direct product, non-abelian, not soluble
Aliases: C3×SL2(𝔽5), C6.A5, C2.(C3×A5), SmallGroup(360,51)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C3×SL2(𝔽5) |
| SL2(𝔽5) — C3×SL2(𝔽5) |
Character table of C3×SL2(𝔽5)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 30 | 12 | 12 | 1 | 1 | 20 | 20 | 20 | 12 | 12 | 30 | 30 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | |
| 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 | 1 | trivial |
| ρ2 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | -1+√5/2 | -1-√5/2 | -2 | -2 | 1 | 1 | 1 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ5 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | -1-√5/2 | -1+√5/2 | -2 | -2 | 1 | 1 | 1 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ6 | 2 | -2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | -1+√5/2 | -1-√5/2 | 1+√-3 | 1-√-3 | 1 | ζ3 | ζ32 | 1+√5/2 | 1-√5/2 | 0 | 0 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | complex faithful |
| ρ7 | 2 | -2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | -1+√5/2 | -1-√5/2 | 1-√-3 | 1+√-3 | 1 | ζ32 | ζ3 | 1+√5/2 | 1-√5/2 | 0 | 0 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | complex faithful |
| ρ8 | 2 | -2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | -1-√5/2 | -1+√5/2 | 1-√-3 | 1+√-3 | 1 | ζ32 | ζ3 | 1-√5/2 | 1+√5/2 | 0 | 0 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | complex faithful |
| ρ9 | 2 | -2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | -1-√5/2 | -1+√5/2 | 1+√-3 | 1-√-3 | 1 | ζ3 | ζ32 | 1-√5/2 | 1+√5/2 | 0 | 0 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | complex faithful |
| ρ10 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | 1-√5/2 | 1+√5/2 | 3 | 3 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1 | -1 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ11 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | 1+√5/2 | 1-√5/2 | 3 | 3 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1 | -1 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ12 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 1+√5/2 | 1-√5/2 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ65 | ζ6 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | complex lifted from C3×A5 |
| ρ13 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 1-√5/2 | 1+√5/2 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ65 | ζ6 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | complex lifted from C3×A5 |
| ρ14 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 1-√5/2 | 1+√5/2 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ6 | ζ65 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | complex lifted from C3×A5 |
| ρ15 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 1+√5/2 | 1-√5/2 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ6 | ζ65 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | complex lifted from C3×A5 |
| ρ16 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 0 | -1 | -1 | 4 | 4 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ17 | 4 | -4 | 4 | 4 | 1 | 1 | 1 | 0 | -1 | -1 | -4 | -4 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ18 | 4 | 4 | -2+2√-3 | -2-2√-3 | ζ3 | ζ32 | 1 | 0 | -1 | -1 | -2-2√-3 | -2+2√-3 | 1 | ζ3 | ζ32 | -1 | -1 | 0 | 0 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×A5 |
| ρ19 | 4 | -4 | -2-2√-3 | -2+2√-3 | ζ32 | ζ3 | 1 | 0 | -1 | -1 | 2-2√-3 | 2+2√-3 | -1 | ζ6 | ζ65 | 1 | 1 | 0 | 0 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | complex faithful |
| ρ20 | 4 | 4 | -2-2√-3 | -2+2√-3 | ζ32 | ζ3 | 1 | 0 | -1 | -1 | -2+2√-3 | -2-2√-3 | 1 | ζ32 | ζ3 | -1 | -1 | 0 | 0 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×A5 |
| ρ21 | 4 | -4 | -2+2√-3 | -2-2√-3 | ζ3 | ζ32 | 1 | 0 | -1 | -1 | 2+2√-3 | 2-2√-3 | -1 | ζ65 | ζ6 | 1 | 1 | 0 | 0 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | complex faithful |
| ρ22 | 5 | 5 | 5 | 5 | -1 | -1 | -1 | 1 | 0 | 0 | 5 | 5 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ23 | 5 | 5 | -5-5√-3/2 | -5+5√-3/2 | ζ6 | ζ65 | -1 | 1 | 0 | 0 | -5+5√-3/2 | -5-5√-3/2 | -1 | ζ6 | ζ65 | 0 | 0 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A5 |
| ρ24 | 5 | 5 | -5+5√-3/2 | -5-5√-3/2 | ζ65 | ζ6 | -1 | 1 | 0 | 0 | -5-5√-3/2 | -5+5√-3/2 | -1 | ζ65 | ζ6 | 0 | 0 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A5 |
| ρ25 | 6 | -6 | 6 | 6 | 0 | 0 | 0 | 0 | 1 | 1 | -6 | -6 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ26 | 6 | -6 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 3+3√-3 | 3-3√-3 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | complex faithful |
| ρ27 | 6 | -6 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 3-3√-3 | 3+3√-3 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | complex faithful |
►On 72 points
Generators in S72
(1 23 24)(2 18 19)(3 13 14)(4 38 39)(5 33 34)(6 28 29)(7 72 43)(8 67 68)(9 62 63)(10 57 58)(11 52 53)(12 47 48)(15 37 65)(16 59 51)(17 45 25)(20 42 70)(21 64 56)(22 50 30)(26 69 61)(27 55 35)(31 44 66)(32 60 40)(36 49 71)(41 54 46)
(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 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
G:=sub<Sym(72)| (1,23,24)(2,18,19)(3,13,14)(4,38,39)(5,33,34)(6,28,29)(7,72,43)(8,67,68)(9,62,63)(10,57,58)(11,52,53)(12,47,48)(15,37,65)(16,59,51)(17,45,25)(20,42,70)(21,64,56)(22,50,30)(26,69,61)(27,55,35)(31,44,66)(32,60,40)(36,49,71)(41,54,46), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;
G:=Group( (1,23,24)(2,18,19)(3,13,14)(4,38,39)(5,33,34)(6,28,29)(7,72,43)(8,67,68)(9,62,63)(10,57,58)(11,52,53)(12,47,48)(15,37,65)(16,59,51)(17,45,25)(20,42,70)(21,64,56)(22,50,30)(26,69,61)(27,55,35)(31,44,66)(32,60,40)(36,49,71)(41,54,46), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );
G=PermutationGroup([[(1,23,24),(2,18,19),(3,13,14),(4,38,39),(5,33,34),(6,28,29),(7,72,43),(8,67,68),(9,62,63),(10,57,58),(11,52,53),(12,47,48),(15,37,65),(16,59,51),(17,45,25),(20,42,70),(21,64,56),(22,50,30),(26,69,61),(27,55,35),(31,44,66),(32,60,40),(36,49,71),(41,54,46)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)]])
Matrix representation of C3×SL2(𝔽5) ►in GL2(𝔽19) generated by
| 4 | 13 |
| 6 | 8 |
| 1 | 13 |
| 13 | 9 |
G:=sub<GL(2,GF(19))| [4,6,13,8],[1,13,13,9] >;
C3×SL2(𝔽5) in GAP, Magma, Sage, TeX
C_3\times {\rm SL}_2({\mathbb F}_5) % in TeX
G:=Group("C3xSL(2,5)"); // GroupNames label
G:=SmallGroup(360,51);
// by ID
G=gap.SmallGroup(360,51);
# by ID
Export
Subgroup lattice of C3×SL2(𝔽5) in TeX
Character table of C3×SL2(𝔽5) in TeX