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## G = S3×He3⋊C2order 324 = 22·34

### Direct product of S3 and He3⋊C2

Aliases: S3×He3⋊C2, He38D6, C335D6, C324S32, (S3×He3)⋊2C2, (S3×C32)⋊2S3, (C3×He3)⋊4C22, He35S31C2, C3.4(S3×C3⋊S3), C31(C2×He3⋊C2), C32.8(C2×C3⋊S3), (C3×S3).2(C3⋊S3), (C3×He3⋊C2)⋊2C2, SmallGroup(324,122)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — S3×He3⋊C2
 Chief series C1 — C3 — C32 — C33 — C3×He3 — S3×He3 — S3×He3⋊C2
 Lower central C3×He3 — S3×He3⋊C2
 Upper central C1 — C3

Generators and relations for S3×He3⋊C2
G = < a,b,c,d,e,f | a3=b2=c3=d3=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 820 in 144 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C3, C3, C22, S3, S3, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, S32, S3×C6, He3⋊C2, He3⋊C2, C2×He3, S3×C32, S3×C32, C3×C3⋊S3, C3×He3, C2×He3⋊C2, C3×S32, S3×He3, C3×He3⋊C2, He35S3, S3×He3⋊C2
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S32, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, S3×C3⋊S3, S3×He3⋊C2

Character table of S3×He3⋊C2

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M size 1 3 9 27 1 1 2 2 2 6 6 6 6 12 12 12 12 3 3 9 9 18 18 18 18 18 18 18 27 27 ρ1 1 1 1 1 1 1 1 1 1 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 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 1 1 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 ρ4 1 1 -1 -1 1 1 1 1 1 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 ρ5 2 2 0 0 2 2 2 2 2 2 -1 -1 -1 -1 2 -1 -1 2 2 0 0 0 0 -1 -1 2 -1 0 0 0 orthogonal lifted from S3 ρ6 2 -2 0 0 2 2 2 2 2 2 -1 -1 -1 -1 2 -1 -1 -2 -2 0 0 0 0 1 1 -2 1 0 0 0 orthogonal lifted from D6 ρ7 2 0 -2 0 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 0 0 -2 -2 1 1 0 0 0 0 1 0 0 orthogonal lifted from D6 ρ8 2 2 0 0 2 2 2 2 2 -1 2 -1 -1 -1 -1 -1 2 2 2 0 0 0 0 -1 -1 -1 2 0 0 0 orthogonal lifted from S3 ρ9 2 0 2 0 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 0 0 2 2 -1 -1 0 0 0 0 -1 0 0 orthogonal lifted from S3 ρ10 2 -2 0 0 2 2 2 2 2 -1 2 -1 -1 -1 -1 -1 2 -2 -2 0 0 0 0 1 1 1 -2 0 0 0 orthogonal lifted from D6 ρ11 2 -2 0 0 2 2 2 2 2 -1 -1 2 -1 2 -1 -1 -1 -2 -2 0 0 0 0 1 -2 1 1 0 0 0 orthogonal lifted from D6 ρ12 2 2 0 0 2 2 2 2 2 -1 -1 2 -1 2 -1 -1 -1 2 2 0 0 0 0 -1 2 -1 -1 0 0 0 orthogonal lifted from S3 ρ13 2 2 0 0 2 2 2 2 2 -1 -1 -1 2 -1 -1 2 -1 2 2 0 0 0 0 2 -1 -1 -1 0 0 0 orthogonal lifted from S3 ρ14 2 -2 0 0 2 2 2 2 2 -1 -1 -1 2 -1 -1 2 -1 -2 -2 0 0 0 0 -2 1 1 1 0 0 0 orthogonal lifted from D6 ρ15 3 3 1 1 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 3 -3-3√-3/2 0 0 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 ζ32 ζ3 0 0 0 0 1 ζ32 ζ3 complex lifted from He3⋊C2 ρ16 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 3 -3-3√-3/2 0 0 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 ζ6 ζ65 0 0 0 0 -1 ζ6 ζ65 complex lifted from He3⋊C2 ρ17 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 3 -3+3√-3/2 0 0 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 ζ3 ζ32 ζ3 ζ32 0 0 0 0 1 ζ65 ζ6 complex lifted from C2×He3⋊C2 ρ18 3 3 1 1 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 3 -3+3√-3/2 0 0 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 ζ3 ζ32 0 0 0 0 1 ζ3 ζ32 complex lifted from He3⋊C2 ρ19 3 -3 -1 1 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 3 -3+3√-3/2 0 0 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 ζ65 ζ6 ζ65 ζ6 0 0 0 0 -1 ζ3 ζ32 complex lifted from C2×He3⋊C2 ρ20 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 3 -3-3√-3/2 0 0 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 ζ32 ζ3 ζ32 ζ3 0 0 0 0 1 ζ6 ζ65 complex lifted from C2×He3⋊C2 ρ21 3 -3 -1 1 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 3 -3-3√-3/2 0 0 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 ζ6 ζ65 ζ6 ζ65 0 0 0 0 -1 ζ32 ζ3 complex lifted from C2×He3⋊C2 ρ22 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 3 -3+3√-3/2 0 0 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 ζ65 ζ6 0 0 0 0 -1 ζ65 ζ6 complex lifted from He3⋊C2 ρ23 4 0 0 0 4 4 -2 -2 -2 -2 4 -2 -2 1 1 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ24 4 0 0 0 4 4 -2 -2 -2 4 -2 -2 -2 1 -2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ25 4 0 0 0 4 4 -2 -2 -2 -2 -2 4 -2 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ26 4 0 0 0 4 4 -2 -2 -2 -2 -2 -2 4 1 1 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ27 6 0 -2 0 -3+3√-3 -3-3√-3 3-3√-3/2 -3 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 1+√-3 1-√-3 ζ32 ζ3 0 0 0 0 1 0 0 complex faithful ρ28 6 0 2 0 -3+3√-3 -3-3√-3 3-3√-3/2 -3 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 -1 0 0 complex faithful ρ29 6 0 2 0 -3-3√-3 -3+3√-3 3+3√-3/2 -3 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 -1 0 0 complex faithful ρ30 6 0 -2 0 -3-3√-3 -3+3√-3 3+3√-3/2 -3 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 1-√-3 1+√-3 ζ3 ζ32 0 0 0 0 1 0 0 complex faithful

Permutation representations of S3×He3⋊C2
On 18 points - transitive group 18T135
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 10)(2 12)(3 11)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 8 5)(2 9 6)(3 7 4)(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)
(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(18,135);

On 27 points - transitive group 27T118
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)(4 6)(7 8)(11 12)(14 15)(16 18)(20 21)(22 24)(25 27)
(1 9 23)(2 7 24)(3 8 22)(4 21 12)(5 19 10)(6 20 11)(13 26 17)(14 27 18)(15 25 16)
(1 10 13)(2 11 14)(3 12 15)(4 25 8)(5 26 9)(6 27 7)(16 22 21)(17 23 19)(18 24 20)
(1 26 17)(2 27 18)(3 25 16)(4 21 15)(5 19 13)(6 20 14)(7 24 11)(8 22 12)(9 23 10)
(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(16 25)(17 26)(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), (2,3)(4,6)(7,8)(11,12)(14,15)(16,18)(20,21)(22,24)(25,27), (1,9,23)(2,7,24)(3,8,22)(4,21,12)(5,19,10)(6,20,11)(13,26,17)(14,27,18)(15,25,16), (1,10,13)(2,11,14)(3,12,15)(4,25,8)(5,26,9)(6,27,7)(16,22,21)(17,23,19)(18,24,20), (1,26,17)(2,27,18)(3,25,16)(4,21,15)(5,19,13)(6,20,14)(7,24,11)(8,22,12)(9,23,10), (4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(16,25)(17,26)(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), (2,3)(4,6)(7,8)(11,12)(14,15)(16,18)(20,21)(22,24)(25,27), (1,9,23)(2,7,24)(3,8,22)(4,21,12)(5,19,10)(6,20,11)(13,26,17)(14,27,18)(15,25,16), (1,10,13)(2,11,14)(3,12,15)(4,25,8)(5,26,9)(6,27,7)(16,22,21)(17,23,19)(18,24,20), (1,26,17)(2,27,18)(3,25,16)(4,21,15)(5,19,13)(6,20,14)(7,24,11)(8,22,12)(9,23,10), (4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(16,25)(17,26)(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)], [(2,3),(4,6),(7,8),(11,12),(14,15),(16,18),(20,21),(22,24),(25,27)], [(1,9,23),(2,7,24),(3,8,22),(4,21,12),(5,19,10),(6,20,11),(13,26,17),(14,27,18),(15,25,16)], [(1,10,13),(2,11,14),(3,12,15),(4,25,8),(5,26,9),(6,27,7),(16,22,21),(17,23,19),(18,24,20)], [(1,26,17),(2,27,18),(3,25,16),(4,21,15),(5,19,13),(6,20,14),(7,24,11),(8,22,12),(9,23,10)], [(4,21),(5,19),(6,20),(7,24),(8,22),(9,23),(16,25),(17,26),(18,27)]])

G:=TransitiveGroup(27,118);

Matrix representation of S3×He3⋊C2 in GL5(𝔽7)

 6 1 0 0 0 6 0 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 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 3 0
,
 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 1 0 0 0 0 0 4 0 0 0 0 0 2
,
 6 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 3 0

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

S3×He3⋊C2 in GAP, Magma, Sage, TeX

S_3\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("S3xHe3:C2");
// GroupNames label

G:=SmallGroup(324,122);
// by ID

G=gap.SmallGroup(324,122);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,735,2164]);
// Polycyclic

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

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