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## G = He3.S3order 162 = 2·34

### 1st non-split extension by He3 of S3 acting faithfully

Aliases: He3.1S3, C9⋊S32C3, (C3×C9)⋊2C6, He3.C32C2, C32.7(C3×S3), C3.3(C32⋊C6), SmallGroup(162,13)

Series: Derived Chief Lower central Upper central

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

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

Character table of He3.S3

 class 1 2 3A 3B 3C 3D 6A 6B 9A 9B 9C 9D 9E size 1 27 2 6 9 9 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 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 linear of order 3 ρ4 1 -1 1 1 ζ3 ζ32 ζ6 ζ65 1 1 1 ζ32 ζ3 linear of order 6 ρ5 1 -1 1 1 ζ32 ζ3 ζ65 ζ6 1 1 1 ζ3 ζ32 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 linear of order 3 ρ7 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 ζ65 ζ6 complex lifted from C3×S3 ρ9 2 0 2 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 ζ6 ζ65 complex lifted from C3×S3 ρ10 6 0 6 -3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 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.S3
On 27 points - transitive group 27T41
Generators in S27
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)
(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 25 10)(2 23 11)(3 21 12)(4 19 13)(5 26 14)(6 24 15)(7 22 16)(8 20 17)(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)(11 18)(12 17)(13 16)(14 15)(19 22)(20 21)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,41);

On 27 points - transitive group 27T72
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 19)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)

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)(10,19)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)>;

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)(10,19)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20) );

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),(10,19),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20)]])

G:=TransitiveGroup(27,72);

He3.S3 is a maximal subgroup of   He3.D6  C9⋊S3⋊C32  C324D9⋊C3  C3≀C3.S3
He3.S3 is a maximal quotient of   He3.Dic3  C32⋊C9⋊C6  C32⋊C9.C6  C322D9.C3  (C3×C9)⋊C18  He3⋊D9  C324D9⋊C3

Matrix representation of He3.S3 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 7 7 18 18 0 0 11 11 0 0 18 18 0 0 0 0 1 0
,
 18 1 0 0 0 0 18 0 0 0 0 0 12 0 0 1 0 0 0 7 18 18 0 0 8 0 0 0 0 1 0 11 0 0 18 18
,
 0 0 18 1 0 0 7 7 17 18 0 0 0 0 12 0 1 0 0 0 12 0 0 1 0 0 8 0 0 0 1 0 8 0 0 0
,
 17 7 0 0 0 0 12 5 0 0 0 0 8 0 5 7 0 0 0 11 12 17 0 0 16 17 0 0 2 14 18 0 0 0 5 7
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 7 7 18 18 0 0 0 0 0 0 1 0 11 11 0 0 18 18

G:=sub<GL(6,GF(19))| [1,0,0,7,11,0,0,1,0,7,11,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,18,12,0,8,0,1,0,0,7,0,11,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,7,0,0,0,1,0,7,0,0,0,0,18,17,12,12,8,8,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[17,12,8,0,16,18,7,5,0,11,17,0,0,0,5,12,0,0,0,0,7,17,0,0,0,0,0,0,2,5,0,0,0,0,14,7],[0,1,0,7,0,11,1,0,0,7,0,11,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;

He3.S3 in GAP, Magma, Sage, TeX

{\rm He}_3.S_3
% in TeX

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

G:=SmallGroup(162,13);
// by ID

G=gap.SmallGroup(162,13);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,1802,187,147,723,728,2704]);
// 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=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*b*c,c*e=e*c,e*d*e=b*d^2>;
// generators/relations

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