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## G = U2(𝔽3)⋊C2order 192 = 26·3

### 6th semidirect product of U2(𝔽3) and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — U2(𝔽3)⋊C2
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — U2(𝔽3) — U2(𝔽3)⋊C2
 Lower central SL2(𝔽3) — U2(𝔽3)⋊C2
 Upper central C1 — C4 — C2×C4

Generators and relations for U2(𝔽3)⋊C2
G = < a,b,c,d,e,f | a4=d3=f2=1, b2=c2=a2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=a2b, dbd-1=a2bc, ebe-1=bc, bf=fb, dcd-1=b, ece-1=a2c, cf=fc, ede-1=d-1, df=fd, fef=a2e >

Subgroups: 251 in 72 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), C2×C12, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C4.Dic3, C2×SL2(𝔽3), C4.A4, C42⋊C22, U2(𝔽3), C2×C4.A4, U2(𝔽3)⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, C2×A4⋊C4, U2(𝔽3)⋊C2

Character table of U2(𝔽3)⋊C2

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 2 6 6 8 1 1 2 6 6 12 12 12 12 8 8 8 12 12 12 12 8 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 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 -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 -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 1 1 linear of order 2 ρ5 1 1 1 -1 -1 1 -1 -1 -1 1 1 i i -i -i 1 1 1 i -i -i i -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 1 -1 -1 1 -1 1 -i i i -i -1 1 -1 i i -i -i -1 1 1 -1 linear of order 4 ρ7 1 1 1 -1 -1 1 -1 -1 -1 1 1 -i -i i i 1 1 1 -i i i -i -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 1 -1 1 -1 -1 1 -1 1 i -i -i i -1 1 -1 -i -i i i -1 1 1 -1 linear of order 4 ρ9 2 2 2 2 2 -1 2 2 2 2 2 0 0 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 2 -1 2 2 -2 -2 2 0 0 0 0 1 -1 1 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 -2 -2 -1 -2 -2 -2 2 2 0 0 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 -2 2 -2 -1 -2 -2 2 -2 2 0 0 0 0 1 -1 1 0 0 0 0 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 -3 1 -1 0 3 3 -3 1 -1 -1 1 -1 1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ14 3 3 3 -1 -1 0 3 3 3 -1 -1 1 1 1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ15 3 3 3 -1 -1 0 3 3 3 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ16 3 3 -3 1 -1 0 3 3 -3 1 -1 1 -1 1 -1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 -3 -1 1 0 -3 -3 3 1 -1 i -i -i i 0 0 0 i i -i -i 0 0 0 0 complex lifted from A4⋊C4 ρ18 3 3 3 1 1 0 -3 -3 -3 -1 -1 -i -i i i 0 0 0 i -i -i i 0 0 0 0 complex lifted from A4⋊C4 ρ19 3 3 -3 -1 1 0 -3 -3 3 1 -1 -i i i -i 0 0 0 -i -i i i 0 0 0 0 complex lifted from A4⋊C4 ρ20 3 3 3 1 1 0 -3 -3 -3 -1 -1 i i -i -i 0 0 0 -i i i -i 0 0 0 0 complex lifted from A4⋊C4 ρ21 4 -4 0 0 0 -2 4i -4i 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2i 0 0 -2i complex faithful ρ22 4 -4 0 0 0 -2 -4i 4i 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2i 0 0 2i complex faithful ρ23 4 -4 0 0 0 1 4i -4i 0 0 0 0 0 0 0 √-3 -1 -√-3 0 0 0 0 -i √3 -√3 i complex faithful ρ24 4 -4 0 0 0 1 -4i 4i 0 0 0 0 0 0 0 √-3 -1 -√-3 0 0 0 0 i -√3 √3 -i complex faithful ρ25 4 -4 0 0 0 1 4i -4i 0 0 0 0 0 0 0 -√-3 -1 √-3 0 0 0 0 -i -√3 √3 i complex faithful ρ26 4 -4 0 0 0 1 -4i 4i 0 0 0 0 0 0 0 -√-3 -1 √-3 0 0 0 0 i √3 -√3 -i complex faithful

Smallest permutation representation of U2(𝔽3)⋊C2
On 32 points
Generators in S32
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 19 5 23)(2 12 6 16)(3 21 7 17)(4 14 8 10)(9 27 13 31)(11 29 15 25)(18 32 22 28)(20 26 24 30)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)
(9 21 31)(10 32 22)(11 23 25)(12 26 24)(13 17 27)(14 28 18)(15 19 29)(16 30 20)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)

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

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

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

Matrix representation of U2(𝔽3)⋊C2 in GL4(𝔽5) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 3 0 3 3 0 0 0 2 0 1 0 0 4
,
 2 0 0 1 0 4 4 0 0 2 1 0 0 0 0 3
,
 2 0 0 3 0 2 4 0 0 2 2 0 1 0 0 2
,
 2 0 0 4 0 0 1 0 0 2 0 0 2 0 0 3
,
 0 2 2 0 1 0 0 3 2 0 0 2 0 3 1 0
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,1,0,3,0,0,0,3,2,0,3,0,0,4],[2,0,0,0,0,4,2,0,0,4,1,0,1,0,0,3],[2,0,0,1,0,2,2,0,0,4,2,0,3,0,0,2],[2,0,0,2,0,0,2,0,0,1,0,0,4,0,0,3],[0,1,2,0,2,0,0,3,2,0,0,1,0,3,2,0] >;

U2(𝔽3)⋊C2 in GAP, Magma, Sage, TeX

{\rm U}_2({\mathbb F}_3)\rtimes C_2
% in TeX

G:=Group("U(2,3):C2");
// GroupNames label

G:=SmallGroup(192,982);
// by ID

G=gap.SmallGroup(192,982);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,1373,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=d^3=f^2=1,b^2=c^2=a^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,e*b*e^-1=b*c,b*f=f*b,d*c*d^-1=b,e*c*e^-1=a^2*c,c*f=f*c,e*d*e^-1=d^-1,d*f=f*d,f*e*f=a^2*e>;
// generators/relations

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