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## G = C22×PSU3(𝔽2)  order 288 = 25·32

### Direct product of C22 and PSU3(𝔽2)

Aliases: C22×PSU3(𝔽2), C622Q8, C32⋊(C22×Q8), C3⋊S3.2C24, C32⋊C4.3C23, C3⋊S3⋊(C2×Q8), (C3×C6)⋊(C2×Q8), (C2×C3⋊S3)⋊4Q8, (C2×C3⋊S3).33C23, (C22×C32⋊C4).10C2, (C2×C32⋊C4).27C22, (C22×C3⋊S3).61C22, SmallGroup(288,1032)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C22×PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — PSU3(𝔽2) — C2×PSU3(𝔽2) — C22×PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — C22×PSU3(𝔽2)
 Upper central C1 — C22

Generators and relations for C22×PSU3(𝔽2)
G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fdf-1=cd=dc, ece-1=d-1, fcf-1=c-1d, ede-1=c, fef-1=e-1 >

Subgroups: 892 in 172 conjugacy classes, 83 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, Q8, C23, C32, D6, C2×C6, C22×C4, C2×Q8, C3⋊S3, C3⋊S3, C3×C6, C22×S3, C22×Q8, C32⋊C4, C2×C3⋊S3, C62, PSU3(𝔽2), C2×C32⋊C4, C22×C3⋊S3, C2×PSU3(𝔽2), C22×C32⋊C4, C22×PSU3(𝔽2)
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, PSU3(𝔽2), C2×PSU3(𝔽2), C22×PSU3(𝔽2)

Character table of C22×PSU3(𝔽2)

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C size 1 1 1 1 9 9 9 9 8 18 18 18 18 18 18 18 18 18 18 18 18 8 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 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 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 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 -1 1 -1 -1 -1 1 linear of order 2 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 -2 2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 symplectic lifted from Q8, Schur index 2 ρ19 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 symplectic lifted from Q8, Schur index 2 ρ20 2 2 -2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 symplectic lifted from Q8, Schur index 2 ρ21 8 -8 8 -8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C2×PSU3(𝔽2) ρ22 8 -8 -8 8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 orthogonal lifted from C2×PSU3(𝔽2) ρ23 8 8 -8 -8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 orthogonal lifted from C2×PSU3(𝔽2) ρ24 8 8 8 8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from PSU3(𝔽2)

Smallest permutation representation of C22×PSU3(𝔽2)
On 36 points
Generators in S36
(1 2)(3 4)(5 10)(6 11)(7 12)(8 9)(13 27)(14 28)(15 25)(16 26)(17 36)(18 33)(19 34)(20 35)(21 31)(22 32)(23 29)(24 30)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 28)(10 25)(11 26)(12 27)(17 23)(18 24)(19 21)(20 22)(29 36)(30 33)(31 34)(32 35)
(1 6 8)(2 11 9)(3 16 14)(4 26 28)(5 31 24)(7 22 29)(10 21 30)(12 32 23)(13 20 36)(15 34 18)(17 27 35)(19 33 25)
(1 12 10)(2 7 5)(3 27 25)(4 13 15)(6 32 21)(8 23 30)(9 29 24)(11 22 31)(14 17 33)(16 35 19)(18 28 36)(20 34 26)
(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 2)(3 4)(5 32 7 30)(6 31 8 29)(9 23 11 21)(10 22 12 24)(13 33 15 35)(14 36 16 34)(17 26 19 28)(18 25 20 27)

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

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

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

Matrix representation of C22×PSU3(𝔽2) in GL12(𝔽13)

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

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

C22×PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_2^2\times {\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C2^2xPSU(3,2)");
// GroupNames label

G:=SmallGroup(288,1032);
// by ID

G=gap.SmallGroup(288,1032);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,253,120,9413,2028,201,12550,1581,622]);
// Polycyclic

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

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