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G = C3×PSU3(𝔽2)  order 216 = 23·33

Direct product of C3 and PSU3(𝔽2)

direct product, non-abelian, soluble, monomial

Aliases: C3×PSU3(𝔽2), C331Q8, C32⋊(C3×Q8), C32⋊C4.2C6, C3⋊S3.2(C2×C6), (C3×C32⋊C4).4C2, (C3×C3⋊S3).2C22, SmallGroup(216,160)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C3×PSU3(𝔽2)
C1C32C3⋊S3C3×C3⋊S3C3×C32⋊C4 — C3×PSU3(𝔽2)
C32C3⋊S3 — C3×PSU3(𝔽2)
C1C3

Generators and relations for C3×PSU3(𝔽2)
 G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=d-1 >

9C2
4C3
8C3
9C4
9C4
9C4
9C6
12S3
4C32
8C32
9Q8
9C12
9C12
9C12
12C3×S3
9C3×Q8

Character table of C3×PSU3(𝔽2)

 class 123A3B3C3D3E4A4B4C6A6B12A12B12C12D12E12F
 size 191188818181899181818181818
ρ1111111111111111111    trivial
ρ211111111-1-1111-1-11-1-1    linear of order 2
ρ31111111-1-1111-1-1-1-111    linear of order 2
ρ41111111-11-111-111-1-1-1    linear of order 2
ρ511ζ3ζ32ζ31ζ32-1-11ζ3ζ32ζ65ζ65ζ6ζ6ζ3ζ32    linear of order 6
ρ611ζ3ζ32ζ31ζ32-11-1ζ3ζ32ζ65ζ3ζ32ζ6ζ65ζ6    linear of order 6
ρ711ζ32ζ3ζ321ζ3-1-11ζ32ζ3ζ6ζ6ζ65ζ65ζ32ζ3    linear of order 6
ρ811ζ32ζ3ζ321ζ3111ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ911ζ3ζ32ζ31ζ321-1-1ζ3ζ32ζ3ζ65ζ6ζ32ζ65ζ6    linear of order 6
ρ1011ζ3ζ32ζ31ζ32111ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ1111ζ32ζ3ζ321ζ31-1-1ζ32ζ3ζ32ζ6ζ65ζ3ζ6ζ65    linear of order 6
ρ1211ζ32ζ3ζ321ζ3-11-1ζ32ζ3ζ6ζ32ζ3ζ65ζ6ζ65    linear of order 6
ρ132-222222000-2-2000000    symplectic lifted from Q8, Schur index 2
ρ142-2-1+-3-1--3-1+-32-1--30001--31+-3000000    complex lifted from C3×Q8
ρ152-2-1--3-1+-3-1--32-1+-30001+-31--3000000    complex lifted from C3×Q8
ρ168088-1-1-100000000000    orthogonal lifted from PSU3(𝔽2)
ρ1780-4+4-3-4-4-3ζ65-1ζ600000000000    complex faithful
ρ1880-4-4-3-4+4-3ζ6-1ζ6500000000000    complex faithful

Permutation representations of C3×PSU3(𝔽2)
On 24 points - transitive group 24T565
Generators in S24
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(2 20 15)(4 13 18)(5 10 21)(6 11 22)(7 23 12)(8 24 9)
(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)

G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;

G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );

G=PermutationGroup([(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(2,20,15),(4,13,18),(5,10,21),(6,11,22),(7,23,12),(8,24,9)], [(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)])

G:=TransitiveGroup(24,565);

On 27 points - transitive group 27T88
Generators in S27
(1 2 3)(4 9 15)(5 10 12)(6 11 13)(7 8 14)(16 22 25)(17 23 26)(18 20 27)(19 21 24)
(1 21 23)(2 24 26)(3 19 17)(4 22 7)(5 20 6)(8 9 25)(10 27 11)(12 18 13)(14 15 16)
(1 22 20)(2 25 27)(3 16 18)(4 5 23)(6 21 7)(8 11 24)(9 10 26)(12 17 15)(13 19 14)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 21 6 23)(5 20 7 22)(8 25 10 27)(9 24 11 26)(12 18 14 16)(13 17 15 19)

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

G:=Group( (1,2,3)(4,9,15)(5,10,12)(6,11,13)(7,8,14)(16,22,25)(17,23,26)(18,20,27)(19,21,24), (1,21,23)(2,24,26)(3,19,17)(4,22,7)(5,20,6)(8,9,25)(10,27,11)(12,18,13)(14,15,16), (1,22,20)(2,25,27)(3,16,18)(4,5,23)(6,21,7)(8,11,24)(9,10,26)(12,17,15)(13,19,14), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,21,6,23)(5,20,7,22)(8,25,10,27)(9,24,11,26)(12,18,14,16)(13,17,15,19) );

G=PermutationGroup([(1,2,3),(4,9,15),(5,10,12),(6,11,13),(7,8,14),(16,22,25),(17,23,26),(18,20,27),(19,21,24)], [(1,21,23),(2,24,26),(3,19,17),(4,22,7),(5,20,6),(8,9,25),(10,27,11),(12,18,13),(14,15,16)], [(1,22,20),(2,25,27),(3,16,18),(4,5,23),(6,21,7),(8,11,24),(9,10,26),(12,17,15),(13,19,14)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,21,6,23),(5,20,7,22),(8,25,10,27),(9,24,11,26),(12,18,14,16),(13,17,15,19)])

G:=TransitiveGroup(27,88);

C3×PSU3(𝔽2) is a maximal subgroup of   C333SD16

Matrix representation of C3×PSU3(𝔽2) in GL8(𝔽13)

30000000
03000000
00300000
00030000
00003000
00000300
00000030
00000003
,
10000000
01000000
00900000
112930000
00003000
1212009900
80006090
011400003
,
90000000
03000000
00100000
811010000
00003000
09009900
70300030
071207009
,
00100000
58180000
01000000
00050000
051205008
00010011
00000005
00080108
,
00001000
12120012800
801208080
000005121
01000000
00000100
00010010
00000810

G:=sub<GL(8,GF(13))| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[1,0,0,11,0,12,8,0,0,1,0,2,0,12,0,11,0,0,9,9,0,0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,3,9,6,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3],[9,0,0,8,0,0,7,0,0,3,0,11,0,9,0,7,0,0,1,0,0,0,3,12,0,0,0,1,0,0,0,0,0,0,0,0,3,9,0,7,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9],[0,5,0,0,0,0,0,0,0,8,1,0,5,0,0,0,1,1,0,0,12,0,0,0,0,8,0,5,0,1,0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,8,1,5,8],[0,12,8,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,1,12,8,0,0,0,0,0,0,8,0,5,0,1,0,8,0,0,8,12,0,0,1,1,0,0,0,1,0,0,0,0] >;

C3×PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_3\times {\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C3xPSU(3,2)");
// GroupNames label

G:=SmallGroup(216,160);
// by ID

G=gap.SmallGroup(216,160);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,3,72,169,79,5044,1090,142,6917,875,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,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

Export

Subgroup lattice of C3×PSU3(𝔽2) in TeX
Character table of C3×PSU3(𝔽2) in TeX

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