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## G = C12.D18order 432 = 24·33

### 16th non-split extension by C12 of D18 acting via D18/C9=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — C12.D18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C9×Dic6 — C12.D18
 Lower central C3×C9 — C3×C18 — C3×C36 — C12.D18
 Upper central C1 — C2 — C4

Generators and relations for C12.D18
G = < a,b,c | a12=1, b18=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a9b17 >

Subgroups: 272 in 62 conjugacy classes, 25 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, Dic6, Dic6, C3×Q8, C3×C9, Dic9, C36, C36, C3×Dic3, C3×C12, C3⋊Q16, C3×C18, C9⋊C8, Dic18, Q8×C9, C324C8, C3×Dic6, C3×Dic6, C3×Dic9, C9×Dic3, C3×C36, C9⋊Q16, C322Q16, C36.S3, C3×Dic18, C9×Dic6, C12.D18
Quotients: C1, C2, C22, S3, D4, D6, Q16, D9, C3⋊D4, D18, S32, C3⋊Q16, C9⋊D4, D6⋊S3, S3×D9, C9⋊Q16, C322Q16, D6⋊D9, C12.D18

Smallest permutation representation of C12.D18
On 144 points
Generators in S144
```(1 127 7 133 13 139 19 109 25 115 31 121)(2 122 32 116 26 110 20 140 14 134 8 128)(3 129 9 135 15 141 21 111 27 117 33 123)(4 124 34 118 28 112 22 142 16 136 10 130)(5 131 11 137 17 143 23 113 29 119 35 125)(6 126 36 120 30 114 24 144 18 138 12 132)(37 95 43 101 49 107 55 77 61 83 67 89)(38 90 68 84 62 78 56 108 50 102 44 96)(39 97 45 103 51 73 57 79 63 85 69 91)(40 92 70 86 64 80 58 74 52 104 46 98)(41 99 47 105 53 75 59 81 65 87 71 93)(42 94 72 88 66 82 60 76 54 106 48 100)
(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 91 115 63 19 73 133 45)(2 62 134 90 20 44 116 108)(3 89 117 61 21 107 135 43)(4 60 136 88 22 42 118 106)(5 87 119 59 23 105 137 41)(6 58 138 86 24 40 120 104)(7 85 121 57 25 103 139 39)(8 56 140 84 26 38 122 102)(9 83 123 55 27 101 141 37)(10 54 142 82 28 72 124 100)(11 81 125 53 29 99 143 71)(12 52 144 80 30 70 126 98)(13 79 127 51 31 97 109 69)(14 50 110 78 32 68 128 96)(15 77 129 49 33 95 111 67)(16 48 112 76 34 66 130 94)(17 75 131 47 35 93 113 65)(18 46 114 74 36 64 132 92)```

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

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

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

48 conjugacy classes

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

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 Q16 D9 C3⋊D4 C3⋊D4 D18 C9⋊D4 S32 C3⋊Q16 C3⋊Q16 D6⋊S3 S3×D9 C9⋊Q16 C32⋊2Q16 D6⋊D9 C12.D18 kernel C12.D18 C36.S3 C3×Dic18 C9×Dic6 Dic18 C3×Dic6 C3×C18 C36 C3×C12 C3×C9 Dic6 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 C12.D18 in GL6(𝔽73)

 28 71 0 0 0 0 64 45 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 4 17 0 0 0 0 72 69 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 20 71 0 0 0 0 2 18
,
 26 41 0 0 0 0 2 6 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 26 39 0 0 0 0 65 47

`G:=sub<GL(6,GF(73))| [28,64,0,0,0,0,71,45,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[4,72,0,0,0,0,17,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,20,2,0,0,0,0,71,18],[26,2,0,0,0,0,41,6,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,26,65,0,0,0,0,39,47] >;`

C12.D18 in GAP, Magma, Sage, TeX

`C_{12}.D_{18}`
`% in TeX`

`G:=Group("C12.D18");`
`// GroupNames label`

`G:=SmallGroup(432,74);`
`// by ID`

`G=gap.SmallGroup(432,74);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^18=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^9*b^17>;`
`// generators/relations`

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