direct product, non-abelian, soluble
Aliases: C3×CSU2(𝔽3), C6.8S4, SL2(𝔽3).C6, Q8.(C3×S3), C2.2(C3×S4), (C3×Q8).4S3, (C3×SL2(𝔽3)).2C2, SmallGroup(144,121)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×CSU2(𝔽3) |
| SL2(𝔽3) — C3×CSU2(𝔽3) |
Generators and relations for C3×CSU2(𝔽3)
G = < a,b,c,d,e | a3=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >
Character table of C3×CSU2(𝔽3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 6 | 12 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 12 | 12 | 6 | 6 | 6 | 6 | |
| 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 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -√2 | √2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | √2 | -√2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | 2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 2 | 0 | -1-√-3 | -1+√-3 | -1 | ζ6 | ζ65 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ11 | 2 | 2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 2 | 0 | -1+√-3 | -1-√-3 | -1 | ζ65 | ζ6 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ12 | 2 | -2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 1+√-3 | 1-√-3 | 1 | ζ32 | ζ3 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ83ζ32-ζ8ζ32 | ζ87ζ32-ζ85ζ32 | ζ87ζ3-ζ85ζ3 | ζ83ζ3-ζ8ζ3 | complex faithful |
| ρ13 | 2 | -2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 1+√-3 | 1-√-3 | 1 | ζ32 | ζ3 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ87ζ32-ζ85ζ32 | ζ83ζ32-ζ8ζ32 | ζ83ζ3-ζ8ζ3 | ζ87ζ3-ζ85ζ3 | complex faithful |
| ρ14 | 2 | -2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 1-√-3 | 1+√-3 | 1 | ζ3 | ζ32 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ83ζ3-ζ8ζ3 | ζ87ζ3-ζ85ζ3 | ζ87ζ32-ζ85ζ32 | ζ83ζ32-ζ8ζ32 | complex faithful |
| ρ15 | 2 | -2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 1-√-3 | 1+√-3 | 1 | ζ3 | ζ32 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ87ζ3-ζ85ζ3 | ζ83ζ3-ζ8ζ3 | ζ83ζ32-ζ8ζ32 | ζ87ζ32-ζ85ζ32 | complex faithful |
| ρ16 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×S4 |
| ρ19 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | complex lifted from C3×S4 |
| ρ20 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×S4 |
| ρ21 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | complex lifted from C3×S4 |
| ρ22 | 4 | -4 | 4 | 4 | 1 | 1 | 1 | 0 | 0 | -4 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ23 | 4 | -4 | -2-2√-3 | -2+2√-3 | ζ3 | ζ32 | 1 | 0 | 0 | 2+2√-3 | 2-2√-3 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | -2+2√-3 | -2-2√-3 | ζ32 | ζ3 | 1 | 0 | 0 | 2-2√-3 | 2+2√-3 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 34 39)(6 35 40)(7 36 37)(8 33 38)(9 17 21)(10 18 22)(11 19 23)(12 20 24)(29 44 48)(30 41 45)(31 42 46)(32 43 47)
(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 45 7 47)(6 48 8 46)(13 19 15 17)(14 18 16 20)(21 25 23 27)(22 28 24 26)(29 33 31 35)(30 36 32 34)(37 43 39 41)(38 42 40 44)
(1 25 13)(2 23 18)(3 27 15)(4 21 20)(5 32 40)(6 34 43)(7 30 38)(8 36 41)(9 24 16)(10 26 19)(11 22 14)(12 28 17)(29 44 48)(31 42 46)(33 37 45)(35 39 47)
(1 29 3 31)(2 35 4 33)(5 24 7 22)(6 28 8 26)(9 30 11 32)(10 34 12 36)(13 44 15 42)(14 40 16 38)(17 41 19 43)(18 39 20 37)(21 45 23 47)(25 48 27 46)
G:=sub<Sym(48)| (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,34,39)(6,35,40)(7,36,37)(8,33,38)(9,17,21)(10,18,22)(11,19,23)(12,20,24)(29,44,48)(30,41,45)(31,42,46)(32,43,47), (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)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,25,23,27)(22,28,24,26)(29,33,31,35)(30,36,32,34)(37,43,39,41)(38,42,40,44), (1,25,13)(2,23,18)(3,27,15)(4,21,20)(5,32,40)(6,34,43)(7,30,38)(8,36,41)(9,24,16)(10,26,19)(11,22,14)(12,28,17)(29,44,48)(31,42,46)(33,37,45)(35,39,47), (1,29,3,31)(2,35,4,33)(5,24,7,22)(6,28,8,26)(9,30,11,32)(10,34,12,36)(13,44,15,42)(14,40,16,38)(17,41,19,43)(18,39,20,37)(21,45,23,47)(25,48,27,46)>;
G:=Group( (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,34,39)(6,35,40)(7,36,37)(8,33,38)(9,17,21)(10,18,22)(11,19,23)(12,20,24)(29,44,48)(30,41,45)(31,42,46)(32,43,47), (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)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,25,23,27)(22,28,24,26)(29,33,31,35)(30,36,32,34)(37,43,39,41)(38,42,40,44), (1,25,13)(2,23,18)(3,27,15)(4,21,20)(5,32,40)(6,34,43)(7,30,38)(8,36,41)(9,24,16)(10,26,19)(11,22,14)(12,28,17)(29,44,48)(31,42,46)(33,37,45)(35,39,47), (1,29,3,31)(2,35,4,33)(5,24,7,22)(6,28,8,26)(9,30,11,32)(10,34,12,36)(13,44,15,42)(14,40,16,38)(17,41,19,43)(18,39,20,37)(21,45,23,47)(25,48,27,46) );
G=PermutationGroup([[(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,34,39),(6,35,40),(7,36,37),(8,33,38),(9,17,21),(10,18,22),(11,19,23),(12,20,24),(29,44,48),(30,41,45),(31,42,46),(32,43,47)], [(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),(5,45,7,47),(6,48,8,46),(13,19,15,17),(14,18,16,20),(21,25,23,27),(22,28,24,26),(29,33,31,35),(30,36,32,34),(37,43,39,41),(38,42,40,44)], [(1,25,13),(2,23,18),(3,27,15),(4,21,20),(5,32,40),(6,34,43),(7,30,38),(8,36,41),(9,24,16),(10,26,19),(11,22,14),(12,28,17),(29,44,48),(31,42,46),(33,37,45),(35,39,47)], [(1,29,3,31),(2,35,4,33),(5,24,7,22),(6,28,8,26),(9,30,11,32),(10,34,12,36),(13,44,15,42),(14,40,16,38),(17,41,19,43),(18,39,20,37),(21,45,23,47),(25,48,27,46)]])
C3×CSU2(𝔽3) is a maximal subgroup of
CSU2(𝔽3)⋊S3 Dic3.5S4 D6.2S4 C32⋊2CSU2(𝔽3)
C3×CSU2(𝔽3) is a maximal quotient of C32.CSU2(𝔽3) C32⋊CSU2(𝔽3)
Matrix representation of C3×CSU2(𝔽3) ►in GL2(𝔽7) generated by
| 2 | 0 |
| 0 | 2 |
| 6 | 5 |
| 1 | 1 |
| 5 | 6 |
| 5 | 2 |
| 2 | 0 |
| 5 | 4 |
| 2 | 5 |
| 6 | 5 |
G:=sub<GL(2,GF(7))| [2,0,0,2],[6,1,5,1],[5,5,6,2],[2,5,0,4],[2,6,5,5] >;
C3×CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_3\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C3xCSU(2,3)"); // GroupNames label
G:=SmallGroup(144,121);
// by ID
G=gap.SmallGroup(144,121);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,218,867,447,117,544,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^3=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=b*c,e*b*e^-1=b^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C3×CSU2(𝔽3) in TeX
Character table of C3×CSU2(𝔽3) in TeX