metabelian, supersoluble, monomial
Aliases: C12.9Dic6, C62.111D4, (C3×C6).35D8, C12.24(C4×S3), C32⋊4C8⋊3C4, (C2×C12).85D6, (C3×C12).11Q8, (C3×C6).15Q16, C6.21(D4⋊S3), C32⋊7(C2.D8), C3⋊3(C6.Q16), (C6×C12).52C22, C6.10(C3⋊Q16), C2.1(C32⋊7D8), C4.1(C32⋊4Q8), C6.16(Dic3⋊C4), C2.1(C32⋊7Q16), C12⋊Dic3.12C2, C2.3(C6.Dic6), C22.12(C32⋊7D4), C4.11(C4×C3⋊S3), C4⋊C4.1(C3⋊S3), (C3×C4⋊C4).17S3, (C3×C6).37(C4⋊C4), (C3×C12).46(C2×C4), (C32×C4⋊C4).4C2, (C2×C6).87(C3⋊D4), (C2×C32⋊4C8).5C2, (C2×C4).34(C2×C3⋊S3), SmallGroup(288,282)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.9Dic6
G = < a,b,c | a12=b12=1, c2=a9b6, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >
Subgroups: 332 in 108 conjugacy classes, 59 normal (21 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2.D8, C3⋊Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C32⋊4C8, C2×C3⋊Dic3, C6×C12, C6×C12, C6.Q16, C2×C32⋊4C8, C12⋊Dic3, C32×C4⋊C4, C12.9Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D8, Q16, C3⋊S3, Dic6, C4×S3, C3⋊D4, C2.D8, C2×C3⋊S3, Dic3⋊C4, D4⋊S3, C3⋊Q16, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C6.Q16, C6.Dic6, C32⋊7D8, C32⋊7Q16, C12.9Dic6
►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 59 232 97 26 255 126 120 164 200 68 191)(2 54 233 104 27 262 127 115 165 195 69 186)(3 49 234 99 28 257 128 110 166 202 70 181)(4 56 235 106 29 264 129 117 167 197 71 188)(5 51 236 101 30 259 130 112 168 204 72 183)(6 58 237 108 31 254 131 119 157 199 61 190)(7 53 238 103 32 261 132 114 158 194 62 185)(8 60 239 98 33 256 121 109 159 201 63 192)(9 55 240 105 34 263 122 116 160 196 64 187)(10 50 229 100 35 258 123 111 161 203 65 182)(11 57 230 107 36 253 124 118 162 198 66 189)(12 52 231 102 25 260 125 113 163 193 67 184)(13 248 174 219 213 91 267 74 136 147 283 38)(14 243 175 226 214 86 268 81 137 154 284 45)(15 250 176 221 215 93 269 76 138 149 285 40)(16 245 177 228 216 88 270 83 139 156 286 47)(17 252 178 223 205 95 271 78 140 151 287 42)(18 247 179 218 206 90 272 73 141 146 288 37)(19 242 180 225 207 85 273 80 142 153 277 44)(20 249 169 220 208 92 274 75 143 148 278 39)(21 244 170 227 209 87 275 82 144 155 279 46)(22 251 171 222 210 94 276 77 133 150 280 41)(23 246 172 217 211 89 265 84 134 145 281 48)(24 241 173 224 212 96 266 79 135 152 282 43)
(1 221 123 146 7 227 129 152)(2 226 124 151 8 220 130 145)(3 219 125 156 9 225 131 150)(4 224 126 149 10 218 132 155)(5 217 127 154 11 223 121 148)(6 222 128 147 12 228 122 153)(13 202 276 108 19 196 270 102)(14 195 265 101 20 201 271 107)(15 200 266 106 21 194 272 100)(16 193 267 99 22 199 273 105)(17 198 268 104 23 204 274 98)(18 203 269 97 24 197 275 103)(25 47 64 85 31 41 70 91)(26 40 65 90 32 46 71 96)(27 45 66 95 33 39 72 89)(28 38 67 88 34 44 61 94)(29 43 68 93 35 37 62 87)(30 48 69 86 36 42 63 92)(49 171 119 142 55 177 113 136)(50 176 120 135 56 170 114 141)(51 169 109 140 57 175 115 134)(52 174 110 133 58 180 116 139)(53 179 111 138 59 173 117 144)(54 172 112 143 60 178 118 137)(73 238 244 167 79 232 250 161)(74 231 245 160 80 237 251 166)(75 236 246 165 81 230 252 159)(76 229 247 158 82 235 241 164)(77 234 248 163 83 240 242 157)(78 239 249 168 84 233 243 162)(181 210 254 277 187 216 260 283)(182 215 255 282 188 209 261 288)(183 208 256 287 189 214 262 281)(184 213 257 280 190 207 263 286)(185 206 258 285 191 212 264 279)(186 211 259 278 192 205 253 284)
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), (1,59,232,97,26,255,126,120,164,200,68,191)(2,54,233,104,27,262,127,115,165,195,69,186)(3,49,234,99,28,257,128,110,166,202,70,181)(4,56,235,106,29,264,129,117,167,197,71,188)(5,51,236,101,30,259,130,112,168,204,72,183)(6,58,237,108,31,254,131,119,157,199,61,190)(7,53,238,103,32,261,132,114,158,194,62,185)(8,60,239,98,33,256,121,109,159,201,63,192)(9,55,240,105,34,263,122,116,160,196,64,187)(10,50,229,100,35,258,123,111,161,203,65,182)(11,57,230,107,36,253,124,118,162,198,66,189)(12,52,231,102,25,260,125,113,163,193,67,184)(13,248,174,219,213,91,267,74,136,147,283,38)(14,243,175,226,214,86,268,81,137,154,284,45)(15,250,176,221,215,93,269,76,138,149,285,40)(16,245,177,228,216,88,270,83,139,156,286,47)(17,252,178,223,205,95,271,78,140,151,287,42)(18,247,179,218,206,90,272,73,141,146,288,37)(19,242,180,225,207,85,273,80,142,153,277,44)(20,249,169,220,208,92,274,75,143,148,278,39)(21,244,170,227,209,87,275,82,144,155,279,46)(22,251,171,222,210,94,276,77,133,150,280,41)(23,246,172,217,211,89,265,84,134,145,281,48)(24,241,173,224,212,96,266,79,135,152,282,43), (1,221,123,146,7,227,129,152)(2,226,124,151,8,220,130,145)(3,219,125,156,9,225,131,150)(4,224,126,149,10,218,132,155)(5,217,127,154,11,223,121,148)(6,222,128,147,12,228,122,153)(13,202,276,108,19,196,270,102)(14,195,265,101,20,201,271,107)(15,200,266,106,21,194,272,100)(16,193,267,99,22,199,273,105)(17,198,268,104,23,204,274,98)(18,203,269,97,24,197,275,103)(25,47,64,85,31,41,70,91)(26,40,65,90,32,46,71,96)(27,45,66,95,33,39,72,89)(28,38,67,88,34,44,61,94)(29,43,68,93,35,37,62,87)(30,48,69,86,36,42,63,92)(49,171,119,142,55,177,113,136)(50,176,120,135,56,170,114,141)(51,169,109,140,57,175,115,134)(52,174,110,133,58,180,116,139)(53,179,111,138,59,173,117,144)(54,172,112,143,60,178,118,137)(73,238,244,167,79,232,250,161)(74,231,245,160,80,237,251,166)(75,236,246,165,81,230,252,159)(76,229,247,158,82,235,241,164)(77,234,248,163,83,240,242,157)(78,239,249,168,84,233,243,162)(181,210,254,277,187,216,260,283)(182,215,255,282,188,209,261,288)(183,208,256,287,189,214,262,281)(184,213,257,280,190,207,263,286)(185,206,258,285,191,212,264,279)(186,211,259,278,192,205,253,284)>;
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), (1,59,232,97,26,255,126,120,164,200,68,191)(2,54,233,104,27,262,127,115,165,195,69,186)(3,49,234,99,28,257,128,110,166,202,70,181)(4,56,235,106,29,264,129,117,167,197,71,188)(5,51,236,101,30,259,130,112,168,204,72,183)(6,58,237,108,31,254,131,119,157,199,61,190)(7,53,238,103,32,261,132,114,158,194,62,185)(8,60,239,98,33,256,121,109,159,201,63,192)(9,55,240,105,34,263,122,116,160,196,64,187)(10,50,229,100,35,258,123,111,161,203,65,182)(11,57,230,107,36,253,124,118,162,198,66,189)(12,52,231,102,25,260,125,113,163,193,67,184)(13,248,174,219,213,91,267,74,136,147,283,38)(14,243,175,226,214,86,268,81,137,154,284,45)(15,250,176,221,215,93,269,76,138,149,285,40)(16,245,177,228,216,88,270,83,139,156,286,47)(17,252,178,223,205,95,271,78,140,151,287,42)(18,247,179,218,206,90,272,73,141,146,288,37)(19,242,180,225,207,85,273,80,142,153,277,44)(20,249,169,220,208,92,274,75,143,148,278,39)(21,244,170,227,209,87,275,82,144,155,279,46)(22,251,171,222,210,94,276,77,133,150,280,41)(23,246,172,217,211,89,265,84,134,145,281,48)(24,241,173,224,212,96,266,79,135,152,282,43), (1,221,123,146,7,227,129,152)(2,226,124,151,8,220,130,145)(3,219,125,156,9,225,131,150)(4,224,126,149,10,218,132,155)(5,217,127,154,11,223,121,148)(6,222,128,147,12,228,122,153)(13,202,276,108,19,196,270,102)(14,195,265,101,20,201,271,107)(15,200,266,106,21,194,272,100)(16,193,267,99,22,199,273,105)(17,198,268,104,23,204,274,98)(18,203,269,97,24,197,275,103)(25,47,64,85,31,41,70,91)(26,40,65,90,32,46,71,96)(27,45,66,95,33,39,72,89)(28,38,67,88,34,44,61,94)(29,43,68,93,35,37,62,87)(30,48,69,86,36,42,63,92)(49,171,119,142,55,177,113,136)(50,176,120,135,56,170,114,141)(51,169,109,140,57,175,115,134)(52,174,110,133,58,180,116,139)(53,179,111,138,59,173,117,144)(54,172,112,143,60,178,118,137)(73,238,244,167,79,232,250,161)(74,231,245,160,80,237,251,166)(75,236,246,165,81,230,252,159)(76,229,247,158,82,235,241,164)(77,234,248,163,83,240,242,157)(78,239,249,168,84,233,243,162)(181,210,254,277,187,216,260,283)(182,215,255,282,188,209,261,288)(183,208,256,287,189,214,262,281)(184,213,257,280,190,207,263,286)(185,206,258,285,191,212,264,279)(186,211,259,278,192,205,253,284) );
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)], [(1,59,232,97,26,255,126,120,164,200,68,191),(2,54,233,104,27,262,127,115,165,195,69,186),(3,49,234,99,28,257,128,110,166,202,70,181),(4,56,235,106,29,264,129,117,167,197,71,188),(5,51,236,101,30,259,130,112,168,204,72,183),(6,58,237,108,31,254,131,119,157,199,61,190),(7,53,238,103,32,261,132,114,158,194,62,185),(8,60,239,98,33,256,121,109,159,201,63,192),(9,55,240,105,34,263,122,116,160,196,64,187),(10,50,229,100,35,258,123,111,161,203,65,182),(11,57,230,107,36,253,124,118,162,198,66,189),(12,52,231,102,25,260,125,113,163,193,67,184),(13,248,174,219,213,91,267,74,136,147,283,38),(14,243,175,226,214,86,268,81,137,154,284,45),(15,250,176,221,215,93,269,76,138,149,285,40),(16,245,177,228,216,88,270,83,139,156,286,47),(17,252,178,223,205,95,271,78,140,151,287,42),(18,247,179,218,206,90,272,73,141,146,288,37),(19,242,180,225,207,85,273,80,142,153,277,44),(20,249,169,220,208,92,274,75,143,148,278,39),(21,244,170,227,209,87,275,82,144,155,279,46),(22,251,171,222,210,94,276,77,133,150,280,41),(23,246,172,217,211,89,265,84,134,145,281,48),(24,241,173,224,212,96,266,79,135,152,282,43)], [(1,221,123,146,7,227,129,152),(2,226,124,151,8,220,130,145),(3,219,125,156,9,225,131,150),(4,224,126,149,10,218,132,155),(5,217,127,154,11,223,121,148),(6,222,128,147,12,228,122,153),(13,202,276,108,19,196,270,102),(14,195,265,101,20,201,271,107),(15,200,266,106,21,194,272,100),(16,193,267,99,22,199,273,105),(17,198,268,104,23,204,274,98),(18,203,269,97,24,197,275,103),(25,47,64,85,31,41,70,91),(26,40,65,90,32,46,71,96),(27,45,66,95,33,39,72,89),(28,38,67,88,34,44,61,94),(29,43,68,93,35,37,62,87),(30,48,69,86,36,42,63,92),(49,171,119,142,55,177,113,136),(50,176,120,135,56,170,114,141),(51,169,109,140,57,175,115,134),(52,174,110,133,58,180,116,139),(53,179,111,138,59,173,117,144),(54,172,112,143,60,178,118,137),(73,238,244,167,79,232,250,161),(74,231,245,160,80,237,251,166),(75,236,246,165,81,230,252,159),(76,229,247,158,82,235,241,164),(77,234,248,163,83,240,242,157),(78,239,249,168,84,233,243,162),(181,210,254,277,187,216,260,283),(182,215,255,282,188,209,261,288),(183,208,256,287,189,214,262,281),(184,213,257,280,190,207,263,286),(185,206,258,285,191,212,264,279),(186,211,259,278,192,205,253,284)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 36 | 36 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | D8 | Q16 | Dic6 | C4×S3 | C3⋊D4 | D4⋊S3 | C3⋊Q16 |
| kernel | C12.9Dic6 | C2×C32⋊4C8 | C12⋊Dic3 | C32×C4⋊C4 | C32⋊4C8 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 2 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C12.9Dic6 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 2 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 24 | 0 | 0 |
| 0 | 0 | 72 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 43 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 41 | 63 | 0 | 0 | 0 | 0 |
| 22 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 41 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,60,72,0,0,0,0,24,13,0,0,0,0,0,0,43,30,0,0,0,0,43,13],[41,22,0,0,0,0,63,32,0,0,0,0,0,0,32,16,0,0,0,0,41,0,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;
C12.9Dic6 in GAP, Magma, Sage, TeX
C_{12}._9{\rm Dic}_6 % in TeX
G:=Group("C12.9Dic6"); // GroupNames label
G:=SmallGroup(288,282);
// by ID
G=gap.SmallGroup(288,282);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^9*b^6,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations