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## G = C4.4PSU3(𝔽2)  order 288 = 25·32

### The central extension by C4 of PSU3(𝔽2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C4.4PSU3(𝔽2)
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4×C32⋊C4 — C4.4PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — C4.4PSU3(𝔽2)
 Upper central C1 — C4

Generators and relations for C4.4PSU3(𝔽2)
G = < a,b,c,d,e | a4=b3=c3=1, d4=a2, e2=a-1d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=a-1d3 >

Character table of C4.4PSU3(𝔽2)

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B size 1 1 9 9 8 1 1 9 9 18 18 18 18 8 18 18 18 18 18 18 18 18 8 8 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 1 -1 -1 1 i -i i i -i -i i -i -1 -1 linear of order 4 ρ6 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -i i -i -i i i -i i -1 -1 linear of order 4 ρ7 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i i -i -i -i -i i i -1 -1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i -i i i i i -i -i -1 -1 linear of order 4 ρ9 1 -1 -1 1 1 -i i -i i -1 1 -i i -1 ζ87 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 ζ85 i -i linear of order 8 ρ10 1 -1 -1 1 1 -i i -i i -1 1 -i i -1 ζ83 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 ζ8 i -i linear of order 8 ρ11 1 -1 -1 1 1 i -i i -i 1 -1 -i i -1 ζ85 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ83 -i i linear of order 8 ρ12 1 -1 -1 1 1 i -i i -i 1 -1 -i i -1 ζ8 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ87 -i i linear of order 8 ρ13 1 -1 -1 1 1 i -i i -i -1 1 i -i -1 ζ85 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 ζ87 -i i linear of order 8 ρ14 1 -1 -1 1 1 i -i i -i -1 1 i -i -1 ζ8 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 ζ83 -i i linear of order 8 ρ15 1 -1 -1 1 1 -i i -i i 1 -1 i -i -1 ζ83 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ85 i -i linear of order 8 ρ16 1 -1 -1 1 1 -i i -i i 1 -1 i -i -1 ζ87 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ8 i -i linear of order 8 ρ17 2 2 -2 -2 2 -2 -2 2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 2 2 -2 -2 0 0 0 0 2 0 0 0 0 0 0 0 0 2 2 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 2 -2 2 -2i 2i 2i -2i 0 0 0 0 -2 0 0 0 0 0 0 0 0 2i -2i complex lifted from M4(2) ρ20 2 -2 2 -2 2 2i -2i -2i 2i 0 0 0 0 -2 0 0 0 0 0 0 0 0 -2i 2i complex lifted from M4(2) ρ21 8 8 0 0 -1 -8 -8 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 orthogonal lifted from C2.PSU3(𝔽2) ρ22 8 8 0 0 -1 8 8 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from PSU3(𝔽2) ρ23 8 -8 0 0 -1 8i -8i 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 i -i complex faithful ρ24 8 -8 0 0 -1 -8i 8i 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -i i complex faithful

Smallest permutation representation of C4.4PSU3(𝔽2)
On 48 points
Generators in S48
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 47 21 43)(18 48 22 44)(19 41 23 45)(20 42 24 46)(25 38 29 34)(26 39 30 35)(27 40 31 36)(28 33 32 37)
(2 35 32)(4 26 37)(6 39 28)(8 30 33)(9 23 43)(10 44 24)(11 45 17)(12 18 46)(13 19 47)(14 48 20)(15 41 21)(16 22 42)
(1 31 34)(3 36 25)(5 27 38)(7 40 29)(9 23 43)(10 24 44)(11 45 17)(12 46 18)(13 19 47)(14 20 48)(15 41 21)(16 42 22)
(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 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 27 45 38)(18 39 46 28)(19 29 47 40)(20 33 48 30)(21 31 41 34)(22 35 42 32)(23 25 43 36)(24 37 44 26)

G:=sub<Sym(48)| (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,47,21,43)(18,48,22,44)(19,41,23,45)(20,42,24,46)(25,38,29,34)(26,39,30,35)(27,40,31,36)(28,33,32,37), (2,35,32)(4,26,37)(6,39,28)(8,30,33)(9,23,43)(10,44,24)(11,45,17)(12,18,46)(13,19,47)(14,48,20)(15,41,21)(16,22,42), (1,31,34)(3,36,25)(5,27,38)(7,40,29)(9,23,43)(10,24,44)(11,45,17)(12,46,18)(13,19,47)(14,20,48)(15,41,21)(16,42,22), (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), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,27,45,38)(18,39,46,28)(19,29,47,40)(20,33,48,30)(21,31,41,34)(22,35,42,32)(23,25,43,36)(24,37,44,26)>;

G:=Group( (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,47,21,43)(18,48,22,44)(19,41,23,45)(20,42,24,46)(25,38,29,34)(26,39,30,35)(27,40,31,36)(28,33,32,37), (2,35,32)(4,26,37)(6,39,28)(8,30,33)(9,23,43)(10,44,24)(11,45,17)(12,18,46)(13,19,47)(14,48,20)(15,41,21)(16,22,42), (1,31,34)(3,36,25)(5,27,38)(7,40,29)(9,23,43)(10,24,44)(11,45,17)(12,46,18)(13,19,47)(14,20,48)(15,41,21)(16,42,22), (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), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,27,45,38)(18,39,46,28)(19,29,47,40)(20,33,48,30)(21,31,41,34)(22,35,42,32)(23,25,43,36)(24,37,44,26) );

G=PermutationGroup([(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,47,21,43),(18,48,22,44),(19,41,23,45),(20,42,24,46),(25,38,29,34),(26,39,30,35),(27,40,31,36),(28,33,32,37)], [(2,35,32),(4,26,37),(6,39,28),(8,30,33),(9,23,43),(10,44,24),(11,45,17),(12,18,46),(13,19,47),(14,48,20),(15,41,21),(16,22,42)], [(1,31,34),(3,36,25),(5,27,38),(7,40,29),(9,23,43),(10,24,44),(11,45,17),(12,46,18),(13,19,47),(14,20,48),(15,41,21),(16,42,22)], [(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)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,27,45,38),(18,39,46,28),(19,29,47,40),(20,33,48,30),(21,31,41,34),(22,35,42,32),(23,25,43,36),(24,37,44,26)])

Matrix representation of C4.4PSU3(𝔽2) in GL10(𝔽73)

 46 0 0 0 0 0 0 0 0 0 0 46 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 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 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 1 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 46 0 1 0 0 25 25 0 1 27 0 72 72
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 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 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 25 1 1 27 0 72 72 0 0 48 0 0 0 0 46 1 0
,
 10 0 0 0 0 0 0 0 0 0 0 63 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 0 0 0 0 0 0 25 25 1 1 27 27 71 72 0 0 0 0 0 0 0 0 72 1 0 0 55 55 66 66 48 48 46 0 0 0 55 55 66 66 49 48 46 0
,
 0 72 0 0 0 0 0 0 0 0 1 0 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 0 0 0 72 1 0 0 25 25 1 1 27 27 71 72 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 66 66 0 0 66 66 72 0 0 0 66 66 0 1 66 66 72 0

G:=sub<GL(10,GF(73))| [46,0,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,25,0,0,0,1,0,0,0,0,0,25,0,0,0,0,72,72,0,0,72,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,27,0,0,0,0,0,0,72,72,46,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,48,0,0,1,0,0,0,0,0,25,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,27,0,0,0,0,0,0,0,72,72,0,46,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[10,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,0,0,0,1,25,0,55,55,0,0,0,0,1,0,25,0,55,55,0,0,1,0,0,0,1,0,66,66,0,0,0,1,0,0,1,0,66,66,0,0,0,0,0,0,27,0,48,49,0,0,0,0,0,0,27,0,48,48,0,0,0,0,0,0,71,72,46,46,0,0,0,0,0,0,72,1,0,0],[0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,1,66,66,0,0,0,0,0,25,1,0,66,66,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,27,0,0,66,66,0,0,0,1,0,27,0,0,66,66,0,0,0,0,72,71,0,0,72,72,0,0,0,0,1,72,0,0,0,0] >;

C4.4PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_4._4{\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C4.4PSU(3,2)");
// GroupNames label

G:=SmallGroup(288,392);
// by ID

G=gap.SmallGroup(288,392);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,85,92,80,9413,2028,691,12550,1581,2372]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^3=1,d^4=a^2,e^2=a^-1*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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^-1*d^3>;
// generators/relations

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