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## G = He3⋊5S3order 162 = 2·34

### 2nd semidirect product of He3 and S3 acting via S3/C3=C2

Aliases: He35S3, C336S3, (C3×He3)⋊5C2, C3⋊(He3⋊C2), C322(C3⋊S3), C3.2(C33⋊C2), SmallGroup(162,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — He3⋊5S3
 Chief series C1 — C3 — C32 — He3 — C3×He3 — He3⋊5S3
 Lower central C3×He3 — He3⋊5S3
 Upper central C1 — C3

Generators and relations for He35S3
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, dad-1=ab=ba, cac-1=ab-1, eae=a-1, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 444 in 97 conjugacy classes, 31 normal (6 characteristic)
C1, C2, C3 [×2], C3 [×13], S3 [×13], C6, C32, C32 [×12], C32 [×8], C3×S3 [×13], C3⋊S3 [×4], He3 [×9], C33 [×4], He3⋊C2 [×9], C3×C3⋊S3 [×4], C3×He3, He35S3
Quotients: C1, C2, S3 [×13], C3⋊S3 [×13], He3⋊C2, C33⋊C2, He35S3

Character table of He35S3

 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 1 1 2 2 2 6 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 2 0 2 2 2 2 2 -1 2 -1 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 2 -1 -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ5 2 0 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 2 0 0 orthogonal lifted from S3 ρ6 2 0 2 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ7 2 0 2 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 -1 -1 -1 -1 -1 2 -1 2 2 -1 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ9 2 0 2 2 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ10 2 0 2 2 -1 -1 -1 -1 -1 -1 2 2 -1 2 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ11 2 0 2 2 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ12 2 0 2 2 -1 -1 -1 2 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ13 2 0 2 2 -1 -1 -1 2 -1 2 -1 -1 -1 -1 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ14 2 0 2 2 2 2 2 -1 -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ15 2 0 2 2 2 2 2 2 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ16 3 1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ32 ζ3 complex lifted from He3⋊C2 ρ17 3 1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ32 complex lifted from He3⋊C2 ρ18 3 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ65 ζ6 complex lifted from He3⋊C2 ρ19 3 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ6 ζ65 complex lifted from He3⋊C2 ρ20 6 0 -3-3√-3 -3+3√-3 3-3√-3/2 3+3√-3/2 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ21 6 0 -3+3√-3 -3-3√-3 3+3√-3/2 3-3√-3/2 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of He35S3
On 18 points - transitive group 18T89
Generators in S18
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 4)(2 6 3)(7 9 8)(10 11 12)(13 15 14)(16 17 18)
(1 15 17)(2 9 11)(3 7 10)(4 13 16)(5 14 18)(6 8 12)
(1 18 13)(2 10 8)(3 12 9)(4 17 14)(5 16 15)(6 11 7)
(1 6)(2 4)(3 5)(7 18)(8 17)(9 16)(10 14)(11 13)(12 15)

G:=sub<Sym(18)| (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,4)(2,6,3)(7,9,8)(10,11,12)(13,15,14)(16,17,18), (1,15,17)(2,9,11)(3,7,10)(4,13,16)(5,14,18)(6,8,12), (1,18,13)(2,10,8)(3,12,9)(4,17,14)(5,16,15)(6,11,7), (1,6)(2,4)(3,5)(7,18)(8,17)(9,16)(10,14)(11,13)(12,15)>;

G:=Group( (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,4)(2,6,3)(7,9,8)(10,11,12)(13,15,14)(16,17,18), (1,15,17)(2,9,11)(3,7,10)(4,13,16)(5,14,18)(6,8,12), (1,18,13)(2,10,8)(3,12,9)(4,17,14)(5,16,15)(6,11,7), (1,6)(2,4)(3,5)(7,18)(8,17)(9,16)(10,14)(11,13)(12,15) );

G=PermutationGroup([(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,4),(2,6,3),(7,9,8),(10,11,12),(13,15,14),(16,17,18)], [(1,15,17),(2,9,11),(3,7,10),(4,13,16),(5,14,18),(6,8,12)], [(1,18,13),(2,10,8),(3,12,9),(4,17,14),(5,16,15),(6,11,7)], [(1,6),(2,4),(3,5),(7,18),(8,17),(9,16),(10,14),(11,13),(12,15)])

G:=TransitiveGroup(18,89);

On 27 points - transitive group 27T74
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 13 11)(2 14 12)(3 15 10)(4 26 8)(5 27 9)(6 25 7)(16 19 22)(17 20 23)(18 21 24)
(2 14 12)(3 10 15)(4 8 26)(6 25 7)(16 19 22)(17 23 20)
(1 24 5)(2 19 25)(3 17 8)(4 15 20)(6 12 16)(7 14 22)(9 11 21)(10 23 26)(13 18 27)
(2 3)(4 22)(5 24)(6 23)(7 20)(8 19)(9 21)(10 12)(14 15)(16 26)(17 25)(18 27)

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

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

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

G:=TransitiveGroup(27,74);

He35S3 is a maximal subgroup of
He35D6  S3×He3⋊C2  C3.C3≀S3  C32⋊C9⋊S3  (C3×He3).C6  C343S3  (C32×C9)⋊8S3  C345S3  He3.C3⋊S3  He3⋊C32S3  He34D9  C9○He34S3  C3413S3  3+ 1+43C2
He35S3 is a maximal quotient of
He36Dic3  C336D9  He34D9  C346S3  C347S3  He3.(C3⋊S3)  C3⋊(He3⋊S3)  (C32×C9).S3  C3≀C3⋊S3  C3413S3

Matrix representation of He35S3 in GL5(𝔽7)

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

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

He35S3 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5S_3
% in TeX

G:=Group("He3:5S3");
// GroupNames label

G:=SmallGroup(162,46);
// by ID

G=gap.SmallGroup(162,46);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,723,253]);
// 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,e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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