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G = PSU3(𝔽2)⋊C4  order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: PSU3(𝔽2)⋊C4, C2.2AΓL1(𝔽9), C3⋊S3.Q16, (C2×F9).1C2, C32⋊C4.2D4, C32⋊(Q8⋊C4), (C3×C6).2SD16, (C2×PSU3(𝔽2)).1C2, (C2×C3⋊S3).2D4, C32⋊C4.2(C2×C4), C3⋊S3.Q8.2C2, C3⋊S3.2(C22⋊C4), (C2×C32⋊C4).2C22, SmallGroup(288,842)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32⋊C4 — PSU3(𝔽2)⋊C4
Chief series
C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C2×F9 — PSU3(𝔽2)⋊C4
Lower central
C32 — C3⋊S3 — C32⋊C4 — PSU3(𝔽2)⋊C4
Upper central
C1 — C2

Generators and relations for PSU3(𝔽2)⋊C4
 G = < a,b,c,d,e | a3=b3=c4=e4=1, d2=c2, dbd-1=ab=ba, cac-1=ebe-1=b-1, dad-1=a-1b, ae=ea, cbc-1=a, dcd-1=ece-1=c-1, ede-1=c-1d >

C1
C2
9C2
9C2
4C3
9C4
9C22
9C4
12C4
18C4
18C4
4C6
12S3
12S3
C32
9Q8
9Q8
9C2×C4
18C2×C4
18C8
18C2×C4
18Q8
4Dic3
12C12
12D6
C3⋊S3
C3⋊S3
C3×C6
9C4⋊C4
9C2×Q8
9C2×C8
12C4×S3
C2×C3⋊S3
C32⋊C4
C32⋊C4
2C32⋊C4
2C32⋊C4
4C3×Dic3
9Q8⋊C4
C2×C32⋊C4
PSU3(𝔽2)
PSU3(𝔽2)
2C6.D6
2PSU3(𝔽2)
2F9
2C2×C32⋊C4
C3⋊S3.Q8
C2×F9
C2×PSU3(𝔽2)
PSU3(𝔽2)⋊C4

Character table of PSU3(𝔽2)⋊C4

 class 12A2B2C34A4B4C4D4E4F68A8B8C8D12A12B
 size 119981212181836368181818182424
ρ1111111111111111111    trivial
ρ211111-1-111-1-111111-1-1    linear of order 2
ρ311111-1-111111-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-11-1-1-1-111    linear of order 2
ρ51-1-111i-i-11-11-1i-ii-ii-i    linear of order 4
ρ61-1-111i-i-111-1-1-ii-iii-i    linear of order 4
ρ71-1-111-ii-111-1-1i-ii-i-ii    linear of order 4
ρ81-1-111-ii-11-11-1-ii-ii-ii    linear of order 4
ρ92-2-222002-200-2000000    orthogonal lifted from D4
ρ102222200-2-2002000000    orthogonal lifted from D4
ρ112-22-22000000-2-√2-√2√2√200    symplectic lifted from Q16, Schur index 2
ρ122-22-22000000-2√2√2-√2-√200    symplectic lifted from Q16, Schur index 2
ρ1322-2-220000002-√-2√-2√-2-√-200    complex lifted from SD16
ρ1422-2-220000002√-2-√-2-√-2√-200    complex lifted from SD16
ρ158800-1-2-20000-1000011    orthogonal lifted from AΓL1(𝔽9)
ρ168800-1220000-10000-1-1    orthogonal lifted from AΓL1(𝔽9)
ρ178-800-12i-2i000010000-ii    complex faithful
ρ188-800-1-2i2i000010000i-i    complex faithful

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

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

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

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

Matrix representation of PSU3(𝔽2)⋊C4 ►in GL10(𝔽73)

1000000000
0100000000
00000001072
00000000172
00010000072
00000000072
00100000072
00000010072
00001000072
00000100072
,
1000000000
0100000000
00072000000
00172000000
00072001000
00072100000
00072010000
00072000001
00072000100
00072000010
,
72100000000
71100000000
0000010000
0000000001
0000000010
0001000000
0000100000
0000001000
0000000100
0010000000
,
126700000000
126100000000
00000072000
00000000720
00000720000
00000007200
00072000000
00000000072
00720000000
00007200000
,
274600000000
04600000000
00072000000
00720000000
00000072000
00000720000
00007200000
00000000720
00000007200
00000000072

G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[72,71,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[12,12,0,0,0,0,0,0,0,0,67,61,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0],[27,0,0,0,0,0,0,0,0,0,46,46,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72] >;
 

PSU3(𝔽2)⋊C4 in GAP, Magma, Sage, TeX 

{\rm PSU}_3({\mathbb F}_2)\rtimes C_4
 
% in TeX
 
G:=Group("PSU(3,2):C4");
 
// GroupNames label
 
G:=SmallGroup(288,842);
 
// by ID
 
G=gap.SmallGroup(288,842);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,421,64,219,100,4037,4716,2371,201,10982,4717,3156,622]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^4=1,d^2=c^2,d*b*d^-1=a*b=b*a,c*a*c^-1=e*b*e^-1=b^-1,d*a*d^-1=a^-1*b,a*e=e*a,c*b*c^-1=a,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=c^-1*d>;
 
// generators/relations
 

Export

Subgroup lattice of PSU3(𝔽2)⋊C4 in TeX
Character table of PSU3(𝔽2)⋊C4 in TeX

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