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G = S3×C3⋊S3  order 108 = 22·33

Direct product of S3 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: S3×C3⋊S3, C32⋊6D6, C33⋊2C22, C3⋊1S32, (C3×S3)⋊S3, C33⋊C2⋊C2, (S3×C32)⋊2C2, C3⋊1(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 12A2B2C3A3B3C3D3E3F3G3H3I6A6B6C6D6E
 size 13927222224444666618
ρ1111111111111111111    trivial
ρ21-11-1111111111-1-1-1-11    linear of order 2
ρ311-1-11111111111111-1    linear of order 2
ρ41-1-11111111111-1-1-1-1-1    linear of order 2
ρ52200-1-1-1222-1-1-12-1-1-10    orthogonal lifted from S3
ρ622002-1-12-1-1-12-1-12-1-10    orthogonal lifted from S3
ρ72-200-1-122-1-1-1-12111-20    orthogonal lifted from D6
ρ82-2002-1-12-1-1-12-11-2110    orthogonal lifted from D6
ρ92200-1-122-1-1-1-12-1-1-120    orthogonal lifted from S3
ρ102-200-1-1-1222-1-1-1-21110    orthogonal lifted from D6
ρ112-200-12-12-1-12-1-111-210    orthogonal lifted from D6
ρ1220-20222-12-1-1-1-100001    orthogonal lifted from D6
ρ132020222-12-1-1-1-10000-1    orthogonal lifted from S3
ρ142200-12-12-1-12-1-1-1-12-10    orthogonal lifted from S3
ρ154000-2-24-2-2111-200000    orthogonal lifted from S32
ρ1640004-2-2-2-211-2100000    orthogonal lifted from S32
ρ174000-2-2-2-24-211100000    orthogonal lifted from S32
ρ184000-24-2-2-21-21100000    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  He3⋊5D6  He3⋊6D6  C33⋊17D6
S3×C3⋊S3 is a maximal quotient of   C33⋊8(C2×C4)  C33⋊6D4  C33⋊7D4  C33⋊8D4  C33⋊4Q8  He3⋊5D6  C33⋊17D6

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

100000
010000
001000
000100
000001
0000-1-1
,
-100000
0-10000
001000
000100
0000-10
000011
,
-1-10000
100000
000100
00-1-100
000010
000001
,
100000
010000
000100
00-1-100
000010
000001
,
-100000
110000
001000
00-1-100
0000-10
00000-1

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
 

Export

Character table of S3×C3⋊S3 in TeX

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