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G = S3xSL2(F3)  order 144 = 24·32

Direct product of S3 and SL2(F3)

direct product, non-abelian, soluble

Aliases: S3xSL2(F3), D6.A4, (S3xQ8):C3, (C3xQ8):C6, C2.3(S3xA4), C6.2(C2xA4), Q8:2(C3xS3), C3:(C2xSL2(F3)), (C3xSL2(F3)):3C2, SL2(Z/6Z), SmallGroup(144,128)

Series: Derived Chief Lower central Upper central

C1C2C3xQ8 — S3xSL2(F3)
C1C2C6C3xQ8C3xSL2(F3) — S3xSL2(F3)
C3xQ8 — S3xSL2(F3)
C1C2

Generators and relations for S3xSL2(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 >

Subgroups: 152 in 39 conjugacy classes, 13 normal (11 characteristic)
Quotients: C1, C2, C3, S3, C6, A4, C3xS3, SL2(F3), C2xA4, C2xSL2(F3), S3xA4, S3xSL2(F3)
3C2
3C2
4C3
8C3
3C22
3C4
9C4
4C6
8C6
12C6
12C6
4C32
9C2xC4
9Q8
3C12
3Dic3
12C2xC6
4C3xC6
4C3xS3
4C3xS3
3C2xQ8
2SL2(F3)
3C4xS3
3Dic6
4S3xC6
3C2xSL2(F3)

Character table of S3xSL2(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 C3xS3
ρ122-22-22ζ6ζ65ζ65ζ600-2ζ32ζ3ζ3ζ32ζ6ζ65ζ32ζ30    complex lifted from SL2(F3)
ρ132200-1-1--3-1+-3ζ65ζ620-1-1--3-1+-3ζ65ζ60000-1    complex lifted from C3xS3
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 C2xA4
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 S3xA4

Permutation representations of S3xSL2(F3)
On 24 points - transitive group 24T249
Generators in S24
(1 9 16)(2 10 13)(3 11 14)(4 12 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 14)(10 15)(11 16)(12 13)(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 18 11 20)(10 17 12 19)
(2 23 24)(4 21 22)(5 6 15)(7 8 13)(10 17 18)(12 19 20)

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

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

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

G:=TransitiveGroup(24,249);

S3xSL2(F3) is a maximal subgroup of   D6.S4  D6.2S4  SL2(F3).11D6  D12.A4  D18.A4  Q8:He3:C2
S3xSL2(F3) is a maximal quotient of   D18.A4  Q8:He3:C2

Matrix representation of S3xSL2(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] >;

S3xSL2(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 S3xSL2(F3) in TeX
Character table of S3xSL2(F3) in TeX

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