metabelian, supersoluble, monomial
Aliases: C36.3S3, C9⋊2Dic6, C12.3D9, C3⋊2Dic18, C6.14D18, C18.14D6, C32.4Dic6, (C3×C9)⋊4Q8, C4.(C9⋊S3), (C3×C36).3C2, (C3×C12).9S3, (C3×C6).50D6, C12.1(C3⋊S3), C9⋊Dic3.2C2, C3.(C32⋊4Q8), (C3×C18).18C22, C6.8(C2×C3⋊S3), C2.3(C2×C9⋊S3), SmallGroup(216,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.D9
G = < a,b,c | a12=b9=1, c2=a6, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 258 in 60 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, Q8, C9, C32, Dic3, C12, C12, C18, C3×C6, Dic6, C3×C9, Dic9, C36, C3⋊Dic3, C3×C12, C3×C18, Dic18, C32⋊4Q8, C9⋊Dic3, C3×C36, C12.D9
Quotients: C1, C2, C22, S3, Q8, D6, D9, C3⋊S3, Dic6, D18, C2×C3⋊S3, C9⋊S3, Dic18, C32⋊4Q8, C2×C9⋊S3, C12.D9
►Regular action on 216 points
Generators in S216
(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)
(1 137 167 28 216 176 111 150 97)(2 138 168 29 205 177 112 151 98)(3 139 157 30 206 178 113 152 99)(4 140 158 31 207 179 114 153 100)(5 141 159 32 208 180 115 154 101)(6 142 160 33 209 169 116 155 102)(7 143 161 34 210 170 117 156 103)(8 144 162 35 211 171 118 145 104)(9 133 163 36 212 172 119 146 105)(10 134 164 25 213 173 120 147 106)(11 135 165 26 214 174 109 148 107)(12 136 166 27 215 175 110 149 108)(13 63 73 48 185 93 56 196 124)(14 64 74 37 186 94 57 197 125)(15 65 75 38 187 95 58 198 126)(16 66 76 39 188 96 59 199 127)(17 67 77 40 189 85 60 200 128)(18 68 78 41 190 86 49 201 129)(19 69 79 42 191 87 50 202 130)(20 70 80 43 192 88 51 203 131)(21 71 81 44 181 89 52 204 132)(22 72 82 45 182 90 53 193 121)(23 61 83 46 183 91 54 194 122)(24 62 84 47 184 92 55 195 123)
(1 20 7 14)(2 19 8 13)(3 18 9 24)(4 17 10 23)(5 16 11 22)(6 15 12 21)(25 54 31 60)(26 53 32 59)(27 52 33 58)(28 51 34 57)(29 50 35 56)(30 49 36 55)(37 111 43 117)(38 110 44 116)(39 109 45 115)(40 120 46 114)(41 119 47 113)(42 118 48 112)(61 100 67 106)(62 99 68 105)(63 98 69 104)(64 97 70 103)(65 108 71 102)(66 107 72 101)(73 151 79 145)(74 150 80 156)(75 149 81 155)(76 148 82 154)(77 147 83 153)(78 146 84 152)(85 213 91 207)(86 212 92 206)(87 211 93 205)(88 210 94 216)(89 209 95 215)(90 208 96 214)(121 141 127 135)(122 140 128 134)(123 139 129 133)(124 138 130 144)(125 137 131 143)(126 136 132 142)(157 201 163 195)(158 200 164 194)(159 199 165 193)(160 198 166 204)(161 197 167 203)(162 196 168 202)(169 187 175 181)(170 186 176 192)(171 185 177 191)(172 184 178 190)(173 183 179 189)(174 182 180 188)
G:=sub<Sym(216)| (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), (1,137,167,28,216,176,111,150,97)(2,138,168,29,205,177,112,151,98)(3,139,157,30,206,178,113,152,99)(4,140,158,31,207,179,114,153,100)(5,141,159,32,208,180,115,154,101)(6,142,160,33,209,169,116,155,102)(7,143,161,34,210,170,117,156,103)(8,144,162,35,211,171,118,145,104)(9,133,163,36,212,172,119,146,105)(10,134,164,25,213,173,120,147,106)(11,135,165,26,214,174,109,148,107)(12,136,166,27,215,175,110,149,108)(13,63,73,48,185,93,56,196,124)(14,64,74,37,186,94,57,197,125)(15,65,75,38,187,95,58,198,126)(16,66,76,39,188,96,59,199,127)(17,67,77,40,189,85,60,200,128)(18,68,78,41,190,86,49,201,129)(19,69,79,42,191,87,50,202,130)(20,70,80,43,192,88,51,203,131)(21,71,81,44,181,89,52,204,132)(22,72,82,45,182,90,53,193,121)(23,61,83,46,183,91,54,194,122)(24,62,84,47,184,92,55,195,123), (1,20,7,14)(2,19,8,13)(3,18,9,24)(4,17,10,23)(5,16,11,22)(6,15,12,21)(25,54,31,60)(26,53,32,59)(27,52,33,58)(28,51,34,57)(29,50,35,56)(30,49,36,55)(37,111,43,117)(38,110,44,116)(39,109,45,115)(40,120,46,114)(41,119,47,113)(42,118,48,112)(61,100,67,106)(62,99,68,105)(63,98,69,104)(64,97,70,103)(65,108,71,102)(66,107,72,101)(73,151,79,145)(74,150,80,156)(75,149,81,155)(76,148,82,154)(77,147,83,153)(78,146,84,152)(85,213,91,207)(86,212,92,206)(87,211,93,205)(88,210,94,216)(89,209,95,215)(90,208,96,214)(121,141,127,135)(122,140,128,134)(123,139,129,133)(124,138,130,144)(125,137,131,143)(126,136,132,142)(157,201,163,195)(158,200,164,194)(159,199,165,193)(160,198,166,204)(161,197,167,203)(162,196,168,202)(169,187,175,181)(170,186,176,192)(171,185,177,191)(172,184,178,190)(173,183,179,189)(174,182,180,188)>;
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), (1,137,167,28,216,176,111,150,97)(2,138,168,29,205,177,112,151,98)(3,139,157,30,206,178,113,152,99)(4,140,158,31,207,179,114,153,100)(5,141,159,32,208,180,115,154,101)(6,142,160,33,209,169,116,155,102)(7,143,161,34,210,170,117,156,103)(8,144,162,35,211,171,118,145,104)(9,133,163,36,212,172,119,146,105)(10,134,164,25,213,173,120,147,106)(11,135,165,26,214,174,109,148,107)(12,136,166,27,215,175,110,149,108)(13,63,73,48,185,93,56,196,124)(14,64,74,37,186,94,57,197,125)(15,65,75,38,187,95,58,198,126)(16,66,76,39,188,96,59,199,127)(17,67,77,40,189,85,60,200,128)(18,68,78,41,190,86,49,201,129)(19,69,79,42,191,87,50,202,130)(20,70,80,43,192,88,51,203,131)(21,71,81,44,181,89,52,204,132)(22,72,82,45,182,90,53,193,121)(23,61,83,46,183,91,54,194,122)(24,62,84,47,184,92,55,195,123), (1,20,7,14)(2,19,8,13)(3,18,9,24)(4,17,10,23)(5,16,11,22)(6,15,12,21)(25,54,31,60)(26,53,32,59)(27,52,33,58)(28,51,34,57)(29,50,35,56)(30,49,36,55)(37,111,43,117)(38,110,44,116)(39,109,45,115)(40,120,46,114)(41,119,47,113)(42,118,48,112)(61,100,67,106)(62,99,68,105)(63,98,69,104)(64,97,70,103)(65,108,71,102)(66,107,72,101)(73,151,79,145)(74,150,80,156)(75,149,81,155)(76,148,82,154)(77,147,83,153)(78,146,84,152)(85,213,91,207)(86,212,92,206)(87,211,93,205)(88,210,94,216)(89,209,95,215)(90,208,96,214)(121,141,127,135)(122,140,128,134)(123,139,129,133)(124,138,130,144)(125,137,131,143)(126,136,132,142)(157,201,163,195)(158,200,164,194)(159,199,165,193)(160,198,166,204)(161,197,167,203)(162,196,168,202)(169,187,175,181)(170,186,176,192)(171,185,177,191)(172,184,178,190)(173,183,179,189)(174,182,180,188) );
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)], [(1,137,167,28,216,176,111,150,97),(2,138,168,29,205,177,112,151,98),(3,139,157,30,206,178,113,152,99),(4,140,158,31,207,179,114,153,100),(5,141,159,32,208,180,115,154,101),(6,142,160,33,209,169,116,155,102),(7,143,161,34,210,170,117,156,103),(8,144,162,35,211,171,118,145,104),(9,133,163,36,212,172,119,146,105),(10,134,164,25,213,173,120,147,106),(11,135,165,26,214,174,109,148,107),(12,136,166,27,215,175,110,149,108),(13,63,73,48,185,93,56,196,124),(14,64,74,37,186,94,57,197,125),(15,65,75,38,187,95,58,198,126),(16,66,76,39,188,96,59,199,127),(17,67,77,40,189,85,60,200,128),(18,68,78,41,190,86,49,201,129),(19,69,79,42,191,87,50,202,130),(20,70,80,43,192,88,51,203,131),(21,71,81,44,181,89,52,204,132),(22,72,82,45,182,90,53,193,121),(23,61,83,46,183,91,54,194,122),(24,62,84,47,184,92,55,195,123)], [(1,20,7,14),(2,19,8,13),(3,18,9,24),(4,17,10,23),(5,16,11,22),(6,15,12,21),(25,54,31,60),(26,53,32,59),(27,52,33,58),(28,51,34,57),(29,50,35,56),(30,49,36,55),(37,111,43,117),(38,110,44,116),(39,109,45,115),(40,120,46,114),(41,119,47,113),(42,118,48,112),(61,100,67,106),(62,99,68,105),(63,98,69,104),(64,97,70,103),(65,108,71,102),(66,107,72,101),(73,151,79,145),(74,150,80,156),(75,149,81,155),(76,148,82,154),(77,147,83,153),(78,146,84,152),(85,213,91,207),(86,212,92,206),(87,211,93,205),(88,210,94,216),(89,209,95,215),(90,208,96,214),(121,141,127,135),(122,140,128,134),(123,139,129,133),(124,138,130,144),(125,137,131,143),(126,136,132,142),(157,201,163,195),(158,200,164,194),(159,199,165,193),(160,198,166,204),(161,197,167,203),(162,196,168,202),(169,187,175,181),(170,186,176,192),(171,185,177,191),(172,184,178,190),(173,183,179,189),(174,182,180,188)]])
C12.D9 is a maximal subgroup of
D36.S3 C3⋊Dic36 C36.D6 C9⋊Dic12 C24.D9 C24⋊D9 C36.17D6 C36.19D6 D9×Dic6 S3×Dic18 D12⋊5D9 D36⋊5S3 C36.70D6 C36.27D6 Q8×C9⋊S3
C12.D9 is a maximal quotient of
C6.Dic18 C36⋊Dic3
57 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D9 | Dic6 | Dic6 | D18 | Dic18 |
| kernel | C12.D9 | C9⋊Dic3 | C3×C36 | C36 | C3×C12 | C3×C9 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 3 | 1 | 1 | 3 | 1 | 9 | 6 | 2 | 9 | 18 |
Matrix representation of C12.D9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 32 |
| 0 | 0 | 5 | 32 |
| 31 | 20 | 0 | 0 |
| 17 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 36 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 9 | 8 |
| 0 | 0 | 36 | 28 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,27,5,0,0,32,32],[31,17,0,0,20,11,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,36,0,0,0,0,0,9,36,0,0,8,28] >;
C12.D9 in GAP, Magma, Sage, TeX
C_{12}.D_9 % in TeX
G:=Group("C12.D9"); // GroupNames label
G:=SmallGroup(216,63);
// by ID
G=gap.SmallGroup(216,63);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,2115,453,1444,5189]);
// Polycyclic
G:=Group<a,b,c|a^12=b^9=1,c^2=a^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations