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## G = He3⋊4S3order 162 = 2·34

### 1st semidirect product of He3 and S3 acting via S3/C3=C2

Aliases: He34S3, C334C6, C334S3, C3⋊(C32⋊C6), (C3×He3)⋊3C2, C323(C3×S3), C321(C3⋊S3), C33⋊C22C3, C3.2(C3×C3⋊S3), SmallGroup(162,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — He3⋊4S3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — He3⋊4S3
 Lower central C33 — He3⋊4S3
 Upper central C1

Generators and relations for He34S3
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 364 in 67 conjugacy classes, 18 normal (9 characteristic)
C1, C2, C3, C3 [×3], C3 [×8], S3 [×7], C6, C32 [×2], C32 [×3], C32 [×13], C3×S3 [×4], C3⋊S3 [×7], He3 [×3], He3 [×3], C33 [×2], C33, C32⋊C6 [×3], C3×C3⋊S3, C33⋊C2, C3×He3, He34S3
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, C32⋊C6 [×3], C3×C3⋊S3, He34S3

Character table of He34S3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 6A 6B size 1 27 2 2 2 2 3 3 6 6 6 6 6 6 6 6 6 6 6 27 27 ρ1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 1 ζ32 ζ32 1 ζ3 ζ3 ζ3 ζ3 1 ζ32 ζ3 linear of order 3 ρ4 1 -1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 1 ζ3 ζ3 1 ζ32 ζ32 ζ32 ζ32 1 ζ65 ζ6 linear of order 6 ρ5 1 -1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 1 ζ32 ζ32 1 ζ3 ζ3 ζ3 ζ3 1 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 1 ζ3 ζ3 1 ζ32 ζ32 ζ32 ζ32 1 ζ3 ζ32 linear of order 3 ρ7 2 0 -1 -1 2 -1 2 2 -1 -1 -1 -1 2 2 -1 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 2 2 2 2 -1 2 -1 -1 -1 -1 2 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ9 2 0 -1 -1 2 -1 2 2 -1 -1 2 2 -1 -1 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ10 2 0 -1 -1 2 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 -1 -1 2 0 0 orthogonal lifted from S3 ρ11 2 0 -1 -1 2 -1 -1-√-3 -1+√-3 ζ6 ζ6 -1 ζ6 -1-√-3 2 ζ65 ζ65 ζ65 -1+√-3 -1 0 0 complex lifted from C3×S3 ρ12 2 0 -1 -1 2 -1 -1+√-3 -1-√-3 ζ65 ζ65 -1 ζ65 -1+√-3 2 ζ6 ζ6 ζ6 -1-√-3 -1 0 0 complex lifted from C3×S3 ρ13 2 0 2 2 2 2 -1-√-3 -1+√-3 ζ6 -1-√-3 -1 ζ6 ζ6 -1 -1+√-3 ζ65 ζ65 ζ65 -1 0 0 complex lifted from C3×S3 ρ14 2 0 -1 -1 2 -1 -1+√-3 -1-√-3 -1+√-3 ζ65 -1 ζ65 ζ65 -1 ζ6 -1-√-3 ζ6 ζ6 2 0 0 complex lifted from C3×S3 ρ15 2 0 -1 -1 2 -1 -1-√-3 -1+√-3 ζ6 ζ6 2 -1-√-3 ζ6 -1 ζ65 ζ65 -1+√-3 ζ65 -1 0 0 complex lifted from C3×S3 ρ16 2 0 -1 -1 2 -1 -1-√-3 -1+√-3 -1-√-3 ζ6 -1 ζ6 ζ6 -1 ζ65 -1+√-3 ζ65 ζ65 2 0 0 complex lifted from C3×S3 ρ17 2 0 2 2 2 2 -1+√-3 -1-√-3 ζ65 -1+√-3 -1 ζ65 ζ65 -1 -1-√-3 ζ6 ζ6 ζ6 -1 0 0 complex lifted from C3×S3 ρ18 2 0 -1 -1 2 -1 -1+√-3 -1-√-3 ζ65 ζ65 2 -1+√-3 ζ65 -1 ζ6 ζ6 -1-√-3 ζ6 -1 0 0 complex lifted from C3×S3 ρ19 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ20 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ21 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6

Permutation representations of He34S3
On 27 points - transitive group 27T61
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 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(2 27 9)(3 7 25)(4 19 23)(5 24 20)(10 18 15)(11 13 16)
(1 14 6)(2 15 4)(3 13 5)(7 16 24)(8 17 22)(9 18 23)(10 19 27)(11 20 25)(12 21 26)
(2 3)(4 13)(5 15)(6 14)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(16 19)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(27,61);

On 27 points - transitive group 27T71
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 27 26)
(1 19 10)(2 20 12)(3 21 11)(4 22 14)(5 23 13)(6 24 15)(7 25 17)(8 26 16)(9 27 18)
(1 4 7)(2 5 8)(3 6 9)(10 14 17)(11 15 18)(12 13 16)(19 22 25)(20 23 26)(21 24 27)
(2 3)(4 7)(5 9)(6 8)(11 12)(13 18)(14 17)(15 16)(20 21)(22 25)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,71);

Matrix representation of He34S3 in GL8(𝔽7)

 6 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 1 1 5 6 0 0 0 0 1 1 6 0 1 0 0 0 1 1 6 0 0 1 0 0 6 6 2 0 0 0 0 0 0 6 2 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 1 0 0 0 0 0 1 6 6 0 0 0 0 2 0 0 0 0 1 0 0 0 5 0 0 6 6
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 6 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 5 5 0 0 6 6
,
 6 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 6 6 0 0 6 6 0 0 6 6 1 0 0 0 0 0 0 0 6 6 0 0

G:=sub<GL(8,GF(7))| [6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,1,1,6,0,0,0,0,1,1,1,6,6,0,0,6,5,6,6,2,2,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,6,0,2,0,0,0,1,0,0,1,0,5,0,0,0,0,0,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,6],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,1,0,0,5,0,0,0,1,1,0,0,5,0,0,0,0,6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,6],[6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,6,6,0,0,0,1,0,1,6,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,6,0,0,0,0,1,6,0,0,0,0,0,0,0,6,0,0] >;

He34S3 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4S_3
% in TeX

G:=Group("He3:4S3");
// GroupNames label

G:=SmallGroup(162,40);
// by ID

G=gap.SmallGroup(162,40);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,182,457,723,2704]);
// Polycyclic

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

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