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## G = S3×C6×Dic3order 432 = 24·33

### Direct product of C6, S3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×C6×Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — S3×C6×Dic3
 Lower central C32 — S3×C6×Dic3
 Upper central C1 — C2×C6

Generators and relations for S3×C6×Dic3
G = < a,b,c,d,e | a6=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 768 in 290 conjugacy classes, 112 normal (36 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C62, S3×C2×C4, C22×Dic3, C22×C12, S3×C32, C32×C6, C32×C6, S3×Dic3, S3×C12, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, S3×C2×C6, C2×C62, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C3×C62, C2×S3×Dic3, S3×C2×C12, Dic3×C2×C6, C3×S3×Dic3, Dic3×C3×C6, C6×C3⋊Dic3, S3×C62, S3×C6×Dic3
Quotients:

Smallest permutation representation of S3×C6×Dic3
On 48 points
Generators in S48
(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)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 23)(8 24)(9 19)(10 20)(11 21)(12 22)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(25 45)(26 46)(27 47)(28 48)(29 43)(30 44)
(1 18 5 16 3 14)(2 13 6 17 4 15)(7 45 9 47 11 43)(8 46 10 48 12 44)(19 27 21 29 23 25)(20 28 22 30 24 26)(31 37 35 41 33 39)(32 38 36 42 34 40)
(1 28 16 24)(2 29 17 19)(3 30 18 20)(4 25 13 21)(5 26 14 22)(6 27 15 23)(7 39 47 35)(8 40 48 36)(9 41 43 31)(10 42 44 32)(11 37 45 33)(12 38 46 34)

G:=sub<Sym(48)| (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), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,23)(8,24)(9,19)(10,20)(11,21)(12,22)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,45,9,47,11,43)(8,46,10,48,12,44)(19,27,21,29,23,25)(20,28,22,30,24,26)(31,37,35,41,33,39)(32,38,36,42,34,40), (1,28,16,24)(2,29,17,19)(3,30,18,20)(4,25,13,21)(5,26,14,22)(6,27,15,23)(7,39,47,35)(8,40,48,36)(9,41,43,31)(10,42,44,32)(11,37,45,33)(12,38,46,34)>;

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), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,23)(8,24)(9,19)(10,20)(11,21)(12,22)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,45,9,47,11,43)(8,46,10,48,12,44)(19,27,21,29,23,25)(20,28,22,30,24,26)(31,37,35,41,33,39)(32,38,36,42,34,40), (1,28,16,24)(2,29,17,19)(3,30,18,20)(4,25,13,21)(5,26,14,22)(6,27,15,23)(7,39,47,35)(8,40,48,36)(9,41,43,31)(10,42,44,32)(11,37,45,33)(12,38,46,34) );

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)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(1,40),(2,41),(3,42),(4,37),(5,38),(6,39),(7,23),(8,24),(9,19),(10,20),(11,21),(12,22),(13,33),(14,34),(15,35),(16,36),(17,31),(18,32),(25,45),(26,46),(27,47),(28,48),(29,43),(30,44)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,45,9,47,11,43),(8,46,10,48,12,44),(19,27,21,29,23,25),(20,28,22,30,24,26),(31,37,35,41,33,39),(32,38,36,42,34,40)], [(1,28,16,24),(2,29,17,19),(3,30,18,20),(4,25,13,21),(5,26,14,22),(6,27,15,23),(7,39,47,35),(8,40,48,36),(9,41,43,31),(10,42,44,32),(11,37,45,33),(12,38,46,34)]])

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6X 6Y ··· 6AF 6AG ··· 6AO 6AP ··· 6BA 12A ··· 12H 12I ··· 12T 12U ··· 12AB order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 3 3 3 3 1 1 2 ··· 2 4 4 4 3 3 3 3 9 9 9 9 1 ··· 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 3 ··· 3 6 ··· 6 9 ··· 9

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 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 S3 D6 Dic3 D6 D6 C3×S3 C3×S3 C4×S3 S3×C6 C3×Dic3 S3×C6 S3×C6 S3×C12 S32 S3×Dic3 C2×S32 C3×S32 C3×S3×Dic3 S32×C6 kernel S3×C6×Dic3 C3×S3×Dic3 Dic3×C3×C6 C6×C3⋊Dic3 S3×C62 C2×S3×Dic3 S3×C3×C6 S3×Dic3 C6×Dic3 C2×C3⋊Dic3 S3×C2×C6 S3×C6 C6×Dic3 S3×C2×C6 C3×Dic3 S3×C6 S3×C6 C62 C2×Dic3 C22×S3 C3×C6 Dic3 D6 D6 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 1 1 2 4 2 2 2 2 4 4 8 4 4 8 1 2 1 2 4 2

Matrix representation of S3×C6×Dic3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 5 5

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,5,0,0,0,0,0,5] >;

S3×C6×Dic3 in GAP, Magma, Sage, TeX

S_3\times C_6\times {\rm Dic}_3
% in TeX

G:=Group("S3xC6xDic3");
// GroupNames label

G:=SmallGroup(432,651);
// by ID

G=gap.SmallGroup(432,651);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,2028,14118]);
// Polycyclic

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

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