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## G = C4⋊PSU3(𝔽2)  order 288 = 25·32

### The semidirect product of C4 and PSU3(𝔽2) acting via PSU3(𝔽2)/C32⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C4⋊PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C2×PSU3(𝔽2) — C4⋊PSU3(𝔽2)
 Lower central C32 — C2×C3⋊S3 — 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=d2, 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=d-1 >

Subgroups: 468 in 76 conjugacy classes, 29 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, C32, Dic3, C12, D6, C42, C4⋊C4, C2×Q8, C3⋊S3, C3×C6, C4×S3, C4⋊Q8, C3⋊Dic3, C3×C12, C32⋊C4, C32⋊C4, C2×C3⋊S3, C4×C3⋊S3, PSU3(𝔽2), C2×C32⋊C4, C2×C32⋊C4, C2.PSU3(𝔽2), C4×C32⋊C4, C4⋊(C32⋊C4), C2×PSU3(𝔽2), C4⋊PSU3(𝔽2)
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4⋊Q8, PSU3(𝔽2), C2×PSU3(𝔽2), C4⋊PSU3(𝔽2)

Character table of C4⋊PSU3(𝔽2)

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

Smallest permutation representation of C4⋊PSU3(𝔽2)
On 36 points
Generators in S36
(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)
(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 11 35)(6 12 36)(7 9 33)(8 10 34)(13 19 28)(14 20 25)(15 17 26)(16 18 27)
(1 4)(2 3)(5 30 14 22)(6 29 15 21)(7 32 16 24)(8 31 13 23)(9 17 27 36)(10 20 28 35)(11 19 25 34)(12 18 26 33)
(1 3)(2 4)(5 33 14 18)(6 34 15 19)(7 35 16 20)(8 36 13 17)(9 31 27 23)(10 32 28 24)(11 29 25 21)(12 30 26 22)

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

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), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,11,35)(6,12,36)(7,9,33)(8,10,34)(13,19,28)(14,20,25)(15,17,26)(16,18,27), (1,4)(2,3)(5,30,14,22)(6,29,15,21)(7,32,16,24)(8,31,13,23)(9,17,27,36)(10,20,28,35)(11,19,25,34)(12,18,26,33), (1,3)(2,4)(5,33,14,18)(6,34,15,19)(7,35,16,20)(8,36,13,17)(9,31,27,23)(10,32,28,24)(11,29,25,21)(12,30,26,22) );

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)], [(1,7,16),(2,8,13),(3,5,14),(4,6,15),(9,18,29),(10,19,30),(11,20,31),(12,17,32),(21,33,27),(22,34,28),(23,35,25),(24,36,26)], [(1,29,21),(2,30,22),(3,31,23),(4,32,24),(5,11,35),(6,12,36),(7,9,33),(8,10,34),(13,19,28),(14,20,25),(15,17,26),(16,18,27)], [(1,4),(2,3),(5,30,14,22),(6,29,15,21),(7,32,16,24),(8,31,13,23),(9,17,27,36),(10,20,28,35),(11,19,25,34),(12,18,26,33)], [(1,3),(2,4),(5,33,14,18),(6,34,15,19),(7,35,16,20),(8,36,13,17),(9,31,27,23),(10,32,28,24),(11,29,25,21),(12,30,26,22)]])

Matrix representation of C4⋊PSU3(𝔽2) in GL10(𝔽13)

 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 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 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 12 12 12 12 12 12 12 12 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
,
 1 0 0 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 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 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 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 0
,
 4 3 0 0 0 0 0 0 0 0 3 9 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0
,
 0 12 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 0 0 0 0 0 1 0 0 0 0 0 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 0 1 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 0 0 1 0

G:=sub<GL(10,GF(13))| [0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,1,0,0,0,12,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[4,3,0,0,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,1,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0] >;

C4⋊PSU3(𝔽2) in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,893);
// by ID

G=gap.SmallGroup(288,893);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,141,64,219,100,9413,2028,362,12550,1581,1203]);
// Polycyclic

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

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