Copied to
clipboard

## G = 3- 1+2.S3order 162 = 2·34

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

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 >

Character table of 3- 1+2.S3

 class 1 2 3A 3B 3C 6A 6B 9A 9B 9C 9D 9E 9F size 1 27 2 3 3 27 27 6 6 6 18 18 18 ρ1 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 linear of order 2 ρ3 2 0 2 2 2 0 0 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ4 2 0 2 2 2 0 0 2 2 2 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 0 0 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 0 0 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ7 3 -1 3 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ8 3 -1 3 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ9 3 1 3 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ10 3 1 3 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ11 6 0 -3 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 orthogonal faithful ρ12 6 0 -3 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 orthogonal faithful ρ13 6 0 -3 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 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)

 0 0 17 12 0 0 0 0 7 5 0 0 0 0 0 0 7 5 0 0 0 0 14 2 17 12 0 0 0 0 7 5 0 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 18 18
,
 0 0 0 0 0 1 0 0 0 0 18 18 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 18 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

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

׿
×
𝔽