Aliases: ΓL2(𝔽4), C3⋊S5, A5⋊S3, (C3×A5)⋊2C2, SmallGroup(360,120)
Series: Chief►Derived ►Lower central ►Upper central
| C3×A5 — ΓL2(𝔽4) |
15C2
30C2
10C3
20C3
6C5
5C22
45C22
45C4
10S3
10S3
15C6
30S3
30C6
60S3
10C32
6D5
6C15
45D4
5A4
5A4
5A4
15D6
15Dic3
30D6
10C3⋊S3
10C3×S3
10C3×S3
18F5
15S4
15C3⋊D4
15S4
15S4
10S32
3S5
Character table of ΓL2(𝔽4)
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4 | 5 | 6A | 6B | 15A | 15B | |
| size | 1 | 15 | 30 | 2 | 20 | 40 | 90 | 24 | 30 | 60 | 24 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 2 | -1 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 4 | 0 | -2 | 4 | 1 | 1 | 0 | -1 | 0 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ5 | 4 | 0 | 2 | 4 | 1 | 1 | 0 | -1 | 0 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ6 | 5 | 1 | 1 | 5 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S5 |
| ρ7 | 5 | 1 | -1 | 5 | -1 | -1 | 1 | 0 | 1 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ8 | 6 | -2 | 0 | 6 | 0 | 0 | 0 | 1 | -2 | 0 | 1 | 1 | orthogonal lifted from S5 |
| ρ9 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 1 | 1 | 0 | -1-√-15/2 | -1+√-15/2 | complex faithful |
| ρ10 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 1 | 1 | 0 | -1+√-15/2 | -1-√-15/2 | complex faithful |
| ρ11 | 8 | 0 | 0 | -4 | 2 | -1 | 0 | -2 | 0 | 0 | 1 | 1 | orthogonal faithful |
| ρ12 | 10 | 2 | 0 | -5 | -2 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | orthogonal faithful |
►On 15 points - transitive group 15T21
Generators in S15
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
(2 5 10)(3 15 9)(4 6 8)(7 13 12)
G:=sub<Sym(15)| (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (2,5,10)(3,15,9)(4,6,8)(7,13,12)>;
G:=Group( (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (2,5,10)(3,15,9)(4,6,8)(7,13,12) );
G=PermutationGroup([[(1,2,3),(4,5,6,7,8,9),(10,11,12,13,14,15)], [(2,5,10),(3,15,9),(4,6,8),(7,13,12)]])
G:=TransitiveGroup(15,21);
►On 15 points - transitive group 15T22
Generators in S15
(1 2)(3 4)(5 6)(7 8 9)(10 11 12 13 14 15)
(1 6 4)(2 7 15)(3 13 12)(5 10 9)(8 11 14)
G:=sub<Sym(15)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12,13,14,15), (1,6,4)(2,7,15)(3,13,12)(5,10,9)(8,11,14)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12,13,14,15), (1,6,4)(2,7,15)(3,13,12)(5,10,9)(8,11,14) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12,13,14,15)], [(1,6,4),(2,7,15),(3,13,12),(5,10,9),(8,11,14)]])
G:=TransitiveGroup(15,22);
►On 18 points - transitive group 18T146
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 13 16)(2 10 9)(3 12 11)(4 15 18)(5 17 7)(6 8 14)
G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,13,16)(2,10,9)(3,12,11)(4,15,18)(5,17,7)(6,8,14)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,13,16)(2,10,9)(3,12,11)(4,15,18)(5,17,7)(6,8,14) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,13,16),(2,10,9),(3,12,11),(4,15,18),(5,17,7),(6,8,14)]])
G:=TransitiveGroup(18,146);
►On 30 points - transitive group 30T89
Generators in S30
(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)
(1 22 17)(2 5 7)(3 18 21)(4 10 11)(6 25 24)(8 15 27)(9 28 14)(12 19 30)(13 29 20)(16 23 26)
G:=sub<Sym(30)| (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), (1,22,17)(2,5,7)(3,18,21)(4,10,11)(6,25,24)(8,15,27)(9,28,14)(12,19,30)(13,29,20)(16,23,26)>;
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), (1,22,17)(2,5,7)(3,18,21)(4,10,11)(6,25,24)(8,15,27)(9,28,14)(12,19,30)(13,29,20)(16,23,26) );
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)], [(1,22,17),(2,5,7),(3,18,21),(4,10,11),(6,25,24),(8,15,27),(9,28,14),(12,19,30),(13,29,20),(16,23,26)]])
G:=TransitiveGroup(30,89);
►On 30 points - transitive group 30T93
Generators in S30
(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)
(1 13 21)(2 4 6)(3 27 15)(5 19 29)(7 22 18)(8 10 12)(9 30 24)(11 16 26)
G:=sub<Sym(30)| (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), (1,13,21)(2,4,6)(3,27,15)(5,19,29)(7,22,18)(8,10,12)(9,30,24)(11,16,26)>;
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), (1,13,21)(2,4,6)(3,27,15)(5,19,29)(7,22,18)(8,10,12)(9,30,24)(11,16,26) );
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)], [(1,13,21),(2,4,6),(3,27,15),(5,19,29),(7,22,18),(8,10,12),(9,30,24),(11,16,26)]])
G:=TransitiveGroup(30,93);
►On 30 points - transitive group 30T101
Generators in S30
(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)
(1 25 12)(2 22 27)(3 10 24)(4 9 28)(5 17 26)(6 11 16)(7 18 21)(8 20 29)(13 14 23)(15 30 19)
G:=sub<Sym(30)| (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), (1,25,12)(2,22,27)(3,10,24)(4,9,28)(5,17,26)(6,11,16)(7,18,21)(8,20,29)(13,14,23)(15,30,19)>;
G:=Group( (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), (1,25,12)(2,22,27)(3,10,24)(4,9,28)(5,17,26)(6,11,16)(7,18,21)(8,20,29)(13,14,23)(15,30,19) );
G=PermutationGroup([[(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)], [(1,25,12),(2,22,27),(3,10,24),(4,9,28),(5,17,26),(6,11,16),(7,18,21),(8,20,29),(13,14,23),(15,30,19)]])
G:=TransitiveGroup(30,101);
Polynomial with Galois group ΓL2(𝔽4) over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T21 | x15-96x13+59x12+5031x11-13182x10-109799x9+487485x8+821655x7-6975764x6+2066256x5+46961715x4-80656366x3-8511147x2+57236934x+26273539 | 3140·1372·16096·156672·7066312·611590523272 |
| 15T22 | x15+6x14+27x13-65x12-1818x11-4800x10+15172x9+83334x8+100197x7-95461x6-358221x5-215697x4+185503x3+195837x2-101025x-138979 | 320·134·16096·25932·28418512 |
Matrix representation of ΓL2(𝔽4) ►in GL4(𝔽2) generated by
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 |
G:=sub<GL(4,GF(2))| [1,1,1,0,0,1,1,0,0,1,0,1,1,0,1,0],[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,1] >;
ΓL2(𝔽4) in GAP, Magma, Sage, TeX
{\rm GammaL}_2({\mathbb F}_4) % in TeX
G:=Group("GammaL(2,4)"); // GroupNames label
G:=SmallGroup(360,120);
// by ID
G=gap.SmallGroup(360,120);
# by ID
Export
Subgroup lattice of ΓL2(𝔽4) in TeX
Character table of ΓL2(𝔽4) in TeX