G = C6.2PSU3(𝔽2) order 432 = 24·33
2nd non-split extension by C6 of PSU3(𝔽2) acting via PSU3(𝔽2)/C32⋊C4=C2
non-abelian, soluble, monomial
Aliases: C6.2PSU3(𝔽2), C33⋊7(C4⋊C4), C3⋊S3.3D12, C32⋊C4⋊2Dic3, (C3×C6).4Dic6, (C32×C6).3Q8, C32⋊3(C4⋊Dic3), C2.2(C33⋊Q8), C3⋊2(C2.PSU3(𝔽2)), (C3×C32⋊C4)⋊2C4, (C3×C3⋊S3).13D4, (C2×C3⋊S3).16D6, (C6×C32⋊C4).6C2, (C2×C32⋊C4).4S3, C3⋊S3.4(C2×Dic3), (C6×C3⋊S3).13C22, (C2×C33⋊C4).5C2, (C3×C3⋊S3).14(C2×C4), SmallGroup(432,593)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.2PSU3(𝔽2)
G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=a3d2, ab=ba, ac=ca, ad=da, eae-1=a-1, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=d-1 >
Subgroups: 528 in 66 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C33, C32⋊C4, C32⋊C4, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, C6×C3⋊S3, C2.PSU3(𝔽2), C6×C32⋊C4, C2×C33⋊C4, C6.2PSU3(𝔽2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, D12, C2×Dic3, C4⋊Dic3, PSU3(𝔽2), C2.PSU3(𝔽2), C33⋊Q8, C6.2PSU3(𝔽2)
Character table of C6.2PSU3(𝔽2)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 2 | 8 | 8 | 8 | 18 | 18 | 54 | 54 | 54 | 54 | 2 | 8 | 8 | 8 | 18 | 18 | 18 | 18 | 18 | 18 | |
| 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 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -i | -i | i | i | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | -1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | -√3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | -1 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | √3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 8 | -8 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -8 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2.PSU3(𝔽2) |
| ρ20 | 8 | 8 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from PSU3(𝔽2) |
| ρ21 | 8 | -8 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1-3√-3/2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 8 | 8 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 1+3√-3/2 | 1-3√-3/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊Q8 |
| ρ23 | 8 | -8 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1+3√-3/2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 8 | 8 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 1-3√-3/2 | 1+3√-3/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊Q8 |
►On 48 points
Generators in S48
(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 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)
(1 22 10 13)(2 23 11 14)(3 24 12 15)(4 19 7 16)(5 20 8 17)(6 21 9 18)(25 37 34 46)(26 38 35 47)(27 39 36 48)(28 40 31 43)(29 41 32 44)(30 42 33 45)
(1 35 7 29)(2 34 8 28)(3 33 9 27)(4 32 10 26)(5 31 11 25)(6 36 12 30)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(18 48 24 42)
G:=sub<Sym(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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34), (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,37,34,46)(26,38,35,47)(27,39,36,48)(28,40,31,43)(29,41,32,44)(30,42,33,45), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,36,12,30)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,48,24,42)>;
G:=Group( (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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34), (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,37,34,46)(26,38,35,47)(27,39,36,48)(28,40,31,43)(29,41,32,44)(30,42,33,45), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,36,12,30)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,48,24,42) );
G=PermutationGroup([[(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)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(37,41,39),(38,42,40),(43,45,47),(44,46,48)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,35,33),(32,36,34)], [(1,22,10,13),(2,23,11,14),(3,24,12,15),(4,19,7,16),(5,20,8,17),(6,21,9,18),(25,37,34,46),(26,38,35,47),(27,39,36,48),(28,40,31,43),(29,41,32,44),(30,42,33,45)], [(1,35,7,29),(2,34,8,28),(3,33,9,27),(4,32,10,26),(5,31,11,25),(6,36,12,30),(13,47,19,41),(14,46,20,40),(15,45,21,39),(16,44,22,38),(17,43,23,37),(18,48,24,42)]])
Matrix representation of C6.2PSU3(𝔽2) ►in GL10(𝔽13)
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
G:=sub<GL(10,GF(13))| [1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,2,11,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,2,11,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,3,7,6,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,3,0,0,0,0,7,6,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,10,11,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[10,7,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,1,5,8,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[7,10,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,2,1,0,0,0,0,5,8,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,1,5,8,0,0,0,0,0,0,0,1,0,0,0,0] >;
C6.2PSU3(𝔽2) in GAP, Magma, Sage, TeX
C_6._2{\rm PSU}_3({\mathbb F}_2) % in TeX
G:=Group("C6.2PSU(3,2)"); // GroupNames label
G:=SmallGroup(432,593);
// by ID
G=gap.SmallGroup(432,593);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,197,92,2804,1691,298,2693,348,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=1,e^2=a^3*d^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
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