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G = C22×PSU3(𝔽2)  order 288 = 25·32

Direct product of C22 and PSU3(𝔽2)

direct product, non-abelian, soluble, monomial, rational

Aliases: C22×PSU3(𝔽2), C62⋊2Q8, 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, (C2×C32⋊C4).27C22, (C22×C32⋊C4).10C2, (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)

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)

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ GL12(𝔽13)

1200000000000
0120000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
100000000000
010000000000
0012000000000
0001200000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
100000000000
010000000000
001000000000
000100000000
0000000001210
0000000001201
0000000001200
0000100001200
0000010001200
0000001001200
0000000101200
0000000011200
,
100000000000
010000000000
001000000000
000100000000
0000012000000
0000112000000
0000012001000
0000012100000
0000012010000
0000012000001
0000012000100
0000012000010
,
130000000000
8120000000000
00121000000000
005100000000
000000100000
000000000100
000001000000
000000001000
000000000001
000010000000
000000010000
000000000010
,
140000000000
6120000000000
0012900000000
007100000000
000000010000
000000000001
000000000010
000001000000
000000100000
000000001000
000000000100
000010000000

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] >;
 

Character table of C22×PSU3(𝔽2)

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C
 size 111199998181818181818181818181818888
ρ1111111111111111111111111    trivial
ρ21-1-111-1-111111-1-111-11-1-1-1-11-1    linear of order 2
ρ3111111111-1-1-1-11111-1-1-1-1111    linear of order 2
ρ41-1-111-1-111-1-1-11-111-1-1111-11-1    linear of order 2
ρ51-11-1-11-111-11-11-11-111-11-11-1-1    linear of order 2
ρ611-1-1-1-1111-11-1-111-1-111-11-1-11    linear of order 2
ρ71-11-1-11-1111-11-1-11-11-11-111-1-1    linear of order 2
ρ811-1-1-1-11111-11111-1-1-1-11-1-1-11    linear of order 2
ρ91-1-111-1-1111-1-111-1-111-1-11-11-1    linear of order 2
ρ101111111111-1-1-1-1-1-1-1111-1111    linear of order 2
ρ111-1-111-1-111-111-11-1-11-111-1-11-1    linear of order 2
ρ12111111111-1111-1-1-1-1-1-1-11111    linear of order 2
ρ1311-1-1-1-1111-1-111-1-11111-1-1-1-11    linear of order 2
ρ141-11-1-11-111-1-11-11-11-11-1111-1-1    linear of order 2
ρ1511-1-1-1-111111-1-1-1-111-1-111-1-11    linear of order 2
ρ161-11-1-11-11111-111-11-1-11-1-11-1-1    linear of order 2
ρ172-22-22-22-220000000000002-2-2    symplectic lifted from Q8, Schur index 2
ρ182-2-22-222-22000000000000-22-2    symplectic lifted from Q8, Schur index 2
ρ192222-2-2-2-22000000000000222    symplectic lifted from Q8, Schur index 2
ρ2022-2-222-2-22000000000000-2-22    symplectic lifted from Q8, Schur index 2
ρ218-88-80000-1000000000000-111    orthogonal lifted from C2×PSU3(𝔽2)
ρ228-8-880000-10000000000001-11    orthogonal lifted from C2×PSU3(𝔽2)
ρ2388-8-80000-100000000000011-1    orthogonal lifted from C2×PSU3(𝔽2)
ρ2488880000-1000000000000-1-1-1    orthogonal lifted from PSU3(𝔽2)

In GAP, Magma, Sage, TeX 

C_2^2\times 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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