Aliases: CU2(𝔽3), C8.8S4, U2(𝔽3)⋊7C2, GL2(𝔽3)⋊3C4, CSU2(𝔽3)⋊3C4, C8.A4⋊5C2, C8○D4⋊3S3, C2.8(C4×S4), C4.28(C2×S4), C4○D4.9D6, Q8.4(C4×S3), C4.6S4.2C2, C4.A4.11C22, SL2(𝔽3).3(C2×C4), SmallGroup(192,963)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — CU2(𝔽3) |
Generators and relations for CU2(𝔽3)
G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >
Subgroups: 215 in 64 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, Dic3, C12, D6, C42, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, C24, SL2(𝔽3), C4×S3, C4×C8, C4≀C2, C8.C4, C8○D4, C8○D4, C4○D8, S3×C8, CSU2(𝔽3), GL2(𝔽3), C4.A4, C8○D8, U2(𝔽3), C8.A4, C4.6S4, CU2(𝔽3)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, CU2(𝔽3)
►On 32 points
(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)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)
(9 27 23)(10 28 24)(11 29 17)(12 30 18)(13 31 19)(14 32 20)(15 25 21)(16 26 22)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
G:=sub<Sym(32)| (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), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18), (9,27,23)(10,28,24)(11,29,17)(12,30,18)(13,31,19)(14,32,20)(15,25,21)(16,26,22), (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)>;
G:=Group( (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), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18), (9,27,23)(10,28,24)(11,29,17)(12,30,18)(13,31,19)(14,32,20)(15,25,21)(16,26,22), (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22) );
G=PermutationGroup([[(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)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,25,21,29),(18,26,22,30),(19,27,23,31),(20,28,24,32)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18)], [(9,27,23),(10,28,24),(11,29,17),(12,30,18),(13,31,19),(14,32,20),(15,25,21),(16,26,22)], [(1,5),(2,6),(3,7),(4,8),(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 6 | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 12 | 8 | 1 | 1 | 6 | ··· | 6 | 12 | 8 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | C4×S3 | CU2(𝔽3) | S4 | C2×S4 | C4×S4 | CU2(𝔽3) |
| kernel | CU2(𝔽3) | U2(𝔽3) | C8.A4 | C4.6S4 | CSU2(𝔽3) | GL2(𝔽3) | C8○D4 | C4○D4 | Q8 | C1 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of CU2(𝔽3) ►in GL2(𝔽17) generated by
| 2 | 0 |
| 0 | 2 |
| 13 | 0 |
| 10 | 4 |
| 13 | 7 |
| 0 | 4 |
| 4 | 2 |
| 15 | 12 |
| 2 | 4 |
| 12 | 15 |
G:=sub<GL(2,GF(17))| [2,0,0,2],[13,10,0,4],[13,0,7,4],[4,15,2,12],[2,12,4,15] >;
CU2(𝔽3) in GAP, Magma, Sage, TeX
{\rm CU}_2({\mathbb F}_3) % in TeX
G:=Group("CU(2,3)"); // GroupNames label
G:=SmallGroup(192,963);
// by ID
G=gap.SmallGroup(192,963);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,36,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=d^3=e^2=1,b^2=c^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^4*c,e*d*e=d^-1>;
// generators/relations