Copied to
clipboard

## G = PSU3(𝔽2)⋊C4order 288 = 25·32

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

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 >

Character table of PSU3(𝔽2)⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 8A 8B 8C 8D 12A 12B size 1 1 9 9 8 12 12 18 18 36 36 8 18 18 18 18 24 24 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 -1 1 1 i -i -1 1 -1 1 -1 i -i i -i i -i linear of order 4 ρ6 1 -1 -1 1 1 i -i -1 1 1 -1 -1 -i i -i i i -i linear of order 4 ρ7 1 -1 -1 1 1 -i i -1 1 1 -1 -1 i -i i -i -i i linear of order 4 ρ8 1 -1 -1 1 1 -i i -1 1 -1 1 -1 -i i -i i -i i linear of order 4 ρ9 2 -2 -2 2 2 0 0 2 -2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 0 0 -2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 0 0 0 0 0 0 -2 -√2 -√2 √2 √2 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 2 0 0 0 0 0 0 -2 √2 √2 -√2 -√2 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 2 -2 -2 2 0 0 0 0 0 0 2 -√-2 √-2 √-2 -√-2 0 0 complex lifted from SD16 ρ14 2 2 -2 -2 2 0 0 0 0 0 0 2 √-2 -√-2 -√-2 √-2 0 0 complex lifted from SD16 ρ15 8 8 0 0 -1 -2 -2 0 0 0 0 -1 0 0 0 0 1 1 orthogonal lifted from AΓL1(𝔽9) ρ16 8 8 0 0 -1 2 2 0 0 0 0 -1 0 0 0 0 -1 -1 orthogonal lifted from AΓL1(𝔽9) ρ17 8 -8 0 0 -1 2i -2i 0 0 0 0 1 0 0 0 0 -i i complex faithful ρ18 8 -8 0 0 -1 -2i 2i 0 0 0 0 1 0 0 0 0 i -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)

 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 0 1 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 72 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 1 0 0 0 72
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 72 0 0 0 1 0 0 0 0 0 72 0 0 0 0 1 0
,
 72 1 0 0 0 0 0 0 0 0 71 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 0 1 0 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0
,
 12 67 0 0 0 0 0 0 0 0 12 61 0 0 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0
,
 27 46 0 0 0 0 0 0 0 0 0 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

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

׿
×
𝔽