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## G = S3×Dic6order 144 = 24·32

### Direct product of S3 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×Dic6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — S3×Dic6
 Lower central C32 — C3×C6 — S3×Dic6
 Upper central C1 — C2 — C4

Generators and relations for S3×Dic6
G = < a,b,c,d | a3=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 82 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, Dic3, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C324Q8, S3×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, S3×Dic6

Character table of S3×Dic6

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H 12I size 1 1 3 3 2 2 4 2 6 6 6 18 18 2 2 4 6 6 2 2 4 4 4 6 6 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ6 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 linear of order 2 ρ8 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ9 2 2 0 0 -1 2 -1 2 2 2 0 0 0 2 -1 -1 0 0 2 2 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 2 -1 -1 -2 0 0 -2 0 0 -1 2 -1 -1 -1 1 1 -2 1 1 1 1 0 0 orthogonal lifted from D6 ρ11 2 2 0 0 -1 2 -1 2 -2 -2 0 0 0 2 -1 -1 0 0 2 2 -1 -1 -1 0 0 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 2 -1 -1 2 0 0 -2 0 0 -1 2 -1 1 1 -1 -1 2 -1 -1 1 1 0 0 orthogonal lifted from D6 ρ13 2 2 0 0 -1 2 -1 -2 2 -2 0 0 0 2 -1 -1 0 0 -2 -2 1 1 1 0 0 1 -1 orthogonal lifted from D6 ρ14 2 2 2 2 2 -1 -1 2 0 0 2 0 0 -1 2 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ15 2 2 -2 -2 2 -1 -1 -2 0 0 2 0 0 -1 2 -1 1 1 1 1 -2 1 1 -1 -1 0 0 orthogonal lifted from D6 ρ16 2 2 0 0 -1 2 -1 -2 -2 2 0 0 0 2 -1 -1 0 0 -2 -2 1 1 1 0 0 -1 1 orthogonal lifted from D6 ρ17 2 -2 -2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 1 -2 1 1 -1 √3 -√3 0 -√3 √3 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ20 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 1 -2 1 1 -1 -√3 √3 0 √3 -√3 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ21 2 -2 2 -2 2 -1 -1 0 0 0 0 0 0 1 -2 1 -1 1 √3 -√3 0 -√3 √3 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ22 2 -2 2 -2 2 -1 -1 0 0 0 0 0 0 1 -2 1 -1 1 -√3 √3 0 √3 -√3 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ23 4 4 0 0 -2 -2 1 -4 0 0 0 0 0 -2 -2 1 0 0 2 2 2 -1 -1 0 0 0 0 orthogonal lifted from C2×S32 ρ24 4 4 0 0 -2 -2 1 4 0 0 0 0 0 -2 -2 1 0 0 -2 -2 -2 1 1 0 0 0 0 orthogonal lifted from S32 ρ25 4 -4 0 0 -2 4 -2 0 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ26 4 -4 0 0 -2 -2 1 0 0 0 0 0 0 2 2 -1 0 0 -2√3 2√3 0 -√3 √3 0 0 0 0 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 -2 -2 1 0 0 0 0 0 0 2 2 -1 0 0 2√3 -2√3 0 √3 -√3 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of S3×Dic6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(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 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)

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

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

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

Matrix representation of S3×Dic6 in GL6(𝔽13)

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

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

S3×Dic6 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_6
% in TeX

G:=Group("S3xDic6");
// GroupNames label

G:=SmallGroup(144,137);
// by ID

G=gap.SmallGroup(144,137);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,116,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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