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## G = D12.D9order 432 = 24·33

### 2nd non-split extension by D12 of D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — D12.D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C9×D12 — D12.D9
 Lower central C3×C9 — C3×C18 — C3×C36 — D12.D9
 Upper central C1 — C2 — C4

Generators and relations for D12.D9
G = < a,b,c,d | a12=b2=c9=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 320 in 68 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, C18, C18, C3×S3, C3×C6, C3⋊C8, Dic6, D12, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, D4.S3, Q82S3, S3×C9, C3×C18, C9⋊C8, Dic18, D4×C9, C324C8, C3×Dic6, C3×D12, C3×Dic9, C3×C36, S3×C18, D4.D9, Dic6⋊S3, C36.S3, C3×Dic18, C9×D12, D12.D9
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊D4, D18, S32, D4.S3, Q82S3, C9⋊D4, D6⋊S3, S3×D9, D4.D9, Dic6⋊S3, D6⋊D9, D12.D9

Smallest permutation representation of D12.D9
On 144 points
Generators in S144
```(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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 132)(9 131)(10 130)(11 129)(12 128)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 60)(23 59)(24 58)(25 110)(26 109)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 75)(38 74)(39 73)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 108)(70 107)(71 106)(72 105)(85 141)(86 140)(87 139)(88 138)(89 137)(90 136)(91 135)(92 134)(93 133)(94 144)(95 143)(96 142)
(1 43 99 9 39 107 5 47 103)(2 44 100 10 40 108 6 48 104)(3 45 101 11 41 97 7 37 105)(4 46 102 12 42 98 8 38 106)(13 110 137 21 118 133 17 114 141)(14 111 138 22 119 134 18 115 142)(15 112 139 23 120 135 19 116 143)(16 113 140 24 109 136 20 117 144)(25 89 49 29 93 53 33 85 57)(26 90 50 30 94 54 34 86 58)(27 91 51 31 95 55 35 87 59)(28 92 52 32 96 56 36 88 60)(61 126 80 65 130 84 69 122 76)(62 127 81 66 131 73 70 123 77)(63 128 82 67 132 74 71 124 78)(64 129 83 68 121 75 72 125 79)
(1 55 7 49)(2 50 8 56)(3 57 9 51)(4 52 10 58)(5 59 11 53)(6 54 12 60)(13 128 19 122)(14 123 20 129)(15 130 21 124)(16 125 22 131)(17 132 23 126)(18 127 24 121)(25 99 31 105)(26 106 32 100)(27 101 33 107)(28 108 34 102)(29 103 35 97)(30 98 36 104)(37 89 43 95)(38 96 44 90)(39 91 45 85)(40 86 46 92)(41 93 47 87)(42 88 48 94)(61 114 67 120)(62 109 68 115)(63 116 69 110)(64 111 70 117)(65 118 71 112)(66 113 72 119)(73 144 79 138)(74 139 80 133)(75 134 81 140)(76 141 82 135)(77 136 83 142)(78 143 84 137)```

`G:=sub<Sym(144)| (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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,132)(9,131)(10,130)(11,129)(12,128)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,110)(26,109)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,75)(38,74)(39,73)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,108)(70,107)(71,106)(72,105)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,135)(92,134)(93,133)(94,144)(95,143)(96,142), (1,43,99,9,39,107,5,47,103)(2,44,100,10,40,108,6,48,104)(3,45,101,11,41,97,7,37,105)(4,46,102,12,42,98,8,38,106)(13,110,137,21,118,133,17,114,141)(14,111,138,22,119,134,18,115,142)(15,112,139,23,120,135,19,116,143)(16,113,140,24,109,136,20,117,144)(25,89,49,29,93,53,33,85,57)(26,90,50,30,94,54,34,86,58)(27,91,51,31,95,55,35,87,59)(28,92,52,32,96,56,36,88,60)(61,126,80,65,130,84,69,122,76)(62,127,81,66,131,73,70,123,77)(63,128,82,67,132,74,71,124,78)(64,129,83,68,121,75,72,125,79), (1,55,7,49)(2,50,8,56)(3,57,9,51)(4,52,10,58)(5,59,11,53)(6,54,12,60)(13,128,19,122)(14,123,20,129)(15,130,21,124)(16,125,22,131)(17,132,23,126)(18,127,24,121)(25,99,31,105)(26,106,32,100)(27,101,33,107)(28,108,34,102)(29,103,35,97)(30,98,36,104)(37,89,43,95)(38,96,44,90)(39,91,45,85)(40,86,46,92)(41,93,47,87)(42,88,48,94)(61,114,67,120)(62,109,68,115)(63,116,69,110)(64,111,70,117)(65,118,71,112)(66,113,72,119)(73,144,79,138)(74,139,80,133)(75,134,81,140)(76,141,82,135)(77,136,83,142)(78,143,84,137)>;`

`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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,132)(9,131)(10,130)(11,129)(12,128)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,110)(26,109)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,75)(38,74)(39,73)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,108)(70,107)(71,106)(72,105)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,135)(92,134)(93,133)(94,144)(95,143)(96,142), (1,43,99,9,39,107,5,47,103)(2,44,100,10,40,108,6,48,104)(3,45,101,11,41,97,7,37,105)(4,46,102,12,42,98,8,38,106)(13,110,137,21,118,133,17,114,141)(14,111,138,22,119,134,18,115,142)(15,112,139,23,120,135,19,116,143)(16,113,140,24,109,136,20,117,144)(25,89,49,29,93,53,33,85,57)(26,90,50,30,94,54,34,86,58)(27,91,51,31,95,55,35,87,59)(28,92,52,32,96,56,36,88,60)(61,126,80,65,130,84,69,122,76)(62,127,81,66,131,73,70,123,77)(63,128,82,67,132,74,71,124,78)(64,129,83,68,121,75,72,125,79), (1,55,7,49)(2,50,8,56)(3,57,9,51)(4,52,10,58)(5,59,11,53)(6,54,12,60)(13,128,19,122)(14,123,20,129)(15,130,21,124)(16,125,22,131)(17,132,23,126)(18,127,24,121)(25,99,31,105)(26,106,32,100)(27,101,33,107)(28,108,34,102)(29,103,35,97)(30,98,36,104)(37,89,43,95)(38,96,44,90)(39,91,45,85)(40,86,46,92)(41,93,47,87)(42,88,48,94)(61,114,67,120)(62,109,68,115)(63,116,69,110)(64,111,70,117)(65,118,71,112)(66,113,72,119)(73,144,79,138)(74,139,80,133)(75,134,81,140)(76,141,82,135)(77,136,83,142)(78,143,84,137) );`

`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)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,132),(9,131),(10,130),(11,129),(12,128),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,60),(23,59),(24,58),(25,110),(26,109),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,111),(37,75),(38,74),(39,73),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,108),(70,107),(71,106),(72,105),(85,141),(86,140),(87,139),(88,138),(89,137),(90,136),(91,135),(92,134),(93,133),(94,144),(95,143),(96,142)], [(1,43,99,9,39,107,5,47,103),(2,44,100,10,40,108,6,48,104),(3,45,101,11,41,97,7,37,105),(4,46,102,12,42,98,8,38,106),(13,110,137,21,118,133,17,114,141),(14,111,138,22,119,134,18,115,142),(15,112,139,23,120,135,19,116,143),(16,113,140,24,109,136,20,117,144),(25,89,49,29,93,53,33,85,57),(26,90,50,30,94,54,34,86,58),(27,91,51,31,95,55,35,87,59),(28,92,52,32,96,56,36,88,60),(61,126,80,65,130,84,69,122,76),(62,127,81,66,131,73,70,123,77),(63,128,82,67,132,74,71,124,78),(64,129,83,68,121,75,72,125,79)], [(1,55,7,49),(2,50,8,56),(3,57,9,51),(4,52,10,58),(5,59,11,53),(6,54,12,60),(13,128,19,122),(14,123,20,129),(15,130,21,124),(16,125,22,131),(17,132,23,126),(18,127,24,121),(25,99,31,105),(26,106,32,100),(27,101,33,107),(28,108,34,102),(29,103,35,97),(30,98,36,104),(37,89,43,95),(38,96,44,90),(39,91,45,85),(40,86,46,92),(41,93,47,87),(42,88,48,94),(61,114,67,120),(62,109,68,115),(63,116,69,110),(64,111,70,117),(65,118,71,112),(66,113,72,119),(73,144,79,138),(74,139,80,133),(75,134,81,140),(76,141,82,135),(77,136,83,142),(78,143,84,137)]])`

48 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36I order 1 2 2 3 3 3 4 4 6 6 6 6 6 8 8 9 9 9 9 9 9 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 size 1 1 12 2 2 4 2 36 2 2 4 12 12 54 54 2 2 2 4 4 4 4 4 4 4 36 36 2 2 2 4 4 4 12 ··· 12 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - - + - - image C1 C2 C2 C2 S3 S3 D4 D6 D6 SD16 D9 C3⋊D4 C3⋊D4 D18 C9⋊D4 S32 Q8⋊2S3 D4.S3 D6⋊S3 S3×D9 D4.D9 Dic6⋊S3 D6⋊D9 D12.D9 kernel D12.D9 C36.S3 C3×Dic18 C9×D12 Dic18 C3×D12 C3×C18 C36 C3×C12 C3×C9 D12 C18 C3×C6 C12 C6 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 3 2 2 3 6 1 1 1 1 3 3 2 3 6

Matrix representation of D12.D9 in GL6(𝔽73)

 69 3 0 0 0 0 43 4 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 20 1 0 0 0 0 39 53 0 0 0 0 0 0 28 3 0 0 0 0 31 45 0 0 0 0 0 0 43 13 0 0 0 0 60 30
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 70 28 0 0 0 0 45 42
,
 50 60 0 0 0 0 52 23 0 0 0 0 0 0 30 13 0 0 0 0 60 43 0 0 0 0 0 0 44 25 0 0 0 0 54 29

`G:=sub<GL(6,GF(73))| [69,43,0,0,0,0,3,4,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[20,39,0,0,0,0,1,53,0,0,0,0,0,0,28,31,0,0,0,0,3,45,0,0,0,0,0,0,43,60,0,0,0,0,13,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,70,45,0,0,0,0,28,42],[50,52,0,0,0,0,60,23,0,0,0,0,0,0,30,60,0,0,0,0,13,43,0,0,0,0,0,0,44,54,0,0,0,0,25,29] >;`

D12.D9 in GAP, Magma, Sage, TeX

`D_{12}.D_9`
`% in TeX`

`G:=Group("D12.D9");`
`// GroupNames label`

`G:=SmallGroup(432,70);`
`// by ID`

`G=gap.SmallGroup(432,70);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^9=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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