metabelian, supersoluble, monomial
Aliases: C12.30Dic6, C6.15(S3×C8), (C2×C24).7S3, (C6×C24).3C2, C3⋊Dic3⋊5C8, C32⋊11(C4⋊C8), (C3×C12).28Q8, C6.8(C8⋊S3), C3⋊3(Dic3⋊C8), (C3×C12).168D4, (C2×C12).418D6, C62.72(C2×C4), C2.1(C24⋊S3), (C3×C6).15M4(2), C12.129(C3⋊D4), C4.8(C32⋊4Q8), C6.18(Dic3⋊C4), (C6×C12).340C22, C4.26(C32⋊7D4), C2.1(C6.Dic6), C2.4(C8×C3⋊S3), (C2×C8).1(C3⋊S3), (C3×C6).35(C2×C8), (C2×C6).47(C4×S3), C22.9(C4×C3⋊S3), (C3×C6).39(C4⋊C4), (C4×C3⋊Dic3).12C2, (C2×C3⋊Dic3).10C4, (C2×C32⋊4C8).17C2, (C2×C4).91(C2×C3⋊S3), SmallGroup(288,289)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.30Dic6
G = < a,b,c | a12=1, b12=a6, c2=a9b6, ab=ba, cac-1=a5, cbc-1=a3b11 >
Subgroups: 308 in 114 conjugacy classes, 61 normal (23 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C42, C2×C8, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C4⋊C8, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4×Dic3, C2×C24, C32⋊4C8, C3×C24, C2×C3⋊Dic3, C6×C12, Dic3⋊C8, C2×C32⋊4C8, C4×C3⋊Dic3, C6×C24, C12.30Dic6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D6, C4⋊C4, C2×C8, M4(2), C3⋊S3, Dic6, C4×S3, C3⋊D4, C4⋊C8, C2×C3⋊S3, S3×C8, C8⋊S3, Dic3⋊C4, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, Dic3⋊C8, C8×C3⋊S3, C24⋊S3, C6.Dic6, C12.30Dic6
►Regular action on 288 points
Generators in S288
(1 99 237 159 90 276 13 111 225 147 78 288)(2 100 238 160 91 277 14 112 226 148 79 265)(3 101 239 161 92 278 15 113 227 149 80 266)(4 102 240 162 93 279 16 114 228 150 81 267)(5 103 217 163 94 280 17 115 229 151 82 268)(6 104 218 164 95 281 18 116 230 152 83 269)(7 105 219 165 96 282 19 117 231 153 84 270)(8 106 220 166 73 283 20 118 232 154 85 271)(9 107 221 167 74 284 21 119 233 155 86 272)(10 108 222 168 75 285 22 120 234 156 87 273)(11 109 223 145 76 286 23 97 235 157 88 274)(12 110 224 146 77 287 24 98 236 158 89 275)(25 180 131 49 212 260 37 192 143 61 200 248)(26 181 132 50 213 261 38 169 144 62 201 249)(27 182 133 51 214 262 39 170 121 63 202 250)(28 183 134 52 215 263 40 171 122 64 203 251)(29 184 135 53 216 264 41 172 123 65 204 252)(30 185 136 54 193 241 42 173 124 66 205 253)(31 186 137 55 194 242 43 174 125 67 206 254)(32 187 138 56 195 243 44 175 126 68 207 255)(33 188 139 57 196 244 45 176 127 69 208 256)(34 189 140 58 197 245 46 177 128 70 209 257)(35 190 141 59 198 246 47 178 129 71 210 258)(36 191 142 60 199 247 48 179 130 72 211 259)
(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)
(1 212 153 186)(2 179 154 205)(3 198 155 172)(4 189 156 215)(5 208 157 182)(6 175 158 201)(7 194 159 192)(8 185 160 211)(9 204 161 178)(10 171 162 197)(11 214 163 188)(12 181 164 207)(13 200 165 174)(14 191 166 193)(15 210 167 184)(16 177 168 203)(17 196 145 170)(18 187 146 213)(19 206 147 180)(20 173 148 199)(21 216 149 190)(22 183 150 209)(23 202 151 176)(24 169 152 195)(25 105 67 90)(26 83 68 98)(27 115 69 76)(28 93 70 108)(29 101 71 86)(30 79 72 118)(31 111 49 96)(32 89 50 104)(33 97 51 82)(34 75 52 114)(35 107 53 92)(36 85 54 100)(37 117 55 78)(38 95 56 110)(39 103 57 88)(40 81 58 120)(41 113 59 74)(42 91 60 106)(43 99 61 84)(44 77 62 116)(45 109 63 94)(46 87 64 102)(47 119 65 80)(48 73 66 112)(121 268 244 235)(122 228 245 285)(123 278 246 221)(124 238 247 271)(125 288 248 231)(126 224 249 281)(127 274 250 217)(128 234 251 267)(129 284 252 227)(130 220 253 277)(131 270 254 237)(132 230 255 287)(133 280 256 223)(134 240 257 273)(135 266 258 233)(136 226 259 283)(137 276 260 219)(138 236 261 269)(139 286 262 229)(140 222 263 279)(141 272 264 239)(142 232 241 265)(143 282 242 225)(144 218 243 275)
G:=sub<Sym(288)| (1,99,237,159,90,276,13,111,225,147,78,288)(2,100,238,160,91,277,14,112,226,148,79,265)(3,101,239,161,92,278,15,113,227,149,80,266)(4,102,240,162,93,279,16,114,228,150,81,267)(5,103,217,163,94,280,17,115,229,151,82,268)(6,104,218,164,95,281,18,116,230,152,83,269)(7,105,219,165,96,282,19,117,231,153,84,270)(8,106,220,166,73,283,20,118,232,154,85,271)(9,107,221,167,74,284,21,119,233,155,86,272)(10,108,222,168,75,285,22,120,234,156,87,273)(11,109,223,145,76,286,23,97,235,157,88,274)(12,110,224,146,77,287,24,98,236,158,89,275)(25,180,131,49,212,260,37,192,143,61,200,248)(26,181,132,50,213,261,38,169,144,62,201,249)(27,182,133,51,214,262,39,170,121,63,202,250)(28,183,134,52,215,263,40,171,122,64,203,251)(29,184,135,53,216,264,41,172,123,65,204,252)(30,185,136,54,193,241,42,173,124,66,205,253)(31,186,137,55,194,242,43,174,125,67,206,254)(32,187,138,56,195,243,44,175,126,68,207,255)(33,188,139,57,196,244,45,176,127,69,208,256)(34,189,140,58,197,245,46,177,128,70,209,257)(35,190,141,59,198,246,47,178,129,71,210,258)(36,191,142,60,199,247,48,179,130,72,211,259), (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), (1,212,153,186)(2,179,154,205)(3,198,155,172)(4,189,156,215)(5,208,157,182)(6,175,158,201)(7,194,159,192)(8,185,160,211)(9,204,161,178)(10,171,162,197)(11,214,163,188)(12,181,164,207)(13,200,165,174)(14,191,166,193)(15,210,167,184)(16,177,168,203)(17,196,145,170)(18,187,146,213)(19,206,147,180)(20,173,148,199)(21,216,149,190)(22,183,150,209)(23,202,151,176)(24,169,152,195)(25,105,67,90)(26,83,68,98)(27,115,69,76)(28,93,70,108)(29,101,71,86)(30,79,72,118)(31,111,49,96)(32,89,50,104)(33,97,51,82)(34,75,52,114)(35,107,53,92)(36,85,54,100)(37,117,55,78)(38,95,56,110)(39,103,57,88)(40,81,58,120)(41,113,59,74)(42,91,60,106)(43,99,61,84)(44,77,62,116)(45,109,63,94)(46,87,64,102)(47,119,65,80)(48,73,66,112)(121,268,244,235)(122,228,245,285)(123,278,246,221)(124,238,247,271)(125,288,248,231)(126,224,249,281)(127,274,250,217)(128,234,251,267)(129,284,252,227)(130,220,253,277)(131,270,254,237)(132,230,255,287)(133,280,256,223)(134,240,257,273)(135,266,258,233)(136,226,259,283)(137,276,260,219)(138,236,261,269)(139,286,262,229)(140,222,263,279)(141,272,264,239)(142,232,241,265)(143,282,242,225)(144,218,243,275)>;
G:=Group( (1,99,237,159,90,276,13,111,225,147,78,288)(2,100,238,160,91,277,14,112,226,148,79,265)(3,101,239,161,92,278,15,113,227,149,80,266)(4,102,240,162,93,279,16,114,228,150,81,267)(5,103,217,163,94,280,17,115,229,151,82,268)(6,104,218,164,95,281,18,116,230,152,83,269)(7,105,219,165,96,282,19,117,231,153,84,270)(8,106,220,166,73,283,20,118,232,154,85,271)(9,107,221,167,74,284,21,119,233,155,86,272)(10,108,222,168,75,285,22,120,234,156,87,273)(11,109,223,145,76,286,23,97,235,157,88,274)(12,110,224,146,77,287,24,98,236,158,89,275)(25,180,131,49,212,260,37,192,143,61,200,248)(26,181,132,50,213,261,38,169,144,62,201,249)(27,182,133,51,214,262,39,170,121,63,202,250)(28,183,134,52,215,263,40,171,122,64,203,251)(29,184,135,53,216,264,41,172,123,65,204,252)(30,185,136,54,193,241,42,173,124,66,205,253)(31,186,137,55,194,242,43,174,125,67,206,254)(32,187,138,56,195,243,44,175,126,68,207,255)(33,188,139,57,196,244,45,176,127,69,208,256)(34,189,140,58,197,245,46,177,128,70,209,257)(35,190,141,59,198,246,47,178,129,71,210,258)(36,191,142,60,199,247,48,179,130,72,211,259), (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), (1,212,153,186)(2,179,154,205)(3,198,155,172)(4,189,156,215)(5,208,157,182)(6,175,158,201)(7,194,159,192)(8,185,160,211)(9,204,161,178)(10,171,162,197)(11,214,163,188)(12,181,164,207)(13,200,165,174)(14,191,166,193)(15,210,167,184)(16,177,168,203)(17,196,145,170)(18,187,146,213)(19,206,147,180)(20,173,148,199)(21,216,149,190)(22,183,150,209)(23,202,151,176)(24,169,152,195)(25,105,67,90)(26,83,68,98)(27,115,69,76)(28,93,70,108)(29,101,71,86)(30,79,72,118)(31,111,49,96)(32,89,50,104)(33,97,51,82)(34,75,52,114)(35,107,53,92)(36,85,54,100)(37,117,55,78)(38,95,56,110)(39,103,57,88)(40,81,58,120)(41,113,59,74)(42,91,60,106)(43,99,61,84)(44,77,62,116)(45,109,63,94)(46,87,64,102)(47,119,65,80)(48,73,66,112)(121,268,244,235)(122,228,245,285)(123,278,246,221)(124,238,247,271)(125,288,248,231)(126,224,249,281)(127,274,250,217)(128,234,251,267)(129,284,252,227)(130,220,253,277)(131,270,254,237)(132,230,255,287)(133,280,256,223)(134,240,257,273)(135,266,258,233)(136,226,259,283)(137,276,260,219)(138,236,261,269)(139,286,262,229)(140,222,263,279)(141,272,264,239)(142,232,241,265)(143,282,242,225)(144,218,243,275) );
G=PermutationGroup([[(1,99,237,159,90,276,13,111,225,147,78,288),(2,100,238,160,91,277,14,112,226,148,79,265),(3,101,239,161,92,278,15,113,227,149,80,266),(4,102,240,162,93,279,16,114,228,150,81,267),(5,103,217,163,94,280,17,115,229,151,82,268),(6,104,218,164,95,281,18,116,230,152,83,269),(7,105,219,165,96,282,19,117,231,153,84,270),(8,106,220,166,73,283,20,118,232,154,85,271),(9,107,221,167,74,284,21,119,233,155,86,272),(10,108,222,168,75,285,22,120,234,156,87,273),(11,109,223,145,76,286,23,97,235,157,88,274),(12,110,224,146,77,287,24,98,236,158,89,275),(25,180,131,49,212,260,37,192,143,61,200,248),(26,181,132,50,213,261,38,169,144,62,201,249),(27,182,133,51,214,262,39,170,121,63,202,250),(28,183,134,52,215,263,40,171,122,64,203,251),(29,184,135,53,216,264,41,172,123,65,204,252),(30,185,136,54,193,241,42,173,124,66,205,253),(31,186,137,55,194,242,43,174,125,67,206,254),(32,187,138,56,195,243,44,175,126,68,207,255),(33,188,139,57,196,244,45,176,127,69,208,256),(34,189,140,58,197,245,46,177,128,70,209,257),(35,190,141,59,198,246,47,178,129,71,210,258),(36,191,142,60,199,247,48,179,130,72,211,259)], [(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)], [(1,212,153,186),(2,179,154,205),(3,198,155,172),(4,189,156,215),(5,208,157,182),(6,175,158,201),(7,194,159,192),(8,185,160,211),(9,204,161,178),(10,171,162,197),(11,214,163,188),(12,181,164,207),(13,200,165,174),(14,191,166,193),(15,210,167,184),(16,177,168,203),(17,196,145,170),(18,187,146,213),(19,206,147,180),(20,173,148,199),(21,216,149,190),(22,183,150,209),(23,202,151,176),(24,169,152,195),(25,105,67,90),(26,83,68,98),(27,115,69,76),(28,93,70,108),(29,101,71,86),(30,79,72,118),(31,111,49,96),(32,89,50,104),(33,97,51,82),(34,75,52,114),(35,107,53,92),(36,85,54,100),(37,117,55,78),(38,95,56,110),(39,103,57,88),(40,81,58,120),(41,113,59,74),(42,91,60,106),(43,99,61,84),(44,77,62,116),(45,109,63,94),(46,87,64,102),(47,119,65,80),(48,73,66,112),(121,268,244,235),(122,228,245,285),(123,278,246,221),(124,238,247,271),(125,288,248,231),(126,224,249,281),(127,274,250,217),(128,234,251,267),(129,284,252,227),(130,220,253,277),(131,270,254,237),(132,230,255,287),(133,280,256,223),(134,240,257,273),(135,266,258,233),(136,226,259,283),(137,276,260,219),(138,236,261,269),(139,286,262,229),(140,222,263,279),(141,272,264,239),(142,232,241,265),(143,282,242,225),(144,218,243,275)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | Dic6 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 |
| kernel | C12.30Dic6 | C2×C32⋊4C8 | C4×C3⋊Dic3 | C6×C24 | C2×C3⋊Dic3 | C3⋊Dic3 | C2×C24 | C3×C12 | C3×C12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 4 | 1 | 1 | 4 | 2 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C12.30Dic6 ►in GL5(𝔽73)
| 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 22 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 66 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 62 | 0 | 0 |
| 0 | 60 | 71 | 0 | 0 |
| 0 | 0 | 0 | 61 | 22 |
| 0 | 0 | 0 | 10 | 12 |
G:=sub<GL(5,GF(73))| [46,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,1],[22,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,7,66,0,0,0,7,14],[1,0,0,0,0,0,2,60,0,0,0,62,71,0,0,0,0,0,61,10,0,0,0,22,12] >;
C12.30Dic6 in GAP, Magma, Sage, TeX
C_{12}._{30}{\rm Dic}_6 % in TeX
G:=Group("C12.30Dic6"); // GroupNames label
G:=SmallGroup(288,289);
// by ID
G=gap.SmallGroup(288,289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9*b^6,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations