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## G = S3×C3⋊S3order 108 = 22·33

### Direct product of S3 and C3⋊S3

Aliases: S3×C3⋊S3, C326D6, C332C22, C31S32, (C3×S3)⋊S3, C33⋊C2⋊C2, (S3×C32)⋊2C2, C31(C2×C3⋊S3), (C3×C3⋊S3)⋊2C2, SmallGroup(108,39)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C3⋊S3
G = < a,b,c,d,e | a3=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 368 in 76 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, C3, C22, S3, S3, C6, C32, C32, C32, D6, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C33, S32, C2×C3⋊S3, S3×C32, C3×C3⋊S3, C33⋊C2, S3×C3⋊S3
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S32, C2×C3⋊S3, S3×C3⋊S3

Character table of S3×C3⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 6E size 1 3 9 27 2 2 2 2 2 4 4 4 4 6 6 6 6 18 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 -1 -1 0 orthogonal lifted from S3 ρ6 2 2 0 0 2 -1 -1 2 -1 -1 -1 2 -1 -1 2 -1 -1 0 orthogonal lifted from S3 ρ7 2 -2 0 0 -1 -1 2 2 -1 -1 -1 -1 2 1 1 1 -2 0 orthogonal lifted from D6 ρ8 2 -2 0 0 2 -1 -1 2 -1 -1 -1 2 -1 1 -2 1 1 0 orthogonal lifted from D6 ρ9 2 2 0 0 -1 -1 2 2 -1 -1 -1 -1 2 -1 -1 -1 2 0 orthogonal lifted from S3 ρ10 2 -2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -2 1 1 1 0 orthogonal lifted from D6 ρ11 2 -2 0 0 -1 2 -1 2 -1 -1 2 -1 -1 1 1 -2 1 0 orthogonal lifted from D6 ρ12 2 0 -2 0 2 2 2 -1 2 -1 -1 -1 -1 0 0 0 0 1 orthogonal lifted from D6 ρ13 2 0 2 0 2 2 2 -1 2 -1 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ14 2 2 0 0 -1 2 -1 2 -1 -1 2 -1 -1 -1 -1 2 -1 0 orthogonal lifted from S3 ρ15 4 0 0 0 -2 -2 4 -2 -2 1 1 1 -2 0 0 0 0 0 orthogonal lifted from S32 ρ16 4 0 0 0 4 -2 -2 -2 -2 1 1 -2 1 0 0 0 0 0 orthogonal lifted from S32 ρ17 4 0 0 0 -2 -2 -2 -2 4 -2 1 1 1 0 0 0 0 0 orthogonal lifted from S32 ρ18 4 0 0 0 -2 4 -2 -2 -2 1 -2 1 1 0 0 0 0 0 orthogonal lifted from S32

Permutation representations of S3×C3⋊S3
On 18 points - transitive group 18T58
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 16)(5 18)(6 17)(7 13)(8 15)(9 14)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)
(1 6 8)(2 4 9)(3 5 7)(10 17 15)(11 18 13)(12 16 14)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 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)(4,16)(5,18)(6,17)(7,13)(8,15)(9,14), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,6,8)(2,4,9)(3,5,7)(10,17,15)(11,18,13)(12,16,14), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)>;

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

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

G:=TransitiveGroup(18,58);

On 27 points - transitive group 27T34
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)
(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 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)
(4 7)(5 8)(6 9)(10 19)(11 20)(12 21)(13 25)(14 26)(15 27)(16 22)(17 23)(18 24)

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

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

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

G:=TransitiveGroup(27,34);

S3×C3⋊S3 is a maximal subgroup of   S33  He3⋊D6  He35D6  He36D6  C3317D6
S3×C3⋊S3 is a maximal quotient of   C338(C2×C4)  C336D4  C337D4  C338D4  C334Q8  He35D6  C3317D6

Matrix representation of S3×C3⋊S3 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 1 1
,
 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

S3×C3⋊S3 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes S_3
% in TeX

G:=Group("S3xC3:S3");
// GroupNames label

G:=SmallGroup(108,39);
// by ID

G=gap.SmallGroup(108,39);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,67,248,1804]);
// Polycyclic

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

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