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## G = Dic3×C27order 324 = 22·34

### Direct product of C27 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C27
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C54 — Dic3×C27
 Lower central C3 — Dic3×C27
 Upper central C1 — C54

Generators and relations for Dic3×C27
G = < a,b,c | a27=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C27
On 108 points
Generators in S108
(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 57 10 66 19 75)(2 58 11 67 20 76)(3 59 12 68 21 77)(4 60 13 69 22 78)(5 61 14 70 23 79)(6 62 15 71 24 80)(7 63 16 72 25 81)(8 64 17 73 26 55)(9 65 18 74 27 56)(28 107 46 98 37 89)(29 108 47 99 38 90)(30 82 48 100 39 91)(31 83 49 101 40 92)(32 84 50 102 41 93)(33 85 51 103 42 94)(34 86 52 104 43 95)(35 87 53 105 44 96)(36 88 54 106 45 97)
(1 54 66 97)(2 28 67 98)(3 29 68 99)(4 30 69 100)(5 31 70 101)(6 32 71 102)(7 33 72 103)(8 34 73 104)(9 35 74 105)(10 36 75 106)(11 37 76 107)(12 38 77 108)(13 39 78 82)(14 40 79 83)(15 41 80 84)(16 42 81 85)(17 43 55 86)(18 44 56 87)(19 45 57 88)(20 46 58 89)(21 47 59 90)(22 48 60 91)(23 49 61 92)(24 50 62 93)(25 51 63 94)(26 52 64 95)(27 53 65 96)

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

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

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

162 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 18A ··· 18F 18G ··· 18L 27A ··· 27R 27S ··· 27AJ 36A ··· 36L 54A ··· 54R 54S ··· 54AJ 108A ··· 108AJ order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 27 ··· 27 27 ··· 27 36 ··· 36 54 ··· 54 54 ··· 54 108 ··· 108 size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 1 ··· 1 2 ··· 2 3 3 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 C4 C6 C9 C12 C18 C27 C36 C54 C108 S3 Dic3 C3×S3 C3×Dic3 S3×C9 C9×Dic3 S3×C27 Dic3×C27 kernel Dic3×C27 C3×C54 C9×Dic3 C3×C27 C3×C18 C3×Dic3 C3×C9 C3×C6 Dic3 C32 C6 C3 C54 C27 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 18 12 18 36 1 1 2 2 6 6 18 18

Matrix representation of Dic3×C27 in GL3(𝔽109) generated by

 78 0 0 0 7 0 0 0 7
,
 108 0 0 0 45 0 0 105 63
,
 33 0 0 0 75 44 0 1 34
G:=sub<GL(3,GF(109))| [78,0,0,0,7,0,0,0,7],[108,0,0,0,45,105,0,0,63],[33,0,0,0,75,1,0,44,34] >;

Dic3×C27 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{27}
% in TeX

G:=Group("Dic3xC27");
// GroupNames label

G:=SmallGroup(324,11);
// by ID

G=gap.SmallGroup(324,11);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,93,7781]);
// Polycyclic

G:=Group<a,b,c|a^27=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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