Aliases: U2(𝔽3)⋊6C2, C4.A4⋊1C4, C4.33(C2×S4), (C2×C4).14S4, C4.2(A4⋊C4), C4○D4.14D6, C4○D4.3Dic3, (C2×Q8).3Dic3, Q8.3(C2×Dic3), C22.6(A4⋊C4), C4.A4.14C22, (C2×SL2(𝔽3))⋊5C4, SL2(𝔽3).6(C2×C4), C2.8(C2×A4⋊C4), (C2×C4.A4).4C2, (C2×C4○D4).2S3, SmallGroup(192,982)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — U2(𝔽3)⋊C2 |
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 |
►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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