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## G = C4.PSU3(𝔽2)  order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C4.PSU3(𝔽2)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — C3⋊S3⋊3C8 — C4.PSU3(𝔽2)
 Lower central C32 — C3×C6 — C3⋊Dic3 — C4.PSU3(𝔽2)
 Upper central C1 — C2 — C4

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

Character table of C4.PSU3(𝔽2)

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 8A 8B 8C 8D 12A 12B size 1 1 9 9 8 2 18 36 36 36 36 8 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 trivial ρ2 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 linear of order 2 ρ4 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 -i -i i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 1 -1 1 -i i i -i 1 -1 -1 1 1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 -1 1 i -i -i i 1 -1 -1 1 1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 1 -1 1 i i -i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 -2 -2 0 0 0 0 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 2 -2 0 0 0 0 2 0 0 0 0 2 2 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 2 -2 2 0 0 0 0 0 0 -2 √-2 -√-2 √-2 -√-2 0 0 complex lifted from SD16 ρ12 2 -2 -2 2 2 0 0 0 0 0 0 -2 √-2 -√-2 -√-2 √-2 0 0 complex lifted from SD16 ρ13 2 -2 -2 2 2 0 0 0 0 0 0 -2 -√-2 √-2 √-2 -√-2 0 0 complex lifted from SD16 ρ14 2 -2 2 -2 2 0 0 0 0 0 0 -2 -√-2 √-2 -√-2 √-2 0 0 complex lifted from SD16 ρ15 8 8 0 0 -1 -8 0 0 0 0 0 -1 0 0 0 0 1 1 orthogonal lifted from C2.PSU3(𝔽2) ρ16 8 8 0 0 -1 8 0 0 0 0 0 -1 0 0 0 0 -1 -1 orthogonal lifted from PSU3(𝔽2) ρ17 8 -8 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 -3i 3i complex faithful ρ18 8 -8 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 3i -3i complex faithful

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

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

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

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

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

 72 2 0 0 0 0 0 0 0 0 72 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 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 0 72 0 0 0 0 0 0 0 0 1 72 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
,
 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 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 72 1 0 0 0 0 0 0 0 0 72 0
,
 71 35 0 0 0 0 0 0 0 0 52 2 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 0 0 0 0 0 0 0 1 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 0 0 1 0 0
,
 61 12 0 0 0 0 0 0 0 0 67 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0

G:=sub<GL(10,GF(73))| [72,72,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[71,52,0,0,0,0,0,0,0,0,35,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[61,67,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C4.PSU3(𝔽2) in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,393);
// by ID

G=gap.SmallGroup(288,393);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,92,219,100,346,80,9413,2028,691,12550,1581,2372]);
// Polycyclic

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

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