metabelian, supersoluble, monomial
Aliases: C12.57D12, C122.4C2, C12.29Dic6, C12⋊1(C3⋊C8), (C3×C12)⋊5C8, C32⋊9(C4⋊C8), C4⋊(C32⋊4C8), (C4×C12).14S3, (C6×C12).19C4, C3⋊2(C12⋊C8), (C3×C12).27Q8, (C2×C12).416D6, (C3×C12).137D4, C42.2(C3⋊S3), C62.102(C2×C4), C6.12(C4⋊Dic3), (C2×C12).12Dic3, C4.16(C12⋊S3), (C3×C6).22M4(2), C6.9(C4.Dic3), C4.7(C32⋊4Q8), (C6×C12).338C22, C2.1(C12⋊Dic3), C2.2(C12.58D6), C6.13(C2×C3⋊C8), (C3×C6).43(C2×C8), (C3×C6).36(C4⋊C4), C2.3(C2×C32⋊4C8), (C2×C4).4(C3⋊Dic3), (C2×C6).44(C2×Dic3), C22.8(C2×C3⋊Dic3), (C2×C32⋊4C8).16C2, (C2×C4).89(C2×C3⋊S3), SmallGroup(288,279)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.57D12
G = < a,b,c | a12=b12=1, c2=a9, ab=ba, cac-1=a5, cbc-1=b-1 >
Subgroups: 244 in 114 conjugacy classes, 81 normal (23 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C32, C12, C12, C2×C6, C42, C2×C8, C3×C6, C3⋊C8, C2×C12, C4⋊C8, C3×C12, C3×C12, C3×C12, C62, C2×C3⋊C8, C4×C12, C32⋊4C8, C6×C12, C12⋊C8, C2×C32⋊4C8, C122, C12.57D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C2×C8, M4(2), C3⋊S3, C3⋊C8, Dic6, D12, C2×Dic3, C4⋊C8, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, C32⋊4C8, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C12⋊C8, C2×C32⋊4C8, C12.58D6, C12⋊Dic3, C12.57D12
►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)
(1 263 153 86 288 102 161 244 135 275 229 195)(2 264 154 87 277 103 162 245 136 276 230 196)(3 253 155 88 278 104 163 246 137 265 231 197)(4 254 156 89 279 105 164 247 138 266 232 198)(5 255 145 90 280 106 165 248 139 267 233 199)(6 256 146 91 281 107 166 249 140 268 234 200)(7 257 147 92 282 108 167 250 141 269 235 201)(8 258 148 93 283 97 168 251 142 270 236 202)(9 259 149 94 284 98 157 252 143 271 237 203)(10 260 150 95 285 99 158 241 144 272 238 204)(11 261 151 96 286 100 159 242 133 273 239 193)(12 262 152 85 287 101 160 243 134 274 240 194)(13 181 78 72 119 176 132 46 215 224 36 57)(14 182 79 61 120 177 121 47 216 225 25 58)(15 183 80 62 109 178 122 48 205 226 26 59)(16 184 81 63 110 179 123 37 206 227 27 60)(17 185 82 64 111 180 124 38 207 228 28 49)(18 186 83 65 112 169 125 39 208 217 29 50)(19 187 84 66 113 170 126 40 209 218 30 51)(20 188 73 67 114 171 127 41 210 219 31 52)(21 189 74 68 115 172 128 42 211 220 32 53)(22 190 75 69 116 173 129 43 212 221 33 54)(23 191 76 70 117 174 130 44 213 222 34 55)(24 192 77 71 118 175 131 45 214 223 35 56)
(1 209 10 206 7 215 4 212)(2 214 11 211 8 208 5 205)(3 207 12 216 9 213 6 210)(13 138 22 135 19 144 16 141)(14 143 23 140 20 137 17 134)(15 136 24 133 21 142 18 139)(25 237 34 234 31 231 28 240)(26 230 35 239 32 236 29 233)(27 235 36 232 33 229 30 238)(37 257 46 254 43 263 40 260)(38 262 47 259 44 256 41 253)(39 255 48 264 45 261 42 258)(49 274 58 271 55 268 52 265)(50 267 59 276 56 273 53 270)(51 272 60 269 57 266 54 275)(61 98 70 107 67 104 64 101)(62 103 71 100 68 97 65 106)(63 108 72 105 69 102 66 99)(73 163 82 160 79 157 76 166)(74 168 83 165 80 162 77 159)(75 161 84 158 81 167 78 164)(85 177 94 174 91 171 88 180)(86 170 95 179 92 176 89 173)(87 175 96 172 93 169 90 178)(109 277 118 286 115 283 112 280)(110 282 119 279 116 288 113 285)(111 287 120 284 117 281 114 278)(121 149 130 146 127 155 124 152)(122 154 131 151 128 148 125 145)(123 147 132 156 129 153 126 150)(181 247 190 244 187 241 184 250)(182 252 191 249 188 246 185 243)(183 245 192 242 189 251 186 248)(193 220 202 217 199 226 196 223)(194 225 203 222 200 219 197 228)(195 218 204 227 201 224 198 221)
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), (1,263,153,86,288,102,161,244,135,275,229,195)(2,264,154,87,277,103,162,245,136,276,230,196)(3,253,155,88,278,104,163,246,137,265,231,197)(4,254,156,89,279,105,164,247,138,266,232,198)(5,255,145,90,280,106,165,248,139,267,233,199)(6,256,146,91,281,107,166,249,140,268,234,200)(7,257,147,92,282,108,167,250,141,269,235,201)(8,258,148,93,283,97,168,251,142,270,236,202)(9,259,149,94,284,98,157,252,143,271,237,203)(10,260,150,95,285,99,158,241,144,272,238,204)(11,261,151,96,286,100,159,242,133,273,239,193)(12,262,152,85,287,101,160,243,134,274,240,194)(13,181,78,72,119,176,132,46,215,224,36,57)(14,182,79,61,120,177,121,47,216,225,25,58)(15,183,80,62,109,178,122,48,205,226,26,59)(16,184,81,63,110,179,123,37,206,227,27,60)(17,185,82,64,111,180,124,38,207,228,28,49)(18,186,83,65,112,169,125,39,208,217,29,50)(19,187,84,66,113,170,126,40,209,218,30,51)(20,188,73,67,114,171,127,41,210,219,31,52)(21,189,74,68,115,172,128,42,211,220,32,53)(22,190,75,69,116,173,129,43,212,221,33,54)(23,191,76,70,117,174,130,44,213,222,34,55)(24,192,77,71,118,175,131,45,214,223,35,56), (1,209,10,206,7,215,4,212)(2,214,11,211,8,208,5,205)(3,207,12,216,9,213,6,210)(13,138,22,135,19,144,16,141)(14,143,23,140,20,137,17,134)(15,136,24,133,21,142,18,139)(25,237,34,234,31,231,28,240)(26,230,35,239,32,236,29,233)(27,235,36,232,33,229,30,238)(37,257,46,254,43,263,40,260)(38,262,47,259,44,256,41,253)(39,255,48,264,45,261,42,258)(49,274,58,271,55,268,52,265)(50,267,59,276,56,273,53,270)(51,272,60,269,57,266,54,275)(61,98,70,107,67,104,64,101)(62,103,71,100,68,97,65,106)(63,108,72,105,69,102,66,99)(73,163,82,160,79,157,76,166)(74,168,83,165,80,162,77,159)(75,161,84,158,81,167,78,164)(85,177,94,174,91,171,88,180)(86,170,95,179,92,176,89,173)(87,175,96,172,93,169,90,178)(109,277,118,286,115,283,112,280)(110,282,119,279,116,288,113,285)(111,287,120,284,117,281,114,278)(121,149,130,146,127,155,124,152)(122,154,131,151,128,148,125,145)(123,147,132,156,129,153,126,150)(181,247,190,244,187,241,184,250)(182,252,191,249,188,246,185,243)(183,245,192,242,189,251,186,248)(193,220,202,217,199,226,196,223)(194,225,203,222,200,219,197,228)(195,218,204,227,201,224,198,221)>;
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), (1,263,153,86,288,102,161,244,135,275,229,195)(2,264,154,87,277,103,162,245,136,276,230,196)(3,253,155,88,278,104,163,246,137,265,231,197)(4,254,156,89,279,105,164,247,138,266,232,198)(5,255,145,90,280,106,165,248,139,267,233,199)(6,256,146,91,281,107,166,249,140,268,234,200)(7,257,147,92,282,108,167,250,141,269,235,201)(8,258,148,93,283,97,168,251,142,270,236,202)(9,259,149,94,284,98,157,252,143,271,237,203)(10,260,150,95,285,99,158,241,144,272,238,204)(11,261,151,96,286,100,159,242,133,273,239,193)(12,262,152,85,287,101,160,243,134,274,240,194)(13,181,78,72,119,176,132,46,215,224,36,57)(14,182,79,61,120,177,121,47,216,225,25,58)(15,183,80,62,109,178,122,48,205,226,26,59)(16,184,81,63,110,179,123,37,206,227,27,60)(17,185,82,64,111,180,124,38,207,228,28,49)(18,186,83,65,112,169,125,39,208,217,29,50)(19,187,84,66,113,170,126,40,209,218,30,51)(20,188,73,67,114,171,127,41,210,219,31,52)(21,189,74,68,115,172,128,42,211,220,32,53)(22,190,75,69,116,173,129,43,212,221,33,54)(23,191,76,70,117,174,130,44,213,222,34,55)(24,192,77,71,118,175,131,45,214,223,35,56), (1,209,10,206,7,215,4,212)(2,214,11,211,8,208,5,205)(3,207,12,216,9,213,6,210)(13,138,22,135,19,144,16,141)(14,143,23,140,20,137,17,134)(15,136,24,133,21,142,18,139)(25,237,34,234,31,231,28,240)(26,230,35,239,32,236,29,233)(27,235,36,232,33,229,30,238)(37,257,46,254,43,263,40,260)(38,262,47,259,44,256,41,253)(39,255,48,264,45,261,42,258)(49,274,58,271,55,268,52,265)(50,267,59,276,56,273,53,270)(51,272,60,269,57,266,54,275)(61,98,70,107,67,104,64,101)(62,103,71,100,68,97,65,106)(63,108,72,105,69,102,66,99)(73,163,82,160,79,157,76,166)(74,168,83,165,80,162,77,159)(75,161,84,158,81,167,78,164)(85,177,94,174,91,171,88,180)(86,170,95,179,92,176,89,173)(87,175,96,172,93,169,90,178)(109,277,118,286,115,283,112,280)(110,282,119,279,116,288,113,285)(111,287,120,284,117,281,114,278)(121,149,130,146,127,155,124,152)(122,154,131,151,128,148,125,145)(123,147,132,156,129,153,126,150)(181,247,190,244,187,241,184,250)(182,252,191,249,188,246,185,243)(183,245,192,242,189,251,186,248)(193,220,202,217,199,226,196,223)(194,225,203,222,200,219,197,228)(195,218,204,227,201,224,198,221) );
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)], [(1,263,153,86,288,102,161,244,135,275,229,195),(2,264,154,87,277,103,162,245,136,276,230,196),(3,253,155,88,278,104,163,246,137,265,231,197),(4,254,156,89,279,105,164,247,138,266,232,198),(5,255,145,90,280,106,165,248,139,267,233,199),(6,256,146,91,281,107,166,249,140,268,234,200),(7,257,147,92,282,108,167,250,141,269,235,201),(8,258,148,93,283,97,168,251,142,270,236,202),(9,259,149,94,284,98,157,252,143,271,237,203),(10,260,150,95,285,99,158,241,144,272,238,204),(11,261,151,96,286,100,159,242,133,273,239,193),(12,262,152,85,287,101,160,243,134,274,240,194),(13,181,78,72,119,176,132,46,215,224,36,57),(14,182,79,61,120,177,121,47,216,225,25,58),(15,183,80,62,109,178,122,48,205,226,26,59),(16,184,81,63,110,179,123,37,206,227,27,60),(17,185,82,64,111,180,124,38,207,228,28,49),(18,186,83,65,112,169,125,39,208,217,29,50),(19,187,84,66,113,170,126,40,209,218,30,51),(20,188,73,67,114,171,127,41,210,219,31,52),(21,189,74,68,115,172,128,42,211,220,32,53),(22,190,75,69,116,173,129,43,212,221,33,54),(23,191,76,70,117,174,130,44,213,222,34,55),(24,192,77,71,118,175,131,45,214,223,35,56)], [(1,209,10,206,7,215,4,212),(2,214,11,211,8,208,5,205),(3,207,12,216,9,213,6,210),(13,138,22,135,19,144,16,141),(14,143,23,140,20,137,17,134),(15,136,24,133,21,142,18,139),(25,237,34,234,31,231,28,240),(26,230,35,239,32,236,29,233),(27,235,36,232,33,229,30,238),(37,257,46,254,43,263,40,260),(38,262,47,259,44,256,41,253),(39,255,48,264,45,261,42,258),(49,274,58,271,55,268,52,265),(50,267,59,276,56,273,53,270),(51,272,60,269,57,266,54,275),(61,98,70,107,67,104,64,101),(62,103,71,100,68,97,65,106),(63,108,72,105,69,102,66,99),(73,163,82,160,79,157,76,166),(74,168,83,165,80,162,77,159),(75,161,84,158,81,167,78,164),(85,177,94,174,91,171,88,180),(86,170,95,179,92,176,89,173),(87,175,96,172,93,169,90,178),(109,277,118,286,115,283,112,280),(110,282,119,279,116,288,113,285),(111,287,120,284,117,281,114,278),(121,149,130,146,127,155,124,152),(122,154,131,151,128,148,125,145),(123,147,132,156,129,153,126,150),(181,247,190,244,187,241,184,250),(182,252,191,249,188,246,185,243),(183,245,192,242,189,251,186,248),(193,220,202,217,199,226,196,223),(194,225,203,222,200,219,197,228),(195,218,204,227,201,224,198,221)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 8A | ··· | 8H | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | Dic3 | D6 | M4(2) | C3⋊C8 | Dic6 | D12 | C4.Dic3 |
| kernel | C12.57D12 | C2×C32⋊4C8 | C122 | C6×C12 | C3×C12 | C4×C12 | C3×C12 | C3×C12 | C2×C12 | C2×C12 | C3×C6 | C12 | C12 | C12 | C6 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 8 | 4 | 2 | 16 | 8 | 8 | 16 |
Matrix representation of C12.57D12 ►in GL4(𝔽73) generated by
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 59 | 7 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 61 | 18 |
| 0 | 0 | 30 | 12 |
G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,27,0,0,0,0,27],[59,66,0,0,7,66,0,0,0,0,72,1,0,0,72,0],[72,72,0,0,0,1,0,0,0,0,61,30,0,0,18,12] >;
C12.57D12 in GAP, Magma, Sage, TeX
C_{12}._{57}D_{12} % in TeX
G:=Group("C12.57D12"); // GroupNames label
G:=SmallGroup(288,279);
// by ID
G=gap.SmallGroup(288,279);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^9,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations