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## G = Dic3×C3×C9order 324 = 22·34

### Direct product of C3×C9 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C3×C9
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C32×C18 — Dic3×C3×C9
 Lower central C3 — Dic3×C3×C9
 Upper central C1 — C3×C18

Generators and relations for Dic3×C3×C9
G = < a,b,c,d | a3=b9=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 140 in 90 conjugacy classes, 50 normal (20 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C9, C32, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, C33, C36, C3×Dic3, C3×Dic3, C3×C12, C3×C18, C3×C18, C3×C18, C32×C6, C32×C9, C9×Dic3, C3×C36, C32×Dic3, C32×C18, Dic3×C3×C9
Quotients: C1, C2, C3, C4, S3, C6, C9, C32, Dic3, C12, C18, C3×S3, C3×C6, C3×C9, C36, C3×Dic3, C3×C12, S3×C9, C3×C18, S3×C32, C9×Dic3, C3×C36, C32×Dic3, S3×C3×C9, Dic3×C3×C9

Smallest permutation representation of Dic3×C3×C9
On 108 points
Generators in S108
(1 53 62)(2 54 63)(3 46 55)(4 47 56)(5 48 57)(6 49 58)(7 50 59)(8 51 60)(9 52 61)(10 31 77)(11 32 78)(12 33 79)(13 34 80)(14 35 81)(15 36 73)(16 28 74)(17 29 75)(18 30 76)(19 95 102)(20 96 103)(21 97 104)(22 98 105)(23 99 106)(24 91 107)(25 92 108)(26 93 100)(27 94 101)(37 71 83)(38 72 84)(39 64 85)(40 65 86)(41 66 87)(42 67 88)(43 68 89)(44 69 90)(45 70 82)
(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)
(1 80 53 13 62 34)(2 81 54 14 63 35)(3 73 46 15 55 36)(4 74 47 16 56 28)(5 75 48 17 57 29)(6 76 49 18 58 30)(7 77 50 10 59 31)(8 78 51 11 60 32)(9 79 52 12 61 33)(19 71 102 37 95 83)(20 72 103 38 96 84)(21 64 104 39 97 85)(22 65 105 40 98 86)(23 66 106 41 99 87)(24 67 107 42 91 88)(25 68 108 43 92 89)(26 69 100 44 93 90)(27 70 101 45 94 82)
(1 67 13 91)(2 68 14 92)(3 69 15 93)(4 70 16 94)(5 71 17 95)(6 72 18 96)(7 64 10 97)(8 65 11 98)(9 66 12 99)(19 57 37 75)(20 58 38 76)(21 59 39 77)(22 60 40 78)(23 61 41 79)(24 62 42 80)(25 63 43 81)(26 55 44 73)(27 56 45 74)(28 101 47 82)(29 102 48 83)(30 103 49 84)(31 104 50 85)(32 105 51 86)(33 106 52 87)(34 107 53 88)(35 108 54 89)(36 100 46 90)

G:=sub<Sym(108)| (1,53,62)(2,54,63)(3,46,55)(4,47,56)(5,48,57)(6,49,58)(7,50,59)(8,51,60)(9,52,61)(10,31,77)(11,32,78)(12,33,79)(13,34,80)(14,35,81)(15,36,73)(16,28,74)(17,29,75)(18,30,76)(19,95,102)(20,96,103)(21,97,104)(22,98,105)(23,99,106)(24,91,107)(25,92,108)(26,93,100)(27,94,101)(37,71,83)(38,72,84)(39,64,85)(40,65,86)(41,66,87)(42,67,88)(43,68,89)(44,69,90)(45,70,82), (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), (1,80,53,13,62,34)(2,81,54,14,63,35)(3,73,46,15,55,36)(4,74,47,16,56,28)(5,75,48,17,57,29)(6,76,49,18,58,30)(7,77,50,10,59,31)(8,78,51,11,60,32)(9,79,52,12,61,33)(19,71,102,37,95,83)(20,72,103,38,96,84)(21,64,104,39,97,85)(22,65,105,40,98,86)(23,66,106,41,99,87)(24,67,107,42,91,88)(25,68,108,43,92,89)(26,69,100,44,93,90)(27,70,101,45,94,82), (1,67,13,91)(2,68,14,92)(3,69,15,93)(4,70,16,94)(5,71,17,95)(6,72,18,96)(7,64,10,97)(8,65,11,98)(9,66,12,99)(19,57,37,75)(20,58,38,76)(21,59,39,77)(22,60,40,78)(23,61,41,79)(24,62,42,80)(25,63,43,81)(26,55,44,73)(27,56,45,74)(28,101,47,82)(29,102,48,83)(30,103,49,84)(31,104,50,85)(32,105,51,86)(33,106,52,87)(34,107,53,88)(35,108,54,89)(36,100,46,90)>;

G:=Group( (1,53,62)(2,54,63)(3,46,55)(4,47,56)(5,48,57)(6,49,58)(7,50,59)(8,51,60)(9,52,61)(10,31,77)(11,32,78)(12,33,79)(13,34,80)(14,35,81)(15,36,73)(16,28,74)(17,29,75)(18,30,76)(19,95,102)(20,96,103)(21,97,104)(22,98,105)(23,99,106)(24,91,107)(25,92,108)(26,93,100)(27,94,101)(37,71,83)(38,72,84)(39,64,85)(40,65,86)(41,66,87)(42,67,88)(43,68,89)(44,69,90)(45,70,82), (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), (1,80,53,13,62,34)(2,81,54,14,63,35)(3,73,46,15,55,36)(4,74,47,16,56,28)(5,75,48,17,57,29)(6,76,49,18,58,30)(7,77,50,10,59,31)(8,78,51,11,60,32)(9,79,52,12,61,33)(19,71,102,37,95,83)(20,72,103,38,96,84)(21,64,104,39,97,85)(22,65,105,40,98,86)(23,66,106,41,99,87)(24,67,107,42,91,88)(25,68,108,43,92,89)(26,69,100,44,93,90)(27,70,101,45,94,82), (1,67,13,91)(2,68,14,92)(3,69,15,93)(4,70,16,94)(5,71,17,95)(6,72,18,96)(7,64,10,97)(8,65,11,98)(9,66,12,99)(19,57,37,75)(20,58,38,76)(21,59,39,77)(22,60,40,78)(23,61,41,79)(24,62,42,80)(25,63,43,81)(26,55,44,73)(27,56,45,74)(28,101,47,82)(29,102,48,83)(30,103,49,84)(31,104,50,85)(32,105,51,86)(33,106,52,87)(34,107,53,88)(35,108,54,89)(36,100,46,90) );

G=PermutationGroup([[(1,53,62),(2,54,63),(3,46,55),(4,47,56),(5,48,57),(6,49,58),(7,50,59),(8,51,60),(9,52,61),(10,31,77),(11,32,78),(12,33,79),(13,34,80),(14,35,81),(15,36,73),(16,28,74),(17,29,75),(18,30,76),(19,95,102),(20,96,103),(21,97,104),(22,98,105),(23,99,106),(24,91,107),(25,92,108),(26,93,100),(27,94,101),(37,71,83),(38,72,84),(39,64,85),(40,65,86),(41,66,87),(42,67,88),(43,68,89),(44,69,90),(45,70,82)], [(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)], [(1,80,53,13,62,34),(2,81,54,14,63,35),(3,73,46,15,55,36),(4,74,47,16,56,28),(5,75,48,17,57,29),(6,76,49,18,58,30),(7,77,50,10,59,31),(8,78,51,11,60,32),(9,79,52,12,61,33),(19,71,102,37,95,83),(20,72,103,38,96,84),(21,64,104,39,97,85),(22,65,105,40,98,86),(23,66,106,41,99,87),(24,67,107,42,91,88),(25,68,108,43,92,89),(26,69,100,44,93,90),(27,70,101,45,94,82)], [(1,67,13,91),(2,68,14,92),(3,69,15,93),(4,70,16,94),(5,71,17,95),(6,72,18,96),(7,64,10,97),(8,65,11,98),(9,66,12,99),(19,57,37,75),(20,58,38,76),(21,59,39,77),(22,60,40,78),(23,61,41,79),(24,62,42,80),(25,63,43,81),(26,55,44,73),(27,56,45,74),(28,101,47,82),(29,102,48,83),(30,103,49,84),(31,104,50,85),(32,105,51,86),(33,106,52,87),(34,107,53,88),(35,108,54,89),(36,100,46,90)]])

162 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 9A ··· 9R 9S ··· 9AJ 12A ··· 12P 18A ··· 18R 18S ··· 18AJ 36A ··· 36AJ order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 ··· 1 2 ··· 2 3 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C3 C4 C6 C6 C9 C12 C12 C18 C36 S3 Dic3 C3×S3 C3×S3 C3×Dic3 C3×Dic3 S3×C9 C9×Dic3 kernel Dic3×C3×C9 C32×C18 C9×Dic3 C32×Dic3 C32×C9 C3×C18 C32×C6 C3×Dic3 C3×C9 C33 C3×C6 C32 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 6 2 2 6 2 18 12 4 18 36 1 1 6 2 6 2 18 18

Matrix representation of Dic3×C3×C9 in GL3(𝔽37) generated by

 10 0 0 0 10 0 0 0 10
,
 16 0 0 0 1 0 0 0 1
,
 36 0 0 0 11 0 0 28 27
,
 6 0 0 0 16 25 0 6 21
G:=sub<GL(3,GF(37))| [10,0,0,0,10,0,0,0,10],[16,0,0,0,1,0,0,0,1],[36,0,0,0,11,28,0,0,27],[6,0,0,0,16,6,0,25,21] >;

Dic3×C3×C9 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\times C_9
% in TeX

G:=Group("Dic3xC3xC9");
// GroupNames label

G:=SmallGroup(324,91);
// by ID

G=gap.SmallGroup(324,91);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,122,7781]);
// Polycyclic

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

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