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## G = He3⋊S3order 162 = 2·34

### 3rd semidirect product of He3 and S3 acting faithfully

Aliases: He33S3, (C3×C9)⋊6S3, He3⋊C33C2, C32.3(C3⋊S3), C3.4(He3⋊C2), SmallGroup(162,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3⋊C3 — He3⋊S3
 Chief series C1 — C3 — C32 — He3 — He3⋊C3 — He3⋊S3
 Lower central He3⋊C3 — He3⋊S3
 Upper central C1

Generators and relations for He3⋊S3
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, dad-1=ab=ba, cac-1=ab-1, ae=ea, bc=cb, bd=db, ebe=b-1, dcd-1=a-1b-1c, ece=ab-1c-1, ede=d-1 >

27C2
3C3
9C3
9C3
9C3
9S3
27S3
27S3
27C6
27S3
3C9
3C32
3C32
3C32
3D9

Character table of He3⋊S3

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

Permutation representations of He3⋊S3
On 27 points - transitive group 27T44
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 4 6)(2 3 5)(7 8 9)(10 12 11)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 26 27)
(1 10 17)(2 21 26)(3 20 27)(4 12 18)(5 19 25)(6 11 16)(7 22 14)(8 24 15)(9 23 13)
(1 25 23)(2 13 10)(3 14 12)(4 26 22)(5 15 11)(6 27 24)(7 17 21)(8 18 20)(9 16 19)
(1 8)(2 3)(4 7)(6 9)(10 14)(11 15)(12 13)(16 24)(17 22)(18 23)(19 27)(20 25)(21 26)

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

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

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

G:=TransitiveGroup(27,44);

On 27 points - transitive group 27T66
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)
(1 12 15)(2 10 13)(3 11 14)(4 26 8)(5 27 9)(6 25 7)(16 21 23)(17 19 24)(18 20 22)
(1 3 13)(2 12 11)(5 27 9)(6 7 25)(10 15 14)(16 19 18)(17 22 23)(20 21 24)
(1 20 9)(2 16 6)(3 24 26)(4 14 19)(5 12 22)(7 13 23)(8 11 17)(10 21 25)(15 18 27)
(4 17)(5 18)(6 16)(7 21)(8 19)(9 20)(10 13)(11 14)(12 15)(22 27)(23 25)(24 26)

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

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

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

G:=TransitiveGroup(27,66);

He3⋊S3 is a maximal subgroup of
He3.2D6  C92⋊S3  C9⋊C9⋊S3  He3⋊(C3×S3)  C3⋊(He3⋊S3)  C3≀C3⋊S3
He3⋊S3 is a maximal quotient of
He3⋊Dic3  (C3×He3)⋊S3  C32⋊C96S3  C3.(He3⋊S3)  (C3×C9)⋊6D9  He32D9  C922S3  C3⋊(He3⋊S3)

Matrix representation of He3⋊S3 in GL6(𝔽19)

 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
,
 0 18 0 0 0 0 1 18 0 0 0 0 7 0 18 18 0 0 0 12 1 0 0 0 11 0 0 0 18 18 0 8 0 0 1 0
,
 7 5 7 5 7 5 2 7 2 7 2 7 10 16 1 12 6 10 7 11 18 5 1 12 12 1 13 9 11 2 4 0 11 2 18 7
,
 5 3 14 2 17 12 16 2 12 14 5 17 8 14 7 8 4 13 15 0 14 13 11 18 6 14 10 4 10 8 7 3 17 9 8 1
,
 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))| [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],[0,1,7,0,11,0,18,18,0,12,0,8,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[7,2,10,7,12,4,5,7,16,11,1,0,7,2,1,18,13,11,5,7,12,5,9,2,7,2,6,1,11,18,5,7,10,12,2,7],[5,16,8,15,6,7,3,2,14,0,14,3,14,12,7,14,10,17,2,14,8,13,4,9,17,5,4,11,10,8,12,17,13,18,8,1],[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\rtimes S_3
% in TeX

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

G:=SmallGroup(162,21);
// by ID

G=gap.SmallGroup(162,21);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,187,728,433,2704]);
// Polycyclic

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

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