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## G = C12.D9order 216 = 23·33

### 3rd non-split extension by C12 of D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C12.D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9⋊Dic3 — C12.D9
 Lower central C3×C9 — C3×C18 — C12.D9
 Upper central C1 — C2 — C4

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 [×3], C4, C4 [×2], C6, C6 [×3], Q8, C9 [×3], C32, Dic3 [×8], C12, C12 [×3], C18 [×3], C3×C6, Dic6 [×4], C3×C9, Dic9 [×6], C36 [×3], C3⋊Dic3 [×2], C3×C12, C3×C18, Dic18 [×3], C324Q8, C9⋊Dic3 [×2], C3×C36, C12.D9
Quotients: C1, C2 [×3], C22, S3 [×4], Q8, D6 [×4], D9 [×3], C3⋊S3, Dic6 [×4], D18 [×3], C2×C3⋊S3, C9⋊S3, Dic18 [×3], C324Q8, C2×C9⋊S3, C12.D9

Smallest permutation representation of 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 90 173 29 139 56 213 45 121)(2 91 174 30 140 57 214 46 122)(3 92 175 31 141 58 215 47 123)(4 93 176 32 142 59 216 48 124)(5 94 177 33 143 60 205 37 125)(6 95 178 34 144 49 206 38 126)(7 96 179 35 133 50 207 39 127)(8 85 180 36 134 51 208 40 128)(9 86 169 25 135 52 209 41 129)(10 87 170 26 136 53 210 42 130)(11 88 171 27 137 54 211 43 131)(12 89 172 28 138 55 212 44 132)(13 191 162 109 150 82 71 200 104)(14 192 163 110 151 83 72 201 105)(15 181 164 111 152 84 61 202 106)(16 182 165 112 153 73 62 203 107)(17 183 166 113 154 74 63 204 108)(18 184 167 114 155 75 64 193 97)(19 185 168 115 156 76 65 194 98)(20 186 157 116 145 77 66 195 99)(21 187 158 117 146 78 67 196 100)(22 188 159 118 147 79 68 197 101)(23 189 160 119 148 80 69 198 102)(24 190 161 120 149 81 70 199 103)
(1 83 7 77)(2 82 8 76)(3 81 9 75)(4 80 10 74)(5 79 11 73)(6 78 12 84)(13 51 19 57)(14 50 20 56)(15 49 21 55)(16 60 22 54)(17 59 23 53)(18 58 24 52)(25 167 31 161)(26 166 32 160)(27 165 33 159)(28 164 34 158)(29 163 35 157)(30 162 36 168)(37 197 43 203)(38 196 44 202)(39 195 45 201)(40 194 46 200)(41 193 47 199)(42 204 48 198)(61 126 67 132)(62 125 68 131)(63 124 69 130)(64 123 70 129)(65 122 71 128)(66 121 72 127)(85 156 91 150)(86 155 92 149)(87 154 93 148)(88 153 94 147)(89 152 95 146)(90 151 96 145)(97 215 103 209)(98 214 104 208)(99 213 105 207)(100 212 106 206)(101 211 107 205)(102 210 108 216)(109 180 115 174)(110 179 116 173)(111 178 117 172)(112 177 118 171)(113 176 119 170)(114 175 120 169)(133 186 139 192)(134 185 140 191)(135 184 141 190)(136 183 142 189)(137 182 143 188)(138 181 144 187)```

`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), (1,90,173,29,139,56,213,45,121)(2,91,174,30,140,57,214,46,122)(3,92,175,31,141,58,215,47,123)(4,93,176,32,142,59,216,48,124)(5,94,177,33,143,60,205,37,125)(6,95,178,34,144,49,206,38,126)(7,96,179,35,133,50,207,39,127)(8,85,180,36,134,51,208,40,128)(9,86,169,25,135,52,209,41,129)(10,87,170,26,136,53,210,42,130)(11,88,171,27,137,54,211,43,131)(12,89,172,28,138,55,212,44,132)(13,191,162,109,150,82,71,200,104)(14,192,163,110,151,83,72,201,105)(15,181,164,111,152,84,61,202,106)(16,182,165,112,153,73,62,203,107)(17,183,166,113,154,74,63,204,108)(18,184,167,114,155,75,64,193,97)(19,185,168,115,156,76,65,194,98)(20,186,157,116,145,77,66,195,99)(21,187,158,117,146,78,67,196,100)(22,188,159,118,147,79,68,197,101)(23,189,160,119,148,80,69,198,102)(24,190,161,120,149,81,70,199,103), (1,83,7,77)(2,82,8,76)(3,81,9,75)(4,80,10,74)(5,79,11,73)(6,78,12,84)(13,51,19,57)(14,50,20,56)(15,49,21,55)(16,60,22,54)(17,59,23,53)(18,58,24,52)(25,167,31,161)(26,166,32,160)(27,165,33,159)(28,164,34,158)(29,163,35,157)(30,162,36,168)(37,197,43,203)(38,196,44,202)(39,195,45,201)(40,194,46,200)(41,193,47,199)(42,204,48,198)(61,126,67,132)(62,125,68,131)(63,124,69,130)(64,123,70,129)(65,122,71,128)(66,121,72,127)(85,156,91,150)(86,155,92,149)(87,154,93,148)(88,153,94,147)(89,152,95,146)(90,151,96,145)(97,215,103,209)(98,214,104,208)(99,213,105,207)(100,212,106,206)(101,211,107,205)(102,210,108,216)(109,180,115,174)(110,179,116,173)(111,178,117,172)(112,177,118,171)(113,176,119,170)(114,175,120,169)(133,186,139,192)(134,185,140,191)(135,184,141,190)(136,183,142,189)(137,182,143,188)(138,181,144,187)>;`

`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), (1,90,173,29,139,56,213,45,121)(2,91,174,30,140,57,214,46,122)(3,92,175,31,141,58,215,47,123)(4,93,176,32,142,59,216,48,124)(5,94,177,33,143,60,205,37,125)(6,95,178,34,144,49,206,38,126)(7,96,179,35,133,50,207,39,127)(8,85,180,36,134,51,208,40,128)(9,86,169,25,135,52,209,41,129)(10,87,170,26,136,53,210,42,130)(11,88,171,27,137,54,211,43,131)(12,89,172,28,138,55,212,44,132)(13,191,162,109,150,82,71,200,104)(14,192,163,110,151,83,72,201,105)(15,181,164,111,152,84,61,202,106)(16,182,165,112,153,73,62,203,107)(17,183,166,113,154,74,63,204,108)(18,184,167,114,155,75,64,193,97)(19,185,168,115,156,76,65,194,98)(20,186,157,116,145,77,66,195,99)(21,187,158,117,146,78,67,196,100)(22,188,159,118,147,79,68,197,101)(23,189,160,119,148,80,69,198,102)(24,190,161,120,149,81,70,199,103), (1,83,7,77)(2,82,8,76)(3,81,9,75)(4,80,10,74)(5,79,11,73)(6,78,12,84)(13,51,19,57)(14,50,20,56)(15,49,21,55)(16,60,22,54)(17,59,23,53)(18,58,24,52)(25,167,31,161)(26,166,32,160)(27,165,33,159)(28,164,34,158)(29,163,35,157)(30,162,36,168)(37,197,43,203)(38,196,44,202)(39,195,45,201)(40,194,46,200)(41,193,47,199)(42,204,48,198)(61,126,67,132)(62,125,68,131)(63,124,69,130)(64,123,70,129)(65,122,71,128)(66,121,72,127)(85,156,91,150)(86,155,92,149)(87,154,93,148)(88,153,94,147)(89,152,95,146)(90,151,96,145)(97,215,103,209)(98,214,104,208)(99,213,105,207)(100,212,106,206)(101,211,107,205)(102,210,108,216)(109,180,115,174)(110,179,116,173)(111,178,117,172)(112,177,118,171)(113,176,119,170)(114,175,120,169)(133,186,139,192)(134,185,140,191)(135,184,141,190)(136,183,142,189)(137,182,143,188)(138,181,144,187) );`

`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)], [(1,90,173,29,139,56,213,45,121),(2,91,174,30,140,57,214,46,122),(3,92,175,31,141,58,215,47,123),(4,93,176,32,142,59,216,48,124),(5,94,177,33,143,60,205,37,125),(6,95,178,34,144,49,206,38,126),(7,96,179,35,133,50,207,39,127),(8,85,180,36,134,51,208,40,128),(9,86,169,25,135,52,209,41,129),(10,87,170,26,136,53,210,42,130),(11,88,171,27,137,54,211,43,131),(12,89,172,28,138,55,212,44,132),(13,191,162,109,150,82,71,200,104),(14,192,163,110,151,83,72,201,105),(15,181,164,111,152,84,61,202,106),(16,182,165,112,153,73,62,203,107),(17,183,166,113,154,74,63,204,108),(18,184,167,114,155,75,64,193,97),(19,185,168,115,156,76,65,194,98),(20,186,157,116,145,77,66,195,99),(21,187,158,117,146,78,67,196,100),(22,188,159,118,147,79,68,197,101),(23,189,160,119,148,80,69,198,102),(24,190,161,120,149,81,70,199,103)], [(1,83,7,77),(2,82,8,76),(3,81,9,75),(4,80,10,74),(5,79,11,73),(6,78,12,84),(13,51,19,57),(14,50,20,56),(15,49,21,55),(16,60,22,54),(17,59,23,53),(18,58,24,52),(25,167,31,161),(26,166,32,160),(27,165,33,159),(28,164,34,158),(29,163,35,157),(30,162,36,168),(37,197,43,203),(38,196,44,202),(39,195,45,201),(40,194,46,200),(41,193,47,199),(42,204,48,198),(61,126,67,132),(62,125,68,131),(63,124,69,130),(64,123,70,129),(65,122,71,128),(66,121,72,127),(85,156,91,150),(86,155,92,149),(87,154,93,148),(88,153,94,147),(89,152,95,146),(90,151,96,145),(97,215,103,209),(98,214,104,208),(99,213,105,207),(100,212,106,206),(101,211,107,205),(102,210,108,216),(109,180,115,174),(110,179,116,173),(111,178,117,172),(112,177,118,171),(113,176,119,170),(114,175,120,169),(133,186,139,192),(134,185,140,191),(135,184,141,190),(136,183,142,189),(137,182,143,188),(138,181,144,187)])`

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  D125D9  D365S3  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`

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