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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 [×2], C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C9, C9, C32, Dic3 [×2], C12 [×2], C2×C6 [×2], C2×C6, C18, C18 [×2], C18 [×3], C3×C6, C3×C6 [×2], C2×Dic3, C2×C12, C3×C9, C36 [×2], C2×C18, C2×C18, C3×Dic3 [×2], C62, C3×C18, C3×C18 [×2], C2×C36, C6×Dic3, C9×Dic3 [×2], C6×C18, Dic3×C18
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C9, Dic3 [×2], C12 [×2], D6, C2×C6, C18 [×3], C3×S3, C2×Dic3, C2×C12, C36 [×2], C2×C18, C3×Dic3 [×2], S3×C6, S3×C9, C2×C36, C6×Dic3, C9×Dic3 [×2], 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 69 7 57 13 63)(2 70 8 58 14 64)(3 71 9 59 15 65)(4 72 10 60 16 66)(5 55 11 61 17 67)(6 56 12 62 18 68)(19 54 31 48 25 42)(20 37 32 49 26 43)(21 38 33 50 27 44)(22 39 34 51 28 45)(23 40 35 52 29 46)(24 41 36 53 30 47)
(1 36 57 47)(2 19 58 48)(3 20 59 49)(4 21 60 50)(5 22 61 51)(6 23 62 52)(7 24 63 53)(8 25 64 54)(9 26 65 37)(10 27 66 38)(11 28 67 39)(12 29 68 40)(13 30 69 41)(14 31 70 42)(15 32 71 43)(16 33 72 44)(17 34 55 45)(18 35 56 46)

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

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

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

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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