direct product, non-abelian, soluble
Aliases: S3×CSU2(𝔽3), D6.3S4, SL2(𝔽3).4D6, Q8.5S32, (S3×Q8).S3, C2.8(S3×S4), C6.5(C2×S4), (C3×Q8).5D6, C6.5S4⋊3C2, (S3×SL2(𝔽3)).C2, C3⋊1(C2×CSU2(𝔽3)), (C3×CSU2(𝔽3))⋊2C2, (C3×SL2(𝔽3)).4C22, SmallGroup(288,848)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — S3×CSU2(𝔽3) |
| C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — S3×CSU2(𝔽3) |
| C3×SL2(𝔽3) — S3×CSU2(𝔽3) |
Generators and relations for S3×CSU2(𝔽3)
G = < a,b,c,d,e,f | a3=b2=c4=e3=1, d2=f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf-1=c-1, ece-1=cd, fcf-1=c2d, ede-1=c, fef-1=e-1 >
Subgroups: 430 in 83 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, Q16, C2×Q8, C3×S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3⋊Dic3, S3×C6, S3×C8, Dic12, C3⋊Q16, C3×Q16, CSU2(𝔽3), CSU2(𝔽3), C2×SL2(𝔽3), S3×Q8, S3×Q8, S3×Dic3, C3×SL2(𝔽3), S3×Q16, C2×CSU2(𝔽3), C3×CSU2(𝔽3), C6.5S4, S3×SL2(𝔽3), S3×CSU2(𝔽3)
Quotients: C1, C2, C22, S3, D6, S4, S32, CSU2(𝔽3), C2×S4, C2×CSU2(𝔽3), S3×S4, S3×CSU2(𝔽3)
Character table of S3×CSU2(𝔽3)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | |
| size | 1 | 1 | 3 | 3 | 2 | 8 | 16 | 6 | 12 | 18 | 36 | 2 | 8 | 16 | 24 | 24 | 6 | 6 | 18 | 18 | 12 | 24 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | 0 | -2 | 0 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | -1 | √2 | -√2 | -√2 | √2 | 0 | 0 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -1 | 1 | -√2 | √2 | -√2 | √2 | 0 | 0 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | -1 | -√2 | √2 | √2 | -√2 | 0 | 0 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -1 | 1 | √2 | -√2 | √2 | -√2 | 0 | 0 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ13 | 3 | 3 | -3 | -3 | 3 | 0 | 0 | -1 | 1 | 1 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | -3 | -3 | 3 | 0 | 0 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | 1 | -1 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ17 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ18 | 4 | -4 | -4 | 4 | 4 | 1 | 1 | 0 | 0 | 0 | 0 | -4 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ19 | 4 | -4 | 4 | -4 | 4 | 1 | 1 | 0 | 0 | 0 | 0 | -4 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ22 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 1 | -1 | 1 | 1 | orthogonal lifted from S3×S4 |
| ρ23 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from S3×S4 |
| ρ24 | 8 | -8 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 14 22)(2 15 23)(3 16 24)(4 13 21)(5 35 43)(6 36 44)(7 33 41)(8 34 42)(9 17 25)(10 18 26)(11 19 27)(12 20 28)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(5 43)(6 44)(7 41)(8 42)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(37 45)(38 46)(39 47)(40 48)
(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 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 11 3 9)(2 10 4 12)(5 47 7 45)(6 46 8 48)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)(29 35 31 33)(30 34 32 36)(37 43 39 41)(38 42 40 44)
(2 11 10)(4 9 12)(5 8 46)(6 48 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)(30 35 34)(32 33 36)(38 43 42)(40 41 44)
(1 29 3 31)(2 33 4 35)(5 23 7 21)(6 28 8 26)(9 30 11 32)(10 36 12 34)(13 43 15 41)(14 37 16 39)(17 38 19 40)(18 44 20 42)(22 45 24 47)(25 46 27 48)
G:=sub<Sym(48)| (1,14,22)(2,15,23)(3,16,24)(4,13,21)(5,35,43)(6,36,44)(7,33,41)(8,34,42)(9,17,25)(10,18,26)(11,19,27)(12,20,28)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (5,43)(6,44)(7,41)(8,42)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(37,45)(38,46)(39,47)(40,48), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,11,3,9)(2,10,4,12)(5,47,7,45)(6,46,8,48)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(29,35,31,33)(30,34,32,36)(37,43,39,41)(38,42,40,44), (2,11,10)(4,9,12)(5,8,46)(6,48,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(30,35,34)(32,33,36)(38,43,42)(40,41,44), (1,29,3,31)(2,33,4,35)(5,23,7,21)(6,28,8,26)(9,30,11,32)(10,36,12,34)(13,43,15,41)(14,37,16,39)(17,38,19,40)(18,44,20,42)(22,45,24,47)(25,46,27,48)>;
G:=Group( (1,14,22)(2,15,23)(3,16,24)(4,13,21)(5,35,43)(6,36,44)(7,33,41)(8,34,42)(9,17,25)(10,18,26)(11,19,27)(12,20,28)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (5,43)(6,44)(7,41)(8,42)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(37,45)(38,46)(39,47)(40,48), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,11,3,9)(2,10,4,12)(5,47,7,45)(6,46,8,48)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(29,35,31,33)(30,34,32,36)(37,43,39,41)(38,42,40,44), (2,11,10)(4,9,12)(5,8,46)(6,48,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(30,35,34)(32,33,36)(38,43,42)(40,41,44), (1,29,3,31)(2,33,4,35)(5,23,7,21)(6,28,8,26)(9,30,11,32)(10,36,12,34)(13,43,15,41)(14,37,16,39)(17,38,19,40)(18,44,20,42)(22,45,24,47)(25,46,27,48) );
G=PermutationGroup([[(1,14,22),(2,15,23),(3,16,24),(4,13,21),(5,35,43),(6,36,44),(7,33,41),(8,34,42),(9,17,25),(10,18,26),(11,19,27),(12,20,28),(29,37,45),(30,38,46),(31,39,47),(32,40,48)], [(5,43),(6,44),(7,41),(8,42),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28),(37,45),(38,46),(39,47),(40,48)], [(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,11,3,9),(2,10,4,12),(5,47,7,45),(6,46,8,48),(13,20,15,18),(14,19,16,17),(21,28,23,26),(22,27,24,25),(29,35,31,33),(30,34,32,36),(37,43,39,41),(38,42,40,44)], [(2,11,10),(4,9,12),(5,8,46),(6,48,7),(13,17,20),(15,19,18),(21,25,28),(23,27,26),(30,35,34),(32,33,36),(38,43,42),(40,41,44)], [(1,29,3,31),(2,33,4,35),(5,23,7,21),(6,28,8,26),(9,30,11,32),(10,36,12,34),(13,43,15,41),(14,37,16,39),(17,38,19,40),(18,44,20,42),(22,45,24,47),(25,46,27,48)]])
Matrix representation of S3×CSU2(𝔽3) ►in GL4(𝔽7) generated by
| 0 | 0 | 5 | 2 |
| 5 | 3 | 0 | 3 |
| 2 | 5 | 0 | 3 |
| 5 | 5 | 1 | 2 |
| 1 | 0 | 0 | 2 |
| 1 | 0 | 3 | 4 |
| 2 | 5 | 0 | 3 |
| 0 | 0 | 0 | 6 |
| 6 | 1 | 1 | 1 |
| 1 | 0 | 4 | 1 |
| 6 | 6 | 3 | 2 |
| 5 | 2 | 1 | 5 |
| 3 | 0 | 1 | 2 |
| 1 | 0 | 6 | 6 |
| 2 | 2 | 5 | 3 |
| 1 | 6 | 3 | 6 |
| 0 | 0 | 5 | 2 |
| 2 | 3 | 0 | 4 |
| 2 | 2 | 0 | 3 |
| 5 | 2 | 1 | 2 |
| 6 | 2 | 0 | 5 |
| 0 | 0 | 3 | 4 |
| 1 | 1 | 3 | 5 |
| 1 | 6 | 3 | 5 |
G:=sub<GL(4,GF(7))| [0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[1,1,2,0,0,0,5,0,0,3,0,0,2,4,3,6],[6,1,6,5,1,0,6,2,1,4,3,1,1,1,2,5],[3,1,2,1,0,0,2,6,1,6,5,3,2,6,3,6],[0,2,2,5,0,3,2,2,5,0,0,1,2,4,3,2],[6,0,1,1,2,0,1,6,0,3,3,3,5,4,5,5] >;
S3×CSU2(𝔽3) in GAP, Magma, Sage, TeX
S_3\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("S3xCSU(2,3)"); // GroupNames label
G:=SmallGroup(288,848);
// by ID
G=gap.SmallGroup(288,848);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,93,675,1271,1908,172,768,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^4=e^3=1,d^2=f^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*d*f^-1=c^-1,e*c*e^-1=c*d,f*c*f^-1=c^2*d,e*d*e^-1=c,f*e*f^-1=e^-1>;
// generators/relations
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