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## G = S3×He3order 162 = 2·34

### Direct product of S3 and He3

Aliases: S3×He3, C333C6, C3⋊(C2×He3), (S3×C32)⋊C3, (C3×He3)⋊2C2, C324(C3×S3), C3.5(S3×C32), C32.7(C3×C6), (C3×S3).2C32, SmallGroup(162,35)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×He3
G = < a,b,c,d,e | a3=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 180 in 61 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C3 [×2], C3 [×9], S3, C6 [×5], C32, C32 [×4], C32 [×12], C3×S3, C3×S3 [×4], C3×C6 [×4], He3, He3 [×4], C33 [×4], C2×He3, S3×C32 [×4], C3×He3, S3×He3
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, He3, C2×He3, S3×C32, S3×He3

Permutation representations of S3×He3
On 18 points - transitive group 18T77
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 11)(2 10)(3 12)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)
(1 2 3)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 18 17)
(4 6 5)(7 8 9)(13 14 15)(16 18 17)

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

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

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

G:=TransitiveGroup(18,77);

On 27 points - transitive group 27T73
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)
(2 3)(5 6)(8 9)(11 12)(14 15)(17 18)(20 21)(23 24)(26 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 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)

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

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

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

G:=TransitiveGroup(27,73);

S3×He3 is a maximal subgroup of   C34⋊C6  D9⋊He3  C3≀C3⋊C6  3+ 1+42C2
S3×He3 is a maximal quotient of   C34⋊C6  D9⋊He3

33 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3M 3N ··· 3U 6A 6B 6C ··· 6J order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 size 1 3 1 1 2 2 2 3 ··· 3 6 ··· 6 3 3 9 ··· 9

33 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + + image C1 C2 C3 C6 S3 C3×S3 He3 C2×He3 S3×He3 kernel S3×He3 C3×He3 S3×C32 C33 He3 C32 S3 C3 C1 # reps 1 1 8 8 1 8 2 2 2

Matrix representation of S3×He3 in GL5(𝔽7)

 0 1 0 0 0 6 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 2 0 0 0 0 0 2 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 3 0 0 0 0 0 5 0

G:=sub<GL(5,GF(7))| [0,6,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,1,0,0] >;

S3×He3 in GAP, Magma, Sage, TeX

S_3\times {\rm He}_3
% in TeX

G:=Group("S3xHe3");
// GroupNames label

G:=SmallGroup(162,35);
// by ID

G=gap.SmallGroup(162,35);
# by ID

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

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

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