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G = ΓL2(𝔽4)  order 360 = 23·32·5

Semilinear group on 𝔽42

Aliases: ΓL2(𝔽4), C3⋊S5, A5⋊S3, (C3×A5)⋊2C2, SmallGroup(360,120)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C3 — C3×A5 — ΓL2(𝔽4)
 Derived series A5 — C3×A5 — ΓL2(𝔽4)
 Lower central C3×A5 — ΓL2(𝔽4)
 Upper central C1

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

Permutation representations of ΓL2(𝔽4)
On 15 points - transitive group 15T21
Generators in S15
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
(1 13 7)(2 6 11)(4 9 15)(10 12 14)

G:=sub<Sym(15)| (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (1,13,7)(2,6,11)(4,9,15)(10,12,14)>;

G:=Group( (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (1,13,7)(2,6,11)(4,9,15)(10,12,14) );

G=PermutationGroup([(1,2,3),(4,5,6,7,8,9),(10,11,12,13,14,15)], [(1,13,7),(2,6,11),(4,9,15),(10,12,14)])

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 4 6)(2 7 10)(3 11 9)(5 14 13)(8 12 15)

G:=sub<Sym(15)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12,13,14,15), (1,4,6)(2,7,10)(3,11,9)(5,14,13)(8,12,15)>;

G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12,13,14,15), (1,4,6)(2,7,10)(3,11,9)(5,14,13)(8,12,15) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8,9),(10,11,12,13,14,15)], [(1,4,6),(2,7,10),(3,11,9),(5,14,13),(8,12,15)])

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 9 17)(2 16 5)(3 6 13)(4 18 10)(7 15 8)(11 14 12)

G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,9,17)(2,16,5)(3,6,13)(4,18,10)(7,15,8)(11,14,12)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,9,17)(2,16,5)(3,6,13)(4,18,10)(7,15,8)(11,14,12) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,9,17),(2,16,5),(3,6,13),(4,18,10),(7,15,8),(11,14,12)])

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 24 13)(2 9 8)(3 5 12)(4 21 26)(6 14 23)(7 17 29)(10 27 20)(11 30 16)(15 25 22)(18 19 28)

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

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

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

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 3 20)(2 13 28)(4 6 10)(5 25 16)(8 30 26)(12 15 17)(14 24 18)(22 27 29)

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), (1,3,20)(2,13,28)(4,6,10)(5,25,16)(8,30,26)(12,15,17)(14,24,18)(22,27,29)>;

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), (1,3,20)(2,13,28)(4,6,10)(5,25,16)(8,30,26)(12,15,17)(14,24,18)(22,27,29) );

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)], [(1,3,20),(2,13,28),(4,6,10),(5,25,16),(8,30,26),(12,15,17),(14,24,18),(22,27,29)])

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 21 25)(2 13 23)(3 29 15)(4 12 22)(5 30 11)(6 28 24)(7 18 26)(8 9 14)(10 20 16)(17 19 27)

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

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

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

G:=TransitiveGroup(30,101);

Polynomial with Galois group ΓL2(𝔽4) over ℚ
actionf(x)Disc(f)
15T21x15-96x13+59x12+5031x11-13182x10-109799x9+487485x8+821655x7-6975764x6+2066256x5+46961715x4-80656366x3-8511147x2+57236934x+262735393140·1372·16096·156672·7066312·611590523272
15T22x15+6x14+27x13-65x12-1818x11-4800x10+15172x9+83334x8+100197x7-95461x6-358221x5-215697x4+185503x3+195837x2-101025x-138979320·134·16096·25932·28418512

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

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