Copied to
clipboard

G = S3×SL2(𝔽3)  order 144 = 24·32

Direct product of S3 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: S3×SL2(𝔽3), D6.A4, (S3×Q8)⋊C3, (C3×Q8)⋊C6, C2.3(S3×A4), C6.2(C2×A4), Q82(C3×S3), C3⋊(C2×SL2(𝔽3)), (C3×SL2(𝔽3))⋊3C2, SL2(ℤ/6ℤ), SmallGroup(144,128)

Series: Derived Chief Lower central Upper central

C1C2C3×Q8 — S3×SL2(𝔽3)
C1C2C6C3×Q8C3×SL2(𝔽3) — S3×SL2(𝔽3)
C3×Q8 — S3×SL2(𝔽3)
C1C2

Generators and relations for S3×SL2(𝔽3)
 G = < a,b,c,d,e | a3=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

3C2
3C2
4C3
8C3
3C22
3C4
9C4
4C6
8C6
12C6
12C6
4C32
9C2×C4
9Q8
3C12
3Dic3
12C2×C6
4C3×C6
4C3×S3
4C3×S3
3C2×Q8
2SL2(𝔽3)
3C4×S3
3Dic6
4S3×C6
3C2×SL2(𝔽3)

Character table of S3×SL2(𝔽3)

 class 12A2B2C3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I12
 size 113324488618244881212121212
ρ1111111111111111111111    trivial
ρ211-1-1111111-111111-1-1-1-11    linear of order 2
ρ311111ζ32ζ3ζ3ζ32111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ31    linear of order 3
ρ411-1-11ζ32ζ3ζ3ζ321-11ζ32ζ3ζ3ζ32ζ6ζ65ζ6ζ651    linear of order 6
ρ511111ζ3ζ32ζ32ζ3111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ321    linear of order 3
ρ611-1-11ζ3ζ32ζ32ζ31-11ζ3ζ32ζ32ζ3ζ65ζ6ζ65ζ61    linear of order 6
ρ72200-122-1-120-122-1-10000-1    orthogonal lifted from S3
ρ82-22-22-1-1-1-100-21111-1-1110    symplectic lifted from SL2(𝔽3), Schur index 2
ρ92-2-222-1-1-1-100-2111111-1-10    symplectic lifted from SL2(𝔽3), Schur index 2
ρ102-22-22ζ65ζ6ζ6ζ6500-2ζ3ζ32ζ32ζ3ζ65ζ6ζ3ζ320    complex lifted from SL2(𝔽3)
ρ112200-1-1+-3-1--3ζ6ζ6520-1-1+-3-1--3ζ6ζ650000-1    complex lifted from C3×S3
ρ122-22-22ζ6ζ65ζ65ζ600-2ζ32ζ3ζ3ζ32ζ6ζ65ζ32ζ30    complex lifted from SL2(𝔽3)
ρ132200-1-1--3-1+-3ζ65ζ620-1-1--3-1+-3ζ65ζ60000-1    complex lifted from C3×S3
ρ142-2-222ζ6ζ65ζ65ζ600-2ζ32ζ3ζ3ζ32ζ32ζ3ζ6ζ650    complex lifted from SL2(𝔽3)
ρ152-2-222ζ65ζ6ζ6ζ6500-2ζ3ζ32ζ32ζ3ζ3ζ32ζ65ζ60    complex lifted from SL2(𝔽3)
ρ1633-3-330000-11300000000-1    orthogonal lifted from C2×A4
ρ17333330000-1-1300000000-1    orthogonal lifted from A4
ρ184-400-2-2-21100222-1-100000    symplectic faithful, Schur index 2
ρ194-400-21+-31--3ζ3ζ32002-1--3-1+-3ζ65ζ600000    complex faithful
ρ204-400-21--31+-3ζ32ζ3002-1+-3-1--3ζ6ζ6500000    complex faithful
ρ216600-30000-20-3000000001    orthogonal lifted from S3×A4

Permutation representations of S3×SL2(𝔽3)
On 24 points - transitive group 24T249
Generators in S24
(1 11 16)(2 12 13)(3 9 14)(4 10 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(1 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 16)(10 13)(11 14)(12 15)(21 23)(22 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 24 3 22)(2 23 4 21)(5 13 7 15)(6 16 8 14)(9 20 11 18)(10 19 12 17)
(2 23 24)(4 21 22)(5 6 15)(7 8 13)(10 19 20)(12 17 18)

G:=sub<Sym(24)| (1,11,16)(2,12,13)(3,9,14)(4,10,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,16)(10,13)(11,14)(12,15)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,13,7,15)(6,16,8,14)(9,20,11,18)(10,19,12,17), (2,23,24)(4,21,22)(5,6,15)(7,8,13)(10,19,20)(12,17,18)>;

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

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

G:=TransitiveGroup(24,249);

S3×SL2(𝔽3) is a maximal subgroup of   D6.S4  D6.2S4  SL2(𝔽3).11D6  D12.A4  D18.A4  Q8⋊He3⋊C2
S3×SL2(𝔽3) is a maximal quotient of   D18.A4  Q8⋊He3⋊C2

Matrix representation of S3×SL2(𝔽3) in GL4(𝔽5) generated by

3221
4411
1333
2343
,
1142
4022
0102
3024
,
0412
0311
1233
4344
,
1314
1204
3401
3312
,
3300
4100
1101
2044
G:=sub<GL(4,GF(5))| [3,4,1,2,2,4,3,3,2,1,3,4,1,1,3,3],[1,4,0,3,1,0,1,0,4,2,0,2,2,2,2,4],[0,0,1,4,4,3,2,3,1,1,3,4,2,1,3,4],[1,1,3,3,3,2,4,3,1,0,0,1,4,4,1,2],[3,4,1,2,3,1,1,0,0,0,0,4,0,0,1,4] >;

S3×SL2(𝔽3) in GAP, Magma, Sage, TeX

S_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("S3xSL(2,3)");
// GroupNames label

G:=SmallGroup(144,128);
// by ID

G=gap.SmallGroup(144,128);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-2,170,230,81,351,165,1444]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^3=1,d^2=c^2,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,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

Export

Subgroup lattice of S3×SL2(𝔽3) in TeX
Character table of S3×SL2(𝔽3) in TeX

׿
×
𝔽