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## G = S32×Dic3order 432 = 24·33

### Direct product of S3, S3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S32×Dic3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S32×C6 — S32×Dic3
 Lower central C33 — S32×Dic3
 Upper central C1 — C2

Generators and relations for S32×Dic3
G = < a,b,c,d,e,f | a3=b2=c3=d2=e6=1, f2=e3, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1244 in 270 conjugacy classes, 74 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C22×C4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C2×C4, C22×Dic3, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, S3×Dic3, C6.D6, S3×C12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C2×S32, S3×C2×C6, C32×Dic3, C3×C3⋊Dic3, C335C4, C3×S32, S3×C3×C6, C6×C3⋊S3, C4×S32, C2×S3×Dic3, C3×S3×Dic3, S3×C3⋊Dic3, Dic3×C3⋊S3, C339(C2×C4), S32×C6, S32×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, S32, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, C4×S32, C2×S3×Dic3, S33, S32×Dic3

Smallest permutation representation of S32×Dic3
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(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 45 47)(44 46 48)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 8 4 11)(2 7 5 10)(3 12 6 9)(13 20 16 23)(14 19 17 22)(15 24 18 21)(25 32 28 35)(26 31 29 34)(27 36 30 33)(37 44 40 47)(38 43 41 46)(39 48 42 45)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G ··· 6N 6O 6P 6Q 6R 6S 6T 6U 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 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 6 6 6 6 6 12 12 12 12 12 12 12 12 12 12 size 1 1 3 3 3 3 9 9 2 2 2 4 4 4 8 3 3 9 9 9 9 27 27 2 2 2 4 4 4 6 ··· 6 8 12 12 12 12 18 18 6 6 6 6 12 12 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + + + - + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 S3 S3 D6 D6 Dic3 D6 D6 C4×S3 S32 S32 S3×Dic3 C2×S32 C4×S32 S33 S32×Dic3 kernel S32×Dic3 C3×S3×Dic3 S3×C3⋊Dic3 Dic3×C3⋊S3 C33⋊9(C2×C4) S32×C6 C3×S32 S3×Dic3 C2×S32 C3×Dic3 C3⋊Dic3 S32 S3×C6 C2×C3⋊S3 C3×S3 Dic3 D6 S3 C6 C3 C2 C1 # reps 1 2 2 1 1 1 8 2 1 2 2 4 4 1 8 1 2 4 3 2 1 1

Matrix representation of S32×Dic3 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 12 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 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 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
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

S32×Dic3 in GAP, Magma, Sage, TeX

S_3^2\times {\rm Dic}_3
% in TeX

G:=Group("S3^2xDic3");
// GroupNames label

G:=SmallGroup(432,594);
// by ID

G=gap.SmallGroup(432,594);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,298,2028,14118]);
// Polycyclic

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

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