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## G = S3×C3×C6order 108 = 22·33

### Direct product of C3×C6 and S3

Aliases: S3×C3×C6, C3⋊C62, C334C22, C6⋊(C3×C6), (C3×C6)⋊3C6, C324(C2×C6), (C32×C6)⋊1C2, SmallGroup(108,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C3×C6
 Chief series C1 — C3 — C32 — C33 — S3×C32 — S3×C3×C6
 Lower central C3 — S3×C3×C6
 Upper central C1 — C3×C6

Generators and relations for S3×C3×C6
G = < a,b,c,d | a3=b6=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 128 in 76 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C33, S3×C6, C62, S3×C32, C32×C6, S3×C3×C6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, S3×C6, C62, S3×C32, S3×C3×C6

Smallest permutation representation of S3×C3×C6
On 36 points
Generators in S36
(1 29 24)(2 30 19)(3 25 20)(4 26 21)(5 27 22)(6 28 23)(7 14 32)(8 15 33)(9 16 34)(10 17 35)(11 18 36)(12 13 31)
(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 25 22)(2 26 23)(3 27 24)(4 28 19)(5 29 20)(6 30 21)(7 36 16)(8 31 17)(9 32 18)(10 33 13)(11 34 14)(12 35 15)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)

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

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

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

S3×C3×C6 is a maximal subgroup of   C336D4  C337D4

54 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 6A ··· 6H 6I ··· 6Q 6R ··· 6AG order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 3 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel S3×C3×C6 S3×C32 C32×C6 S3×C6 C3×S3 C3×C6 C3×C6 C32 C6 C3 # reps 1 2 1 8 16 8 1 1 8 8

Matrix representation of S3×C3×C6 in GL3(𝔽7) generated by

 1 0 0 0 4 0 0 0 4
,
 5 0 0 0 2 0 0 0 2
,
 1 0 0 0 2 0 0 0 4
,
 6 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(7))| [1,0,0,0,4,0,0,0,4],[5,0,0,0,2,0,0,0,2],[1,0,0,0,2,0,0,0,4],[6,0,0,0,0,1,0,1,0] >;

S3×C3×C6 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_6
% in TeX

G:=Group("S3xC3xC6");
// GroupNames label

G:=SmallGroup(108,42);
// by ID

G=gap.SmallGroup(108,42);
# by ID

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

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

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