metabelian, supersoluble, monomial
Aliases: C6.4Dic12, C62.81D4, (C2×C24).8S3, (C6×C24).4C2, C12.61(C4×S3), (C2×C6).32D12, (C3×C6).10Q16, C32⋊4Q8⋊5C4, C6.5(C24⋊C2), C6.24(D6⋊C4), (C2×C12).375D6, (C3×C12).150D4, (C3×C6).18SD16, C2.1(C24⋊2S3), C3⋊2(C2.Dic12), C12.114(C3⋊D4), C2.1(C32⋊5Q16), (C6×C12).293C22, C12⋊Dic3.1C2, C32⋊11(Q8⋊C4), C4.19(C32⋊7D4), C22.7(C12⋊S3), C2.7(C6.11D12), C4.7(C4×C3⋊S3), (C2×C8).2(C3⋊S3), (C3×C12).92(C2×C4), (C2×C32⋊4Q8).1C2, (C3×C6).55(C22⋊C4), (C2×C4).71(C2×C3⋊S3), SmallGroup(288,291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.4Dic12
G = < a,b,c | a6=b24=1, c2=a3b12, ab=ba, cac-1=a-1, cbc-1=a3b-1 >
Subgroups: 484 in 126 conjugacy classes, 59 normal (21 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C24, Dic6, C2×Dic3, C2×C12, Q8⋊C4, C3⋊Dic3, C3×C12, C62, C4⋊Dic3, C2×C24, C2×Dic6, C3×C24, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C2.Dic12, C12⋊Dic3, C6×C24, C2×C32⋊4Q8, C6.4Dic12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, C3⋊S3, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×C3⋊S3, C24⋊C2, Dic12, D6⋊C4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C2.Dic12, C24⋊2S3, C32⋊5Q16, C6.11D12, C6.4Dic12
►Regular action on 288 points
Generators in S288
(1 244 219 112 183 160)(2 245 220 113 184 161)(3 246 221 114 185 162)(4 247 222 115 186 163)(5 248 223 116 187 164)(6 249 224 117 188 165)(7 250 225 118 189 166)(8 251 226 119 190 167)(9 252 227 120 191 168)(10 253 228 97 192 145)(11 254 229 98 169 146)(12 255 230 99 170 147)(13 256 231 100 171 148)(14 257 232 101 172 149)(15 258 233 102 173 150)(16 259 234 103 174 151)(17 260 235 104 175 152)(18 261 236 105 176 153)(19 262 237 106 177 154)(20 263 238 107 178 155)(21 264 239 108 179 156)(22 241 240 109 180 157)(23 242 217 110 181 158)(24 243 218 111 182 159)(25 65 211 88 283 143)(26 66 212 89 284 144)(27 67 213 90 285 121)(28 68 214 91 286 122)(29 69 215 92 287 123)(30 70 216 93 288 124)(31 71 193 94 265 125)(32 72 194 95 266 126)(33 49 195 96 267 127)(34 50 196 73 268 128)(35 51 197 74 269 129)(36 52 198 75 270 130)(37 53 199 76 271 131)(38 54 200 77 272 132)(39 55 201 78 273 133)(40 56 202 79 274 134)(41 57 203 80 275 135)(42 58 204 81 276 136)(43 59 205 82 277 137)(44 60 206 83 278 138)(45 61 207 84 279 139)(46 62 208 85 280 140)(47 63 209 86 281 141)(48 64 210 87 282 142)
(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)
(1 131 100 211)(2 198 101 142)(3 129 102 209)(4 196 103 140)(5 127 104 207)(6 194 105 138)(7 125 106 205)(8 216 107 136)(9 123 108 203)(10 214 109 134)(11 121 110 201)(12 212 111 132)(13 143 112 199)(14 210 113 130)(15 141 114 197)(16 208 115 128)(17 139 116 195)(18 206 117 126)(19 137 118 193)(20 204 119 124)(21 135 120 215)(22 202 97 122)(23 133 98 213)(24 200 99 144)(25 219 76 148)(26 159 77 230)(27 217 78 146)(28 157 79 228)(29 239 80 168)(30 155 81 226)(31 237 82 166)(32 153 83 224)(33 235 84 164)(34 151 85 222)(35 233 86 162)(36 149 87 220)(37 231 88 160)(38 147 89 218)(39 229 90 158)(40 145 91 240)(41 227 92 156)(42 167 93 238)(43 225 94 154)(44 165 95 236)(45 223 96 152)(46 163 73 234)(47 221 74 150)(48 161 75 232)(49 260 279 187)(50 174 280 247)(51 258 281 185)(52 172 282 245)(53 256 283 183)(54 170 284 243)(55 254 285 181)(56 192 286 241)(57 252 287 179)(58 190 288 263)(59 250 265 177)(60 188 266 261)(61 248 267 175)(62 186 268 259)(63 246 269 173)(64 184 270 257)(65 244 271 171)(66 182 272 255)(67 242 273 169)(68 180 274 253)(69 264 275 191)(70 178 276 251)(71 262 277 189)(72 176 278 249)
G:=sub<Sym(288)| (1,244,219,112,183,160)(2,245,220,113,184,161)(3,246,221,114,185,162)(4,247,222,115,186,163)(5,248,223,116,187,164)(6,249,224,117,188,165)(7,250,225,118,189,166)(8,251,226,119,190,167)(9,252,227,120,191,168)(10,253,228,97,192,145)(11,254,229,98,169,146)(12,255,230,99,170,147)(13,256,231,100,171,148)(14,257,232,101,172,149)(15,258,233,102,173,150)(16,259,234,103,174,151)(17,260,235,104,175,152)(18,261,236,105,176,153)(19,262,237,106,177,154)(20,263,238,107,178,155)(21,264,239,108,179,156)(22,241,240,109,180,157)(23,242,217,110,181,158)(24,243,218,111,182,159)(25,65,211,88,283,143)(26,66,212,89,284,144)(27,67,213,90,285,121)(28,68,214,91,286,122)(29,69,215,92,287,123)(30,70,216,93,288,124)(31,71,193,94,265,125)(32,72,194,95,266,126)(33,49,195,96,267,127)(34,50,196,73,268,128)(35,51,197,74,269,129)(36,52,198,75,270,130)(37,53,199,76,271,131)(38,54,200,77,272,132)(39,55,201,78,273,133)(40,56,202,79,274,134)(41,57,203,80,275,135)(42,58,204,81,276,136)(43,59,205,82,277,137)(44,60,206,83,278,138)(45,61,207,84,279,139)(46,62,208,85,280,140)(47,63,209,86,281,141)(48,64,210,87,282,142), (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), (1,131,100,211)(2,198,101,142)(3,129,102,209)(4,196,103,140)(5,127,104,207)(6,194,105,138)(7,125,106,205)(8,216,107,136)(9,123,108,203)(10,214,109,134)(11,121,110,201)(12,212,111,132)(13,143,112,199)(14,210,113,130)(15,141,114,197)(16,208,115,128)(17,139,116,195)(18,206,117,126)(19,137,118,193)(20,204,119,124)(21,135,120,215)(22,202,97,122)(23,133,98,213)(24,200,99,144)(25,219,76,148)(26,159,77,230)(27,217,78,146)(28,157,79,228)(29,239,80,168)(30,155,81,226)(31,237,82,166)(32,153,83,224)(33,235,84,164)(34,151,85,222)(35,233,86,162)(36,149,87,220)(37,231,88,160)(38,147,89,218)(39,229,90,158)(40,145,91,240)(41,227,92,156)(42,167,93,238)(43,225,94,154)(44,165,95,236)(45,223,96,152)(46,163,73,234)(47,221,74,150)(48,161,75,232)(49,260,279,187)(50,174,280,247)(51,258,281,185)(52,172,282,245)(53,256,283,183)(54,170,284,243)(55,254,285,181)(56,192,286,241)(57,252,287,179)(58,190,288,263)(59,250,265,177)(60,188,266,261)(61,248,267,175)(62,186,268,259)(63,246,269,173)(64,184,270,257)(65,244,271,171)(66,182,272,255)(67,242,273,169)(68,180,274,253)(69,264,275,191)(70,178,276,251)(71,262,277,189)(72,176,278,249)>;
G:=Group( (1,244,219,112,183,160)(2,245,220,113,184,161)(3,246,221,114,185,162)(4,247,222,115,186,163)(5,248,223,116,187,164)(6,249,224,117,188,165)(7,250,225,118,189,166)(8,251,226,119,190,167)(9,252,227,120,191,168)(10,253,228,97,192,145)(11,254,229,98,169,146)(12,255,230,99,170,147)(13,256,231,100,171,148)(14,257,232,101,172,149)(15,258,233,102,173,150)(16,259,234,103,174,151)(17,260,235,104,175,152)(18,261,236,105,176,153)(19,262,237,106,177,154)(20,263,238,107,178,155)(21,264,239,108,179,156)(22,241,240,109,180,157)(23,242,217,110,181,158)(24,243,218,111,182,159)(25,65,211,88,283,143)(26,66,212,89,284,144)(27,67,213,90,285,121)(28,68,214,91,286,122)(29,69,215,92,287,123)(30,70,216,93,288,124)(31,71,193,94,265,125)(32,72,194,95,266,126)(33,49,195,96,267,127)(34,50,196,73,268,128)(35,51,197,74,269,129)(36,52,198,75,270,130)(37,53,199,76,271,131)(38,54,200,77,272,132)(39,55,201,78,273,133)(40,56,202,79,274,134)(41,57,203,80,275,135)(42,58,204,81,276,136)(43,59,205,82,277,137)(44,60,206,83,278,138)(45,61,207,84,279,139)(46,62,208,85,280,140)(47,63,209,86,281,141)(48,64,210,87,282,142), (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), (1,131,100,211)(2,198,101,142)(3,129,102,209)(4,196,103,140)(5,127,104,207)(6,194,105,138)(7,125,106,205)(8,216,107,136)(9,123,108,203)(10,214,109,134)(11,121,110,201)(12,212,111,132)(13,143,112,199)(14,210,113,130)(15,141,114,197)(16,208,115,128)(17,139,116,195)(18,206,117,126)(19,137,118,193)(20,204,119,124)(21,135,120,215)(22,202,97,122)(23,133,98,213)(24,200,99,144)(25,219,76,148)(26,159,77,230)(27,217,78,146)(28,157,79,228)(29,239,80,168)(30,155,81,226)(31,237,82,166)(32,153,83,224)(33,235,84,164)(34,151,85,222)(35,233,86,162)(36,149,87,220)(37,231,88,160)(38,147,89,218)(39,229,90,158)(40,145,91,240)(41,227,92,156)(42,167,93,238)(43,225,94,154)(44,165,95,236)(45,223,96,152)(46,163,73,234)(47,221,74,150)(48,161,75,232)(49,260,279,187)(50,174,280,247)(51,258,281,185)(52,172,282,245)(53,256,283,183)(54,170,284,243)(55,254,285,181)(56,192,286,241)(57,252,287,179)(58,190,288,263)(59,250,265,177)(60,188,266,261)(61,248,267,175)(62,186,268,259)(63,246,269,173)(64,184,270,257)(65,244,271,171)(66,182,272,255)(67,242,273,169)(68,180,274,253)(69,264,275,191)(70,178,276,251)(71,262,277,189)(72,176,278,249) );
G=PermutationGroup([[(1,244,219,112,183,160),(2,245,220,113,184,161),(3,246,221,114,185,162),(4,247,222,115,186,163),(5,248,223,116,187,164),(6,249,224,117,188,165),(7,250,225,118,189,166),(8,251,226,119,190,167),(9,252,227,120,191,168),(10,253,228,97,192,145),(11,254,229,98,169,146),(12,255,230,99,170,147),(13,256,231,100,171,148),(14,257,232,101,172,149),(15,258,233,102,173,150),(16,259,234,103,174,151),(17,260,235,104,175,152),(18,261,236,105,176,153),(19,262,237,106,177,154),(20,263,238,107,178,155),(21,264,239,108,179,156),(22,241,240,109,180,157),(23,242,217,110,181,158),(24,243,218,111,182,159),(25,65,211,88,283,143),(26,66,212,89,284,144),(27,67,213,90,285,121),(28,68,214,91,286,122),(29,69,215,92,287,123),(30,70,216,93,288,124),(31,71,193,94,265,125),(32,72,194,95,266,126),(33,49,195,96,267,127),(34,50,196,73,268,128),(35,51,197,74,269,129),(36,52,198,75,270,130),(37,53,199,76,271,131),(38,54,200,77,272,132),(39,55,201,78,273,133),(40,56,202,79,274,134),(41,57,203,80,275,135),(42,58,204,81,276,136),(43,59,205,82,277,137),(44,60,206,83,278,138),(45,61,207,84,279,139),(46,62,208,85,280,140),(47,63,209,86,281,141),(48,64,210,87,282,142)], [(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)], [(1,131,100,211),(2,198,101,142),(3,129,102,209),(4,196,103,140),(5,127,104,207),(6,194,105,138),(7,125,106,205),(8,216,107,136),(9,123,108,203),(10,214,109,134),(11,121,110,201),(12,212,111,132),(13,143,112,199),(14,210,113,130),(15,141,114,197),(16,208,115,128),(17,139,116,195),(18,206,117,126),(19,137,118,193),(20,204,119,124),(21,135,120,215),(22,202,97,122),(23,133,98,213),(24,200,99,144),(25,219,76,148),(26,159,77,230),(27,217,78,146),(28,157,79,228),(29,239,80,168),(30,155,81,226),(31,237,82,166),(32,153,83,224),(33,235,84,164),(34,151,85,222),(35,233,86,162),(36,149,87,220),(37,231,88,160),(38,147,89,218),(39,229,90,158),(40,145,91,240),(41,227,92,156),(42,167,93,238),(43,225,94,154),(44,165,95,236),(45,223,96,152),(46,163,73,234),(47,221,74,150),(48,161,75,232),(49,260,279,187),(50,174,280,247),(51,258,281,185),(52,172,282,245),(53,256,283,183),(54,170,284,243),(55,254,285,181),(56,192,286,241),(57,252,287,179),(58,190,288,263),(59,250,265,177),(60,188,266,261),(61,248,267,175),(62,186,268,259),(63,246,269,173),(64,184,270,257),(65,244,271,171),(66,182,272,255),(67,242,273,169),(68,180,274,253),(69,264,275,191),(70,178,276,251),(71,262,277,189),(72,176,278,249)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | SD16 | Q16 | C4×S3 | C3⋊D4 | D12 | C24⋊C2 | Dic12 |
| kernel | C6.4Dic12 | C12⋊Dic3 | C6×C24 | C2×C32⋊4Q8 | C32⋊4Q8 | C2×C24 | 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 | 16 | 16 |
Matrix representation of C6.4Dic12 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 67 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 43 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 4 | 0 | 0 |
| 0 | 0 | 17 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 26 |
| 0 | 0 | 0 | 0 | 34 | 65 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,67,0,0,0,0,12,0,0,0,0,0,0,0,43,30,0,0,0,0,43,13],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,58,17,0,0,0,0,4,15,0,0,0,0,0,0,8,34,0,0,0,0,26,65] >;
C6.4Dic12 in GAP, Magma, Sage, TeX
C_6._4{\rm Dic}_{12} % in TeX
G:=Group("C6.4Dic12"); // GroupNames label
G:=SmallGroup(288,291);
// by ID
G=gap.SmallGroup(288,291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,92,422,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=1,c^2=a^3*b^12,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations