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## G = S3×D6⋊S3order 432 = 24·33

### Direct product of S3 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — S3×D6⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S32×C6 — S3×D6⋊S3
 Lower central C33 — C32×C6 — S3×D6⋊S3
 Upper central C1 — C2

Generators and relations for S3×D6⋊S3
G = < a,b,c,d,e,f | a3=b2=c6=d2=e3=f2=1, 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, fdf=c3d, fef=e-1 >

Subgroups: 1508 in 270 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, D6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, S3×C32, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, D6⋊S3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C2×C3⋊Dic3, C327D4, C2×S32, S3×C2×C6, C3×C3⋊Dic3, C335C4, C3×S32, S3×C3×C6, S3×C3×C6, C6×C3⋊S3, C2×D6⋊S3, S3×C3⋊D4, C3×D6⋊S3, S3×C3⋊Dic3, C336D4, C339D4, S32×C6, S3×D6⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, S32, S3×D4, C2×C3⋊D4, D6⋊S3, C2×S32, C2×D6⋊S3, S3×C3⋊D4, S33, S3×D6⋊S3

Smallest permutation representation of S3×D6⋊S3
On 48 points
Generators in S48
(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 10)(2 9)(3 8)(4 7)(5 12)(6 11)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)(25 34)(26 33)(27 32)(28 31)(29 36)(30 35)(37 43)(38 48)(39 47)(40 46)(41 45)(42 44)
(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G ··· 6N 6O 6P ··· 6W 6X 6Y 6Z 6AA 12 order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 ··· 6 6 6 ··· 6 6 6 6 6 12 size 1 1 3 3 6 6 18 18 2 2 2 4 4 4 8 18 54 2 2 2 4 4 4 6 ··· 6 8 12 ··· 12 18 18 18 18 36

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C3⋊D4 S32 S3×D4 D6⋊S3 C2×S32 S3×C3⋊D4 S33 S3×D6⋊S3 kernel S3×D6⋊S3 C3×D6⋊S3 S3×C3⋊Dic3 C33⋊6D4 C33⋊9D4 S32×C6 D6⋊S3 C2×S32 S3×C32 C3⋊Dic3 S3×C6 C2×C3⋊S3 C3×S3 D6 C32 S3 C6 C3 C2 C1 # reps 1 1 1 2 1 2 1 2 2 1 6 2 8 3 1 2 3 4 1 1

Matrix representation of S3×D6⋊S3 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 11 4 0 0 0 0 9 2 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 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 12 0 0 0 0 0 12 1 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×D6⋊S3 in GAP, Magma, Sage, TeX

S_3\times D_6\rtimes S_3
% in TeX

G:=Group("S3xD6:S3");
// GroupNames label

G:=SmallGroup(432,597);
// by ID

G=gap.SmallGroup(432,597);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,298,2028,14118]);
// Polycyclic

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

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