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G = 3- 1+2.S3  order 162 = 2·34

The non-split extension by 3- 1+2 of S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: 3- 1+2.S3, (C3×C9).2S3, C3.He3⋊C2, C32.4(C3⋊S3), C3.5(He3⋊C2), SmallGroup(162,22)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3.He3 — 3- 1+2.S3
Chief series
C1 — C3 — C32 — C3×C9 — C3.He3 — 3- 1+2.S3
Lower central
C3.He3 — 3- 1+2.S3
Upper central
C1

Generators and relations for 3- 1+2.S3
 G = < a,b,c,d | a9=b3=d2=1, c3=a6, bab-1=a4, cac-1=a7b, dad=a5b-1, cbc-1=a3b, bd=db, dcd=a3c2 >

C1
27C2
C3
3C3
9S3
27C6
C32
3C9
3C9
3C9
3C9
3D9
3D9
3D9
3D9
9C3×S3
3- 1+2
C3×C9
3- 1+2
3- 1+2
3C9⋊C6
3C3×D9
3C9⋊C6
3C9⋊C6
C3.He3
3- 1+2.S3

Character table of 3- 1+2.S3

 class 123A3B3C6A6B9A9B9C9D9E9F
 size 1272332727666181818
ρ11111111111111    trivial
ρ21-1111-1-1111111    linear of order 2
ρ32022200-1-1-1-1-12    orthogonal lifted from S3
ρ42022200222-1-1-1    orthogonal lifted from S3
ρ52022200-1-1-12-1-1    orthogonal lifted from S3
ρ62022200-1-1-1-12-1    orthogonal lifted from S3
ρ73-13-3+3√-3/2-3-3√-3/2ζ65ζ6000000    complex lifted from He3⋊C2
ρ83-13-3-3√-3/2-3+3√-3/2ζ6ζ65000000    complex lifted from He3⋊C2
ρ9313-3-3√-3/2-3+3√-3/2ζ32ζ3000000    complex lifted from He3⋊C2
ρ10313-3+3√-3/2-3-3√-3/2ζ3ζ32000000    complex lifted from He3⋊C2
ρ1160-30000-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000    orthogonal faithful
ρ1260-300002ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000    orthogonal faithful
ρ1360-30000ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000    orthogonal faithful

Permutation representations of 3- 1+2.S3
►On 27 points - transitive group 27T43
Generators in S27
(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)

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

(2 3)(4 7)(5 9)(6 8)(10 22)(11 27)(12 20)(13 19)(14 24)(15 26)(16 25)(17 21)(18 23)
 
G:=sub<Sym(27)| (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), (1,23,15,7,20,12,4,26,18)(2,21,16,8,27,13,5,24,10)(3,22,14,9,19,11,6,25,17), (2,3)(4,7)(5,9)(6,8)(10,22)(11,27)(12,20)(13,19)(14,24)(15,26)(16,25)(17,21)(18,23)>;
 
G:=Group( (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), (1,23,15,7,20,12,4,26,18)(2,21,16,8,27,13,5,24,10)(3,22,14,9,19,11,6,25,17), (2,3)(4,7)(5,9)(6,8)(10,22)(11,27)(12,20)(13,19)(14,24)(15,26)(16,25)(17,21)(18,23) );
 
G=PermutationGroup([[(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)], [(1,23,15,7,20,12,4,26,18),(2,21,16,8,27,13,5,24,10),(3,22,14,9,19,11,6,25,17)], [(2,3),(4,7),(5,9),(6,8),(10,22),(11,27),(12,20),(13,19),(14,24),(15,26),(16,25),(17,21),(18,23)]])
 
G:=TransitiveGroup(27,43);
 

3- 1+2.S3 is a maximal subgroup of   C92.S3  C9⋊C9.S3  C9⋊C9.3S3  C3.He3⋊C6  (C32×C9).S3  C3≀C3⋊S3
3- 1+2.S3 is a maximal quotient of   3- 1+2.Dic3  C33.(C3⋊S3)  C3.(C33⋊S3)  C3.(He3⋊S3)  (C3×C9)⋊6D9  3- 1+2⋊D9  (C32×C9).S3

Matrix representation of 3- 1+2.S3 ►in GL6(𝔽19)

00171200
007500
000075
0000142
17120000
750000
,
100000
010000
00181800
001000
000001
00001818
,
000001
00001818
100000
010000
001000
000100
,
100000
18180000
000001
000010
000100
001000

G:=sub<GL(6,GF(19))| [0,0,0,0,17,7,0,0,0,0,12,5,17,7,0,0,0,0,12,5,0,0,0,0,0,0,7,14,0,0,0,0,5,2,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,18,0,0,0,0,1,18,0,0,0,0],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
 

3- 1+2.S3 in GAP, Magma, Sage, TeX 

3_-^{1+2}.S_3
 
% in TeX
 
G:=Group("ES-(3,1).S3");
 
// GroupNames label
 
G:=SmallGroup(162,22);
 
// by ID
 
G=gap.SmallGroup(162,22);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,581,546,992,187,282,2523,728,2704]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^9=b^3=d^2=1,c^3=a^6,b*a*b^-1=a^4,c*a*c^-1=a^7*b,d*a*d=a^5*b^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^3*c^2>;
 
// generators/relations
 

Export

Subgroup lattice of 3- 1+2.S3 in TeX
Character table of 3- 1+2.S3 in TeX

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