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## G = C6.PSU3(𝔽2)  order 432 = 24·33

### 1st non-split extension by C6 of PSU3(𝔽2) acting via PSU3(𝔽2)/C32⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C6.PSU3(𝔽2)
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C33⋊C4 — C6.PSU3(𝔽2)
 Lower central C33 — C3×C3⋊S3 — C6.PSU3(𝔽2)
 Upper central C1 — C2

Generators and relations for C6.PSU3(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=a3d-1 >

Subgroups: 528 in 66 conjugacy classes, 21 normal (19 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, S3×C6, C2×C3⋊S3, Dic3⋊C4, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, C33⋊C4, C6×C3⋊S3, C2.PSU3(𝔽2), C6×C32⋊C4, C2×C33⋊C4, C6.PSU3(𝔽2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, Dic3⋊C4, PSU3(𝔽2), C2.PSU3(𝔽2), C33⋊Q8, C6.PSU3(𝔽2)

Character table of C6.PSU3(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 -i i i 1 -1 -i -1 -1 -1 -1 1 -1 i i -i -i linear of order 4 ρ6 1 -1 -1 1 1 1 1 1 -i i -i -1 1 i -1 -1 -1 -1 1 -1 i i -i -i linear of order 4 ρ7 1 -1 -1 1 1 1 1 1 i -i i -1 1 -i -1 -1 -1 -1 1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 i -i -i 1 -1 i -1 -1 -1 -1 1 -1 -i -i i i 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 -2 -2 0 0 0 0 -1 -1 -1 2 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 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 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 symplectic lifted from Dic6, Schur index 2 ρ14 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 ρ15 2 -2 -2 2 -1 -1 2 -1 2i -2i 0 0 0 0 1 1 1 -2 -1 1 i i -i -i complex lifted from C4×S3 ρ16 2 -2 -2 2 -1 -1 2 -1 -2i 2i 0 0 0 0 1 1 1 -2 -1 1 -i -i i i complex lifted from C4×S3 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 complex lifted from C3⋊D4 ρ18 2 -2 2 -2 -1 -1 2 -1 0 0 0 0 0 0 1 1 1 -2 1 -1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 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

Smallest permutation representation of C6.PSU3(𝔽2)
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 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 33 35)(32 34 36)
(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 40 34 43)(26 41 35 44)(27 42 36 45)(28 37 31 46)(29 38 32 47)(30 39 33 48)
(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), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(37,39,41)(38,40,42)(43,47,45)(44,48,46), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,33,35)(32,34,36), (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,40,34,43)(26,41,35,44)(27,42,36,45)(28,37,31,46)(29,38,32,47)(30,39,33,48), (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), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(37,39,41)(38,40,42)(43,47,45)(44,48,46), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,33,35)(32,34,36), (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,40,34,43)(26,41,35,44)(27,42,36,45)(28,37,31,46)(29,38,32,47)(30,39,33,48), (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)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(37,39,41),(38,40,42),(43,47,45),(44,48,46)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,29,27),(26,30,28),(31,33,35),(32,34,36)], [(1,22,10,13),(2,23,11,14),(3,24,12,15),(4,19,7,16),(5,20,8,17),(6,21,9,18),(25,40,34,43),(26,41,35,44),(27,42,36,45),(28,37,31,46),(29,38,32,47),(30,39,33,48)], [(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.PSU3(𝔽2) in GL8(𝔽13)

 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 4 0 0 0 9 9 0 0 0 4 0 0 6 10 4 0 0 0 4 0 2 6 4 0 0 0 0 4
,
 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 4 0 9 3 0 0 0 0 0 0 0 0 1 0 0 0 12 3 0 0 0 1 0 0 6 0 4 0 10 0 3 0 0 7 0 0 12 0 0 9
,
 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 4 4 9 0 0 0 0 0 0 0 0 9 0 0 0 9 0 0 0 4 3 0 0 8 12 10 0 12 0 1 0 7 2 10 0 1 0 0 1
,
 0 0 1 0 0 0 0 0 12 1 1 8 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 6 5 12 0 12 0 0 8 8 1 6 12 0 0 1 1 3 2 12 9 0 0 0 12 11 3 5 8 0 12 0 1
,
 0 0 0 0 1 0 0 0 1 1 0 0 12 8 0 0 5 4 12 0 1 0 8 0 0 0 0 0 0 12 12 1 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 0 9 1 0 0 0 0 0 0 8 1 0

G:=sub<GL(8,GF(13))| [10,0,0,0,0,9,6,2,0,10,0,0,0,9,10,6,0,0,10,0,0,0,4,4,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[9,0,0,4,0,12,6,0,0,3,0,0,0,3,0,7,0,0,9,9,0,0,4,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,10,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9],[9,0,0,0,0,9,8,7,0,3,0,4,0,0,12,2,0,0,3,4,0,0,10,10,0,0,0,9,0,0,0,0,0,0,0,0,9,4,12,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,6,8,3,11,0,1,12,0,5,1,2,3,1,1,0,0,12,6,12,5,0,8,0,12,0,12,9,8,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,8,1,12,1],[0,1,5,0,0,0,0,0,0,1,4,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,1,12,1,0,0,0,0,0,0,8,0,12,0,12,9,8,0,0,8,12,0,0,1,1,0,0,0,1,0,0,0,0] >;

C6.PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_6.{\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C6.PSU(3,2)");
// GroupNames label

G:=SmallGroup(432,592);
// by ID

G=gap.SmallGroup(432,592);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,85,92,2804,1691,298,2693,348,1027,14118]);
// 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=a^3*d^-1>;
// generators/relations

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