metabelian, supersoluble, monomial
Aliases: C12.10Dic6, C62.112D4, C12.25(C4×S3), C32⋊4C8⋊4C4, (C2×C12).86D6, (C3×C12).12Q8, C32⋊8(C4.Q8), (C3×C6).28SD16, C6.11(D4.S3), (C6×C12).53C22, C3⋊3(C12.Q8), C4.2(C32⋊4Q8), C6.17(Dic3⋊C4), C6.11(Q8⋊2S3), C12⋊Dic3.13C2, C2.1(C32⋊9SD16), C2.1(C32⋊11SD16), C2.4(C6.Dic6), C22.13(C32⋊7D4), C4.12(C4×C3⋊S3), C4⋊C4.2(C3⋊S3), (C3×C4⋊C4).18S3, (C3×C6).38(C4⋊C4), (C3×C12).47(C2×C4), (C32×C4⋊C4).5C2, (C2×C6).88(C3⋊D4), (C2×C32⋊4C8).6C2, (C2×C4).35(C2×C3⋊S3), SmallGroup(288,283)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.10Dic6
G = < a,b,c | a12=b12=1, c2=a9b6, bab-1=a7, cac-1=a5, cbc-1=a3b-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, C4.Q8, C3⋊Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C32⋊4C8, C2×C3⋊Dic3, C6×C12, C6×C12, C12.Q8, C2×C32⋊4C8, C12⋊Dic3, C32×C4⋊C4, C12.10Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, SD16, C3⋊S3, Dic6, C4×S3, C3⋊D4, C4.Q8, C2×C3⋊S3, Dic3⋊C4, D4.S3, Q8⋊2S3, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C12.Q8, C6.Dic6, C32⋊9SD16, C32⋊11SD16, C12.10Dic6
►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 82 64 195 114 252 145 139 213 41 269 169)(2 77 65 202 115 247 146 134 214 48 270 176)(3 84 66 197 116 242 147 141 215 43 271 171)(4 79 67 204 117 249 148 136 216 38 272 178)(5 74 68 199 118 244 149 143 205 45 273 173)(6 81 69 194 119 251 150 138 206 40 274 180)(7 76 70 201 120 246 151 133 207 47 275 175)(8 83 71 196 109 241 152 140 208 42 276 170)(9 78 72 203 110 248 153 135 209 37 265 177)(10 73 61 198 111 243 154 142 210 44 266 172)(11 80 62 193 112 250 155 137 211 39 267 179)(12 75 63 200 113 245 156 144 212 46 268 174)(13 88 35 278 238 187 158 53 256 130 224 99)(14 95 36 285 239 182 159 60 257 125 225 106)(15 90 25 280 240 189 160 55 258 132 226 101)(16 85 26 287 229 184 161 50 259 127 227 108)(17 92 27 282 230 191 162 57 260 122 228 103)(18 87 28 277 231 186 163 52 261 129 217 98)(19 94 29 284 232 181 164 59 262 124 218 105)(20 89 30 279 233 188 165 54 263 131 219 100)(21 96 31 286 234 183 166 49 264 126 220 107)(22 91 32 281 235 190 167 56 253 121 221 102)(23 86 33 288 236 185 168 51 254 128 222 97)(24 93 34 283 237 192 157 58 255 123 223 104)
(1 29 154 259 7 35 148 253)(2 34 155 264 8 28 149 258)(3 27 156 257 9 33 150 263)(4 32 145 262 10 26 151 256)(5 25 146 255 11 31 152 261)(6 30 147 260 12 36 153 254)(13 216 167 64 19 210 161 70)(14 209 168 69 20 215 162 63)(15 214 157 62 21 208 163 68)(16 207 158 67 22 213 164 61)(17 212 159 72 23 206 165 66)(18 205 160 65 24 211 166 71)(37 188 200 97 43 182 194 103)(38 181 201 102 44 187 195 108)(39 186 202 107 45 192 196 101)(40 191 203 100 46 185 197 106)(41 184 204 105 47 190 198 99)(42 189 193 98 48 183 199 104)(49 74 93 140 55 80 87 134)(50 79 94 133 56 73 88 139)(51 84 95 138 57 78 89 144)(52 77 96 143 58 83 90 137)(53 82 85 136 59 76 91 142)(54 75 86 141 60 81 92 135)(109 217 273 240 115 223 267 234)(110 222 274 233 116 228 268 239)(111 227 275 238 117 221 269 232)(112 220 276 231 118 226 270 237)(113 225 265 236 119 219 271 230)(114 218 266 229 120 224 272 235)(121 172 278 252 127 178 284 246)(122 177 279 245 128 171 285 251)(123 170 280 250 129 176 286 244)(124 175 281 243 130 169 287 249)(125 180 282 248 131 174 288 242)(126 173 283 241 132 179 277 247)
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), (1,82,64,195,114,252,145,139,213,41,269,169)(2,77,65,202,115,247,146,134,214,48,270,176)(3,84,66,197,116,242,147,141,215,43,271,171)(4,79,67,204,117,249,148,136,216,38,272,178)(5,74,68,199,118,244,149,143,205,45,273,173)(6,81,69,194,119,251,150,138,206,40,274,180)(7,76,70,201,120,246,151,133,207,47,275,175)(8,83,71,196,109,241,152,140,208,42,276,170)(9,78,72,203,110,248,153,135,209,37,265,177)(10,73,61,198,111,243,154,142,210,44,266,172)(11,80,62,193,112,250,155,137,211,39,267,179)(12,75,63,200,113,245,156,144,212,46,268,174)(13,88,35,278,238,187,158,53,256,130,224,99)(14,95,36,285,239,182,159,60,257,125,225,106)(15,90,25,280,240,189,160,55,258,132,226,101)(16,85,26,287,229,184,161,50,259,127,227,108)(17,92,27,282,230,191,162,57,260,122,228,103)(18,87,28,277,231,186,163,52,261,129,217,98)(19,94,29,284,232,181,164,59,262,124,218,105)(20,89,30,279,233,188,165,54,263,131,219,100)(21,96,31,286,234,183,166,49,264,126,220,107)(22,91,32,281,235,190,167,56,253,121,221,102)(23,86,33,288,236,185,168,51,254,128,222,97)(24,93,34,283,237,192,157,58,255,123,223,104), (1,29,154,259,7,35,148,253)(2,34,155,264,8,28,149,258)(3,27,156,257,9,33,150,263)(4,32,145,262,10,26,151,256)(5,25,146,255,11,31,152,261)(6,30,147,260,12,36,153,254)(13,216,167,64,19,210,161,70)(14,209,168,69,20,215,162,63)(15,214,157,62,21,208,163,68)(16,207,158,67,22,213,164,61)(17,212,159,72,23,206,165,66)(18,205,160,65,24,211,166,71)(37,188,200,97,43,182,194,103)(38,181,201,102,44,187,195,108)(39,186,202,107,45,192,196,101)(40,191,203,100,46,185,197,106)(41,184,204,105,47,190,198,99)(42,189,193,98,48,183,199,104)(49,74,93,140,55,80,87,134)(50,79,94,133,56,73,88,139)(51,84,95,138,57,78,89,144)(52,77,96,143,58,83,90,137)(53,82,85,136,59,76,91,142)(54,75,86,141,60,81,92,135)(109,217,273,240,115,223,267,234)(110,222,274,233,116,228,268,239)(111,227,275,238,117,221,269,232)(112,220,276,231,118,226,270,237)(113,225,265,236,119,219,271,230)(114,218,266,229,120,224,272,235)(121,172,278,252,127,178,284,246)(122,177,279,245,128,171,285,251)(123,170,280,250,129,176,286,244)(124,175,281,243,130,169,287,249)(125,180,282,248,131,174,288,242)(126,173,283,241,132,179,277,247)>;
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), (1,82,64,195,114,252,145,139,213,41,269,169)(2,77,65,202,115,247,146,134,214,48,270,176)(3,84,66,197,116,242,147,141,215,43,271,171)(4,79,67,204,117,249,148,136,216,38,272,178)(5,74,68,199,118,244,149,143,205,45,273,173)(6,81,69,194,119,251,150,138,206,40,274,180)(7,76,70,201,120,246,151,133,207,47,275,175)(8,83,71,196,109,241,152,140,208,42,276,170)(9,78,72,203,110,248,153,135,209,37,265,177)(10,73,61,198,111,243,154,142,210,44,266,172)(11,80,62,193,112,250,155,137,211,39,267,179)(12,75,63,200,113,245,156,144,212,46,268,174)(13,88,35,278,238,187,158,53,256,130,224,99)(14,95,36,285,239,182,159,60,257,125,225,106)(15,90,25,280,240,189,160,55,258,132,226,101)(16,85,26,287,229,184,161,50,259,127,227,108)(17,92,27,282,230,191,162,57,260,122,228,103)(18,87,28,277,231,186,163,52,261,129,217,98)(19,94,29,284,232,181,164,59,262,124,218,105)(20,89,30,279,233,188,165,54,263,131,219,100)(21,96,31,286,234,183,166,49,264,126,220,107)(22,91,32,281,235,190,167,56,253,121,221,102)(23,86,33,288,236,185,168,51,254,128,222,97)(24,93,34,283,237,192,157,58,255,123,223,104), (1,29,154,259,7,35,148,253)(2,34,155,264,8,28,149,258)(3,27,156,257,9,33,150,263)(4,32,145,262,10,26,151,256)(5,25,146,255,11,31,152,261)(6,30,147,260,12,36,153,254)(13,216,167,64,19,210,161,70)(14,209,168,69,20,215,162,63)(15,214,157,62,21,208,163,68)(16,207,158,67,22,213,164,61)(17,212,159,72,23,206,165,66)(18,205,160,65,24,211,166,71)(37,188,200,97,43,182,194,103)(38,181,201,102,44,187,195,108)(39,186,202,107,45,192,196,101)(40,191,203,100,46,185,197,106)(41,184,204,105,47,190,198,99)(42,189,193,98,48,183,199,104)(49,74,93,140,55,80,87,134)(50,79,94,133,56,73,88,139)(51,84,95,138,57,78,89,144)(52,77,96,143,58,83,90,137)(53,82,85,136,59,76,91,142)(54,75,86,141,60,81,92,135)(109,217,273,240,115,223,267,234)(110,222,274,233,116,228,268,239)(111,227,275,238,117,221,269,232)(112,220,276,231,118,226,270,237)(113,225,265,236,119,219,271,230)(114,218,266,229,120,224,272,235)(121,172,278,252,127,178,284,246)(122,177,279,245,128,171,285,251)(123,170,280,250,129,176,286,244)(124,175,281,243,130,169,287,249)(125,180,282,248,131,174,288,242)(126,173,283,241,132,179,277,247) );
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)], [(1,82,64,195,114,252,145,139,213,41,269,169),(2,77,65,202,115,247,146,134,214,48,270,176),(3,84,66,197,116,242,147,141,215,43,271,171),(4,79,67,204,117,249,148,136,216,38,272,178),(5,74,68,199,118,244,149,143,205,45,273,173),(6,81,69,194,119,251,150,138,206,40,274,180),(7,76,70,201,120,246,151,133,207,47,275,175),(8,83,71,196,109,241,152,140,208,42,276,170),(9,78,72,203,110,248,153,135,209,37,265,177),(10,73,61,198,111,243,154,142,210,44,266,172),(11,80,62,193,112,250,155,137,211,39,267,179),(12,75,63,200,113,245,156,144,212,46,268,174),(13,88,35,278,238,187,158,53,256,130,224,99),(14,95,36,285,239,182,159,60,257,125,225,106),(15,90,25,280,240,189,160,55,258,132,226,101),(16,85,26,287,229,184,161,50,259,127,227,108),(17,92,27,282,230,191,162,57,260,122,228,103),(18,87,28,277,231,186,163,52,261,129,217,98),(19,94,29,284,232,181,164,59,262,124,218,105),(20,89,30,279,233,188,165,54,263,131,219,100),(21,96,31,286,234,183,166,49,264,126,220,107),(22,91,32,281,235,190,167,56,253,121,221,102),(23,86,33,288,236,185,168,51,254,128,222,97),(24,93,34,283,237,192,157,58,255,123,223,104)], [(1,29,154,259,7,35,148,253),(2,34,155,264,8,28,149,258),(3,27,156,257,9,33,150,263),(4,32,145,262,10,26,151,256),(5,25,146,255,11,31,152,261),(6,30,147,260,12,36,153,254),(13,216,167,64,19,210,161,70),(14,209,168,69,20,215,162,63),(15,214,157,62,21,208,163,68),(16,207,158,67,22,213,164,61),(17,212,159,72,23,206,165,66),(18,205,160,65,24,211,166,71),(37,188,200,97,43,182,194,103),(38,181,201,102,44,187,195,108),(39,186,202,107,45,192,196,101),(40,191,203,100,46,185,197,106),(41,184,204,105,47,190,198,99),(42,189,193,98,48,183,199,104),(49,74,93,140,55,80,87,134),(50,79,94,133,56,73,88,139),(51,84,95,138,57,78,89,144),(52,77,96,143,58,83,90,137),(53,82,85,136,59,76,91,142),(54,75,86,141,60,81,92,135),(109,217,273,240,115,223,267,234),(110,222,274,233,116,228,268,239),(111,227,275,238,117,221,269,232),(112,220,276,231,118,226,270,237),(113,225,265,236,119,219,271,230),(114,218,266,229,120,224,272,235),(121,172,278,252,127,178,284,246),(122,177,279,245,128,171,285,251),(123,170,280,250,129,176,286,244),(124,175,281,243,130,169,287,249),(125,180,282,248,131,174,288,242),(126,173,283,241,132,179,277,247)]])
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 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | SD16 | Dic6 | C4×S3 | C3⋊D4 | D4.S3 | Q8⋊2S3 |
| kernel | C12.10Dic6 | C2×C32⋊4C8 | C12⋊Dic3 | C32×C4⋊C4 | C32⋊4C8 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C12.10Dic6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 66 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 51 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 44 | 54 | 0 | 0 | 0 | 0 |
| 25 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 14 | 0 | 0 |
| 0 | 0 | 68 | 54 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 67 | 12 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,14,66,0,0,0,0,7,7,0,0,0,0,0,0,41,30,0,0,0,0,51,32],[44,25,0,0,0,0,54,29,0,0,0,0,0,0,19,68,0,0,0,0,14,54,0,0,0,0,0,0,0,67,0,0,0,0,12,12] >;
C12.10Dic6 in GAP, Magma, Sage, TeX
C_{12}._{10}{\rm Dic}_6 % in TeX
G:=Group("C12.10Dic6"); // GroupNames label
G:=SmallGroup(288,283);
// by ID
G=gap.SmallGroup(288,283);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,365,36,346,80,2693,9414]);
// 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^3*b^-1>;
// generators/relations