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G = S3×D11  order 132 = 22·3·11

Direct product of S3 and D11

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×D11, D33⋊C2, C3⋊1D22, C11⋊1D6, C33⋊C22, (S3×C11)⋊C2, (C3×D11)⋊C2, SmallGroup(132,5)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — S3×D11
Chief series
C1 — C11 — C33 — C3×D11 — S3×D11
Lower central
C33 — S3×D11
Upper central
C1

Generators and relations for S3×D11
 G = < a,b,c,d | a3=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

C1
3C2
11C2
33C2
C3
C11
33C22
S3
11C6
11S3
D11
3C22
3D11
C33
11D6
3D22
C3×D11
D33
S3×C11
S3×D11

Character table of S3×D11

 class 12A2B2C3611A11B11C11D11E22A22B22C22D22E33A33B33C33D33E
 size 131133222222226666644444
ρ1111111111111111111111    trivial
ρ21-11-11111111-1-1-1-1-111111    linear of order 2
ρ31-1-111-111111-1-1-1-1-111111    linear of order 2
ρ411-1-11-1111111111111111    linear of order 2
ρ520-20-112222200000-1-1-1-1-1    orthogonal lifted from D6
ρ62020-1-12222200000-1-1-1-1-1    orthogonal lifted from S3
ρ72-20020ζ117+ζ114ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ118+ζ113-ζ116-ζ115-ζ1110-ζ11-ζ119-ζ112-ζ118-ζ113-ζ117-ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114    orthogonal lifted from D22
ρ8220020ζ119+ζ112ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112    orthogonal lifted from D11
ρ9220020ζ116+ζ115ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115    orthogonal lifted from D11
ρ102-20020ζ1110+ζ11ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ119+ζ112-ζ117-ζ114-ζ118-ζ113-ζ116-ζ115-ζ119-ζ112-ζ1110-ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11    orthogonal lifted from D22
ρ11220020ζ118+ζ113ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113    orthogonal lifted from D11
ρ122-20020ζ119+ζ112ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ117+ζ114-ζ118-ζ113-ζ116-ζ115-ζ1110-ζ11-ζ117-ζ114-ζ119-ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112    orthogonal lifted from D22
ρ132-20020ζ118+ζ113ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ116+ζ115-ζ1110-ζ11-ζ119-ζ112-ζ117-ζ114-ζ116-ζ115-ζ118-ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113    orthogonal lifted from D22
ρ14220020ζ117+ζ114ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114    orthogonal lifted from D11
ρ152-20020ζ116+ζ115ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ1110+ζ11-ζ119-ζ112-ζ117-ζ114-ζ118-ζ113-ζ1110-ζ11-ζ116-ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115    orthogonal lifted from D22
ρ16220020ζ1110+ζ11ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11    orthogonal lifted from D11
ρ174000-202ζ119+2ζ1122ζ116+2ζ1152ζ1110+2ζ112ζ118+2ζ1132ζ117+2ζ11400000-ζ118-ζ113-ζ116-ζ115-ζ1110-ζ11-ζ117-ζ114-ζ119-ζ112    orthogonal faithful
ρ184000-202ζ116+2ζ1152ζ117+2ζ1142ζ118+2ζ1132ζ119+2ζ1122ζ1110+2ζ1100000-ζ119-ζ112-ζ117-ζ114-ζ118-ζ113-ζ1110-ζ11-ζ116-ζ115    orthogonal faithful
ρ194000-202ζ118+2ζ1132ζ119+2ζ1122ζ117+2ζ1142ζ1110+2ζ112ζ116+2ζ11500000-ζ1110-ζ11-ζ119-ζ112-ζ117-ζ114-ζ116-ζ115-ζ118-ζ113    orthogonal faithful
ρ204000-202ζ1110+2ζ112ζ118+2ζ1132ζ116+2ζ1152ζ117+2ζ1142ζ119+2ζ11200000-ζ117-ζ114-ζ118-ζ113-ζ116-ζ115-ζ119-ζ112-ζ1110-ζ11    orthogonal faithful
ρ214000-202ζ117+2ζ1142ζ1110+2ζ112ζ119+2ζ1122ζ116+2ζ1152ζ118+2ζ11300000-ζ116-ζ115-ζ1110-ζ11-ζ119-ζ112-ζ118-ζ113-ζ117-ζ114    orthogonal faithful

Smallest permutation representation of S3×D11
►On 33 points
Generators in S33
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)

(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)

(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 28 29 30 31 32 33)

(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)
 
G:=sub<Sym(33)| (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)>;
 
G:=Group( (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32) );
 
G=PermutationGroup([[(1,21,32),(2,22,33),(3,12,23),(4,13,24),(5,14,25),(6,15,26),(7,16,27),(8,17,28),(9,18,29),(10,19,30),(11,20,31)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33)], [(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32)]])
 

S3×D11 is a maximal subgroup of   D33⋊S3
S3×D11 is a maximal quotient of   D33⋊C4  C33⋊D4  C3⋊D44  C11⋊D12  C33⋊Q8  D33⋊S3

Matrix representation of S3×D11 ►in GL4(𝔽67) generated by

1000
0100
00066
00166
,
66000
06600
0001
0010
,
0100
662400
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(67))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,66,66],[66,0,0,0,0,66,0,0,0,0,0,1,0,0,1,0],[0,66,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
 

S3×D11 in GAP, Magma, Sage, TeX 

S_3\times D_{11}
 
% in TeX
 
G:=Group("S3xD11");
 
// GroupNames label
 
G:=SmallGroup(132,5);
 
// by ID
 
G=gap.SmallGroup(132,5);
 
# by ID
 
G:=PCGroup([4,-2,-2,-3,-11,54,1923]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of S3×D11 in TeX
Character table of S3×D11 in TeX

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