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## G = S3×3- 1+2order 162 = 2·34

### Direct product of S3 and 3- 1+2

Aliases: S3×3- 1+2, C33.2C6, (S3×C9)⋊C3, C92(C3×S3), (C3×C9)⋊7C6, (S3×C32).C3, C32.8(C3×C6), C3.6(S3×C32), C3⋊(C2×3- 1+2), (C3×S3).3C32, C32.10(C3×S3), (C3×3- 1+2)⋊3C2, SmallGroup(162,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×3- 1+2
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — S3×3- 1+2
 Lower central C3 — C32 — S3×3- 1+2
 Upper central C1 — C3 — 3- 1+2

Generators and relations for S3×3- 1+2
G = < a,b,c,d | a3=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Permutation representations of S3×3- 1+2
On 18 points - transitive group 18T84
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)

G:=sub<Sym(18)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,8,5)(3,6,9)(10,13,16)(12,18,15)>;

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,8,5)(3,6,9)(10,13,16)(12,18,15) );

G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,10)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15)]])

G:=TransitiveGroup(18,84);

On 27 points - transitive group 27T75
Generators in S27
(1 11 24)(2 12 25)(3 13 26)(4 14 27)(5 15 19)(6 16 20)(7 17 21)(8 18 22)(9 10 23)
(10 23)(11 24)(12 25)(13 26)(14 27)(15 19)(16 20)(17 21)(18 22)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)

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

G:=Group( (1,11,24)(2,12,25)(3,13,26)(4,14,27)(5,15,19)(6,16,20)(7,17,21)(8,18,22)(9,10,23), (10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26) );

G=PermutationGroup([[(1,11,24),(2,12,25),(3,13,26),(4,14,27),(5,15,19),(6,16,20),(7,17,21),(8,18,22),(9,10,23)], [(10,23),(11,24),(12,25),(13,26),(14,27),(15,19),(16,20),(17,21),(18,22)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26)]])

G:=TransitiveGroup(27,75);

S3×3- 1+2 is a maximal subgroup of
C34.C6  C9⋊He3⋊C2  D9⋊3- 1+2  C927C6  C928C6  He3.C3⋊C6  He3.(C3×C6)  C3≀C3.C6  3- 1+42C2
S3×3- 1+2 is a maximal quotient of
C34.C6  C9⋊He3⋊C2  D9⋊3- 1+2  C927C6  C928C6

33 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 1 2 2 2 3 3 6 6 3 3 9 9 3 ··· 3 6 ··· 6 9 ··· 9

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 S3×3- 1+2 kernel S3×3- 1+2 C3×3- 1+2 S3×C9 S3×C32 C3×C9 C33 3- 1+2 C9 C32 S3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 2 2 2

Matrix representation of S3×3- 1+2 in GL5(𝔽19)

 0 1 0 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 7 0 0 0 0 0 7 0 0 0 0 0 0 18 12 0 0 7 18 12 0 0 0 10 1
,
 11 0 0 0 0 0 11 0 0 0 0 0 1 0 18 0 0 0 7 11 0 0 0 0 11

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,18,18,10,0,0,12,12,1],[11,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,18,11,11] >;

S3×3- 1+2 in GAP, Magma, Sage, TeX

S_3\times 3_-^{1+2}
% in TeX

G:=Group("S3xES-(3,1)");
// GroupNames label

G:=SmallGroup(162,37);
// by ID

G=gap.SmallGroup(162,37);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,187,57,2704]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^9=d^3=1,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^4>;
// generators/relations

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