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G = S3×3- 1+2  order 162 = 2·34

Direct product of S3 and 3- 1+2

direct product, metabelian, supersoluble, monomial

Aliases: S3×3- 1+2, C33.2C6, (S3×C9)⋊C3, C9⋊2(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 >

C1
3C2
C3
C3
2C3
3C3
6C3
S3
3C6
9C6
C9
C32
C9
C32
C9
2C32
2C9
2C9
2C9
3C32
6C32
C3×S3
3C3×C6
3C18
3C18
3C18
3C3×S3
C3×C9
C3×C9
C3×C9
C33
3- 1+2
23- 1+2
23- 1+2
23- 1+2
23- 1+2
S3×C9
S3×C9
S3×C32
S3×C9
3C2×3- 1+2
C3×3- 1+2
S3×3- 1+2

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  C92⋊7C6  C92⋊8C6  He3.C3⋊C6  He3.(C3×C6)  C3≀C3.C6  3- 1+4⋊2C2
S3×3- 1+2 is a maximal quotient of
 C34.C6  C9⋊He3⋊C2  D9⋊3- 1+2  C92⋊7C6  C92⋊8C6

33 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H3I6A6B6C6D9A···9F9G···9L18A···18F
order1233333333366669···99···918···18
size1311222336633993···36···69···9

33 irreducible representations

dim111111222336
type+++
imageC1C2C3C3C6C6S3C3×S3C3×S33- 1+2C2×3- 1+2S3×3- 1+2
kernelS3×3- 1+2C3×3- 1+2S3×C9S3×C32C3×C9C333- 1+2C9C32S3C3C1
# reps116262162222

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

01000
1818000
00100
00010
00001
,
01000
10000
00100
00010
00001
,
70000
07000
0001812
0071812
000101
,
110000
011000
001018
000711
000011

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
 

Export

Subgroup lattice of S3×3- 1+2 in TeX

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