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## G = S3×CSU2(𝔽3)  order 288 = 25·32

### Direct product of S3 and CSU2(𝔽3)

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, C6.5S43C2, (S3×SL2(𝔽3)).C2, C31(C2×CSU2(𝔽3)), (C3×CSU2(𝔽3))⋊2C2, (C3×SL2(𝔽3)).4C22, SmallGroup(288,848)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — S3×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — S3×CSU2(𝔽3)
 Lower central C3×SL2(𝔽3) — S3×CSU2(𝔽3)
 Upper central C1 — C2

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

Smallest permutation representation of S3×CSU2(𝔽3)
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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