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## G = Dic3×C18order 216 = 23·33

### Direct product of C18 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C18
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C9×Dic3 — Dic3×C18
 Lower central C3 — Dic3×C18
 Upper central C1 — C2×C18

Generators and relations for Dic3×C18
G = < a,b,c | a18=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 86 in 58 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, C62, C3×C18, C3×C18, C2×C36, C6×Dic3, C9×Dic3, C6×C18, Dic3×C18
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C9, Dic3, C12, D6, C2×C6, C18, C3×S3, C2×Dic3, C2×C12, C36, C2×C18, C3×Dic3, S3×C6, S3×C9, C2×C36, C6×Dic3, C9×Dic3, S3×C18, Dic3×C18

Smallest permutation representation of Dic3×C18
On 72 points
Generators in S72
(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)
(1 49 7 37 13 43)(2 50 8 38 14 44)(3 51 9 39 15 45)(4 52 10 40 16 46)(5 53 11 41 17 47)(6 54 12 42 18 48)(19 71 31 65 25 59)(20 72 32 66 26 60)(21 55 33 67 27 61)(22 56 34 68 28 62)(23 57 35 69 29 63)(24 58 36 70 30 64)
(1 22 37 68)(2 23 38 69)(3 24 39 70)(4 25 40 71)(5 26 41 72)(6 27 42 55)(7 28 43 56)(8 29 44 57)(9 30 45 58)(10 31 46 59)(11 32 47 60)(12 33 48 61)(13 34 49 62)(14 35 50 63)(15 36 51 64)(16 19 52 65)(17 20 53 66)(18 21 54 67)

G:=sub<Sym(72)| (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), (1,49,7,37,13,43)(2,50,8,38,14,44)(3,51,9,39,15,45)(4,52,10,40,16,46)(5,53,11,41,17,47)(6,54,12,42,18,48)(19,71,31,65,25,59)(20,72,32,66,26,60)(21,55,33,67,27,61)(22,56,34,68,28,62)(23,57,35,69,29,63)(24,58,36,70,30,64), (1,22,37,68)(2,23,38,69)(3,24,39,70)(4,25,40,71)(5,26,41,72)(6,27,42,55)(7,28,43,56)(8,29,44,57)(9,30,45,58)(10,31,46,59)(11,32,47,60)(12,33,48,61)(13,34,49,62)(14,35,50,63)(15,36,51,64)(16,19,52,65)(17,20,53,66)(18,21,54,67)>;

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), (1,49,7,37,13,43)(2,50,8,38,14,44)(3,51,9,39,15,45)(4,52,10,40,16,46)(5,53,11,41,17,47)(6,54,12,42,18,48)(19,71,31,65,25,59)(20,72,32,66,26,60)(21,55,33,67,27,61)(22,56,34,68,28,62)(23,57,35,69,29,63)(24,58,36,70,30,64), (1,22,37,68)(2,23,38,69)(3,24,39,70)(4,25,40,71)(5,26,41,72)(6,27,42,55)(7,28,43,56)(8,29,44,57)(9,30,45,58)(10,31,46,59)(11,32,47,60)(12,33,48,61)(13,34,49,62)(14,35,50,63)(15,36,51,64)(16,19,52,65)(17,20,53,66)(18,21,54,67) );

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)], [(1,49,7,37,13,43),(2,50,8,38,14,44),(3,51,9,39,15,45),(4,52,10,40,16,46),(5,53,11,41,17,47),(6,54,12,42,18,48),(19,71,31,65,25,59),(20,72,32,66,26,60),(21,55,33,67,27,61),(22,56,34,68,28,62),(23,57,35,69,29,63),(24,58,36,70,30,64)], [(1,22,37,68),(2,23,38,69),(3,24,39,70),(4,25,40,71),(5,26,41,72),(6,27,42,55),(7,28,43,56),(8,29,44,57),(9,30,45,58),(10,31,46,59),(11,32,47,60),(12,33,48,61),(13,34,49,62),(14,35,50,63),(15,36,51,64),(16,19,52,65),(17,20,53,66),(18,21,54,67)]])

Dic3×C18 is a maximal subgroup of   Dic9⋊Dic3  C18.Dic6  Dic3⋊Dic9  D18⋊Dic3  C6.18D36  D18.3D6  S3×C2×C36

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 9A ··· 9F 9G ··· 9L 12A ··· 12H 18A ··· 18R 18S ··· 18AJ 36A ··· 36X order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 2 2 2 3 3 3 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 S3×C9 C9×Dic3 S3×C18 kernel Dic3×C18 C9×Dic3 C6×C18 C6×Dic3 C3×C18 C3×Dic3 C62 C2×Dic3 C3×C6 Dic3 C2×C6 C6 C2×C18 C18 C18 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 6 8 12 6 24 1 2 1 2 4 2 6 12 6

Matrix representation of Dic3×C18 in GL3(𝔽37) generated by

 25 0 0 0 9 0 0 0 9
,
 1 0 0 0 11 34 0 0 27
,
 1 0 0 0 4 6 0 28 33
G:=sub<GL(3,GF(37))| [25,0,0,0,9,0,0,0,9],[1,0,0,0,11,0,0,34,27],[1,0,0,0,4,28,0,6,33] >;

Dic3×C18 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{18}
% in TeX

G:=Group("Dic3xC18");
// GroupNames label

G:=SmallGroup(216,56);
// by ID

G=gap.SmallGroup(216,56);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,122,5189]);
// Polycyclic

G:=Group<a,b,c|a^18=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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