160)(2,181,97,159)(3,192,98,158)(4,191,99,157)(5,190,100,168)(6,189,101,167)(7,188,102,166)(8,187,103,165)(9,186,104,164)(10,185,105,163)(11,184,106,162)(12,183,107,161)(13,233,249,281)(14,232,250,280)(15,231,251,279)(16,230,252,278)(17,229,241,277)(18,240,242,288)(19,239,243,287)(20,238,244,286)(21,237,245,285)(22,236,246,284)(23,235,247,283)(24,234,248,282)(25,201,116,173)(26,200,117,172)(27,199,118,171)(28,198,119,170)(29,197,120,169)(30,196,109,180)(31,195,110,179)(32,194,111,178)(33,193,112,177)(34,204,113,176)(35,203,114,175)(36,202,115,174)(37,149,223,209)(38,148,224,208)(39,147,225,207)(40,146,226,206)(41,145,227,205)(42,156,228,216)(43,155,217,215)(44,154,218,214)(45,153,219,213)(46,152,220,212)(47,151,221,211)(48,150,222,210)(49,129,96,68)(50,128,85,67)(51,127,86,66)(52,126,87,65)(53,125,88,64)(54,124,89,63)(55,123,90,62)(56,122,91,61)(57,121,92,72)(58,132,93,71)(59,131,94,70)(60,130,95,69)(73,140,267,263)(74,139,268,262)(75,138,269,261)(76,137,270,260)(77,136,271,259)(78,135,272,258)(79,134,273,257)(80,133,274,256)(81,144,275,255)(82,143,276,254)(83,142,265,253)(84,141,266,264) );
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)], [(1,32,123,48,234,136,108,111,62,222,282,259),(2,33,124,37,235,137,97,112,63,223,283,260),(3,34,125,38,236,138,98,113,64,224,284,261),(4,35,126,39,237,139,99,114,65,225,285,262),(5,36,127,40,238,140,100,115,66,226,286,263),(6,25,128,41,239,141,101,116,67,227,287,264),(7,26,129,42,240,142,102,117,68,228,288,253),(8,27,130,43,229,143,103,118,69,217,277,254),(9,28,131,44,230,144,104,119,70,218,278,255),(10,29,132,45,231,133,105,120,71,219,279,256),(11,30,121,46,232,134,106,109,72,220,280,257),(12,31,122,47,233,135,107,110,61,221,281,258),(13,211,56,179,161,272,249,151,91,195,183,78),(14,212,57,180,162,273,250,152,92,196,184,79),(15,213,58,169,163,274,251,153,93,197,185,80),(16,214,59,170,164,275,252,154,94,198,186,81),(17,215,60,171,165,276,241,155,95,199,187,82),(18,216,49,172,166,265,242,156,96,200,188,83),(19,205,50,173,167,266,243,145,85,201,189,84),(20,206,51,174,168,267,244,146,86,202,190,73),(21,207,52,175,157,268,245,147,87,203,191,74),(22,208,53,176,158,269,246,148,88,204,192,75),(23,209,54,177,159,270,247,149,89,193,181,76),(24,210,55,178,160,271,248,150,90,194,182,77)], [(1,182,108,160),(2,181,97,159),(3,192,98,158),(4,191,99,157),(5,190,100,168),(6,189,101,167),(7,188,102,166),(8,187,103,165),(9,186,104,164),(10,185,105,163),(11,184,106,162),(12,183,107,161),(13,233,249,281),(14,232,250,280),(15,231,251,279),(16,230,252,278),(17,229,241,277),(18,240,242,288),(19,239,243,287),(20,238,244,286),(21,237,245,285),(22,236,246,284),(23,235,247,283),(24,234,248,282),(25,201,116,173),(26,200,117,172),(27,199,118,171),(28,198,119,170),(29,197,120,169),(30,196,109,180),(31,195,110,179),(32,194,111,178),(33,193,112,177),(34,204,113,176),(35,203,114,175),(36,202,115,174),(37,149,223,209),(38,148,224,208),(39,147,225,207),(40,146,226,206),(41,145,227,205),(42,156,228,216),(43,155,217,215),(44,154,218,214),(45,153,219,213),(46,152,220,212),(47,151,221,211),(48,150,222,210),(49,129,96,68),(50,128,85,67),(51,127,86,66),(52,126,87,65),(53,125,88,64),(54,124,89,63),(55,123,90,62),(56,122,91,61),(57,121,92,72),(58,132,93,71),(59,131,94,70),(60,130,95,69),(73,140,267,263),(74,139,268,262),(75,138,269,261),(76,137,270,260),(77,136,271,259),(78,135,272,258),(79,134,273,257),(80,133,274,256),(81,144,275,255),(82,143,276,254),(83,142,265,253),(84,141,266,264)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | Dic6 | D12 |
| kernel | C12⋊6Dic6 | C12⋊Dic3 | C122 | C2×C32⋊4Q8 | C4×C12 | C3×C12 | C3×C12 | C2×C12 | C12 | C12 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 4 | 12 | 32 | 16 |
Matrix representation of C12⋊6Dic6 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 6 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 11 | 10 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 7 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 5 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [3,6,0,0,0,0,0,9,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,4,11,0,0,0,0,0,10],[10,7,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,7,0,0,0,0,0,7],[2,5,0,0,0,0,2,11,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,2,5] >;
C12⋊6Dic6 in GAP, Magma, Sage, TeX
C_{12}\rtimes_6{\rm Dic}_6 % in TeX
G:=Group("C12:6Dic6"); // GroupNames label
G:=SmallGroup(288,726);
// by ID
G=gap.SmallGroup(288,726);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations