Copied to
clipboard

G = S3×D11order 132 = 22·3·11

Direct product of S3 and D11

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

Aliases: S3×D11, D33⋊C2, C31D22, C111D6, C33⋊C22, (S3×C11)⋊C2, (C3×D11)⋊C2, SmallGroup(132,5)

Series: Derived Chief Lower central Upper central

C1C33 — S3×D11
C1C11C33C3×D11 — S3×D11
C33 — S3×D11
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 >

3C2
11C2
33C2
33C22
11C6
11S3
3C22
3D11
11D6
3D22

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
ρ72-20020ζ117114ζ111011ζ119112ζ116115ζ118113116115111011119112118113117114ζ116115ζ111011ζ119112ζ118113ζ117114    orthogonal lifted from D22
ρ8220020ζ119112ζ116115ζ111011ζ118113ζ117114ζ118113ζ116115ζ111011ζ117114ζ119112ζ118113ζ116115ζ111011ζ117114ζ119112    orthogonal lifted from D11
ρ9220020ζ116115ζ117114ζ118113ζ119112ζ111011ζ119112ζ117114ζ118113ζ111011ζ116115ζ119112ζ117114ζ118113ζ111011ζ116115    orthogonal lifted from D11
ρ102-20020ζ111011ζ118113ζ116115ζ117114ζ119112117114118113116115119112111011ζ117114ζ118113ζ116115ζ119112ζ111011    orthogonal lifted from D22
ρ11220020ζ118113ζ119112ζ117114ζ111011ζ116115ζ111011ζ119112ζ117114ζ116115ζ118113ζ111011ζ119112ζ117114ζ116115ζ118113    orthogonal lifted from D11
ρ122-20020ζ119112ζ116115ζ111011ζ118113ζ117114118113116115111011117114119112ζ118113ζ116115ζ111011ζ117114ζ119112    orthogonal lifted from D22
ρ132-20020ζ118113ζ119112ζ117114ζ111011ζ116115111011119112117114116115118113ζ111011ζ119112ζ117114ζ116115ζ118113    orthogonal lifted from D22
ρ14220020ζ117114ζ111011ζ119112ζ116115ζ118113ζ116115ζ111011ζ119112ζ118113ζ117114ζ116115ζ111011ζ119112ζ118113ζ117114    orthogonal lifted from D11
ρ152-20020ζ116115ζ117114ζ118113ζ119112ζ111011119112117114118113111011116115ζ119112ζ117114ζ118113ζ111011ζ116115    orthogonal lifted from D22
ρ16220020ζ111011ζ118113ζ116115ζ117114ζ119112ζ117114ζ118113ζ116115ζ119112ζ111011ζ117114ζ118113ζ116115ζ119112ζ111011    orthogonal lifted from D11
ρ174000-20119+2ζ112116+2ζ1151110+2ζ11118+2ζ113117+2ζ11400000118113116115111011117114119112    orthogonal faithful
ρ184000-20116+2ζ115117+2ζ114118+2ζ113119+2ζ1121110+2ζ1100000119112117114118113111011116115    orthogonal faithful
ρ194000-20118+2ζ113119+2ζ112117+2ζ1141110+2ζ11116+2ζ11500000111011119112117114116115118113    orthogonal faithful
ρ204000-201110+2ζ11118+2ζ113116+2ζ115117+2ζ114119+2ζ11200000117114118113116115119112111011    orthogonal faithful
ρ214000-20117+2ζ1141110+2ζ11119+2ζ112116+2ζ115118+2ζ11300000116115111011119112118113117114    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

׿
×
𝔽