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## G = S3×C18order 108 = 22·33

### Direct product of C18 and S3

Aliases: S3×C18, C6⋊C18, C3⋊(C2×C18), (S3×C6).C3, (C3×S3).C6, (C3×C18)⋊1C2, C6.9(C3×S3), C3.4(S3×C6), (C3×C6).6C6, (C3×C9)⋊2C22, C32.2(C2×C6), SmallGroup(108,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C18
 Chief series C1 — C3 — C32 — C3×C9 — S3×C9 — S3×C18
 Lower central C3 — S3×C18
 Upper central C1 — C18

Generators and relations for S3×C18
G = < a,b,c | a18=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C18
On 36 points
Generators in S36
(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)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 19)(16 20)(17 21)(18 22)

G:=sub<Sym(36)| (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), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,19)(16,20)(17,21)(18,22)>;

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), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,19)(16,20)(17,21)(18,22) );

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)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,19),(16,20),(17,21),(18,22)])

S3×C18 is a maximal subgroup of   D6⋊D9  C9⋊D12

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9L 18A ··· 18F 18G ··· 18L 18M ··· 18X order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 3 3 1 1 2 2 2 1 1 2 2 2 3 3 3 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D6 C3×S3 S3×C6 S3×C9 S3×C18 kernel S3×C18 S3×C9 C3×C18 S3×C6 C3×S3 C3×C6 D6 S3 C6 C18 C9 C6 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 1 1 2 2 6 6

Matrix representation of S3×C18 in GL2(𝔽19) generated by

 13 0 0 13
,
 7 0 0 11
,
 0 12 8 0
G:=sub<GL(2,GF(19))| [13,0,0,13],[7,0,0,11],[0,8,12,0] >;

S3×C18 in GAP, Magma, Sage, TeX

S_3\times C_{18}
% in TeX

G:=Group("S3xC18");
// GroupNames label

G:=SmallGroup(108,24);
// by ID

G=gap.SmallGroup(108,24);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,57,1804]);
// Polycyclic

G:=Group<a,b,c|a^18=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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