Copied to
clipboard

## 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 [×3], C2 [×4], C3, C4 [×12], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4 [×18], Q8 [×16], C23, C32, D6 [×6], C2×C6, C22×C4 [×3], C2×Q8 [×12], C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C22×S3, C22×Q8, C32⋊C4 [×12], C2×C3⋊S3 [×6], C62, PSU3(𝔽2) [×16], C2×C32⋊C4 [×18], C22×C3⋊S3, C2×PSU3(𝔽2) [×12], C22×C32⋊C4 [×3], C22×PSU3(𝔽2)
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8, PSU3(𝔽2), C2×PSU3(𝔽2) [×3], 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 9)(6 10)(7 11)(8 12)(13 24)(14 21)(15 22)(16 23)(17 28)(18 25)(19 26)(20 27)(29 36)(30 33)(31 34)(32 35)
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 24)(10 21)(11 22)(12 23)(17 35)(18 36)(19 33)(20 34)(25 29)(26 30)(27 31)(28 32)
(1 6 8)(2 10 12)(3 14 16)(4 21 23)(5 20 26)(7 28 18)(9 27 19)(11 17 25)(13 34 30)(15 32 36)(22 35 29)(24 31 33)
(1 11 9)(2 7 5)(3 22 24)(4 15 13)(6 17 27)(8 25 19)(10 28 20)(12 18 26)(14 35 31)(16 29 33)(21 32 34)(23 36 30)
(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 17 7 19)(6 20 8 18)(9 28 11 26)(10 27 12 25)(13 35 15 33)(14 34 16 36)(21 31 23 29)(22 30 24 32)

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

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

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

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

Export

׿
×
𝔽