Copied to
clipboard

## G = He3.3S3order 162 = 2·34

### 3rd non-split extension by He3 of S3 acting faithfully

Aliases: He3.3S3, 3- 1+22S3, (C3×C9)⋊5S3, He3.C31C2, C32.2(C3⋊S3), C3.3(He3⋊C2), SmallGroup(162,20)

Series: Derived Chief Lower central Upper central

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

Generators and relations for He3.3S3
G = < a,b,c,d,e | a3=b3=c3=e2=1, d3=ebe=b-1, ab=ba, cac-1=eae=ab-1, ad=da, bc=cb, bd=db, dcd-1=a-1bc, ece=c-1, ede=bd2 >

Character table of He3.3S3

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

Permutation representations of He3.3S3
On 27 points - transitive group 27T42
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(19 25 22)(20 26 23)(21 27 24)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(1 22 13)(2 26 17)(3 21 12)(4 25 16)(5 20 11)(6 24 15)(7 19 10)(8 23 14)(9 27 18)
(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 9)(3 8)(4 7)(5 6)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 27)(18 26)

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

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

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

G:=TransitiveGroup(27,42);

On 27 points - transitive group 27T68
Generators in S27
(1 11 27)(2 12 19)(3 13 20)(4 14 21)(5 15 22)(6 16 23)(7 17 24)(8 18 25)(9 10 26)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(2 19 15)(3 13 26)(5 22 18)(6 16 20)(8 25 12)(9 10 23)(11 17 14)(21 24 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 9)(3 8)(4 7)(5 6)(10 15)(11 14)(12 13)(16 18)(19 23)(20 22)(24 27)(25 26)

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

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

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

G:=TransitiveGroup(27,68);

He3.3S3 is a maximal subgroup of
He3.D6  He3.C32C6  He3.(C3⋊S3)  C3≀C3⋊S3
He3.3S3 is a maximal quotient of
He3.3Dic3  (C3×He3).S3  C33.(C3⋊S3)  C32⋊C96S3  C3.(He3⋊S3)  C32⋊C9.10S3  (C3×C9)⋊5D9  He32D9  3- 1+2⋊D9  He3.(C3⋊S3)

Matrix representation of He3.3S3 in GL6(𝔽19)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 0 1 0 0 0 0 18 18 0 0 0 0 0 0 0 1 0 0 0 0 18 18 0 0 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 18 18 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 18 18
,
 18 15 18 15 4 3 4 3 4 3 16 1 4 3 18 15 18 15 16 1 4 3 4 3 18 15 4 3 18 15 4 3 16 1 4 3
,
 1 0 0 0 0 0 18 18 0 0 0 0 0 0 18 18 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(19))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[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],[18,4,4,16,18,4,15,3,3,1,15,3,18,4,18,4,4,16,15,3,15,3,3,1,4,16,18,4,18,4,3,1,15,3,15,3],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

He3.3S3 in GAP, Magma, Sage, TeX

{\rm He}_3._3S_3
% in TeX

G:=Group("He3.3S3");
// GroupNames label

G:=SmallGroup(162,20);
// by ID

G=gap.SmallGroup(162,20);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,1802,187,147,2523,728,2704]);
// Polycyclic

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

Export

׿
×
𝔽