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G = ΓL2(𝔽4)  order 360 = 23·32·5

Semilinear group on 𝔽42

non-abelian, not soluble

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

Series: ChiefDerived Lower central Upper central

C1C3C3×A5 — ΓL2(𝔽4)
A5C3×A5 — ΓL2(𝔽4)
C3×A5 — ΓL2(𝔽4)
C1

15C2
30C2
10C3
20C3
6C5
5C22
45C22
45C4
10S3
10S3
15C6
30S3
30C6
60S3
10C32
6D5
6C15
45D4
5A4
5A4
5C2×C6
5A4
15D6
15Dic3
30D6
10C3⋊S3
10C3×S3
10C3×S3
18F5
6C3×D5
15S4
15C3⋊D4
15S4
15S4
5C3×A4
10S32
6C3⋊F5
5C3⋊S4
3S5

Character table of ΓL2(𝔽4)

 class 12A2B3A3B3C456A6B15A15B
 size 1153022040902430602424
ρ1111111111111    trivial
ρ211-1111-111-111    linear of order 2
ρ3220-12-102-10-1-1    orthogonal lifted from S3
ρ440-24110-101-1-1    orthogonal lifted from S5
ρ54024110-10-1-1-1    orthogonal lifted from S5
ρ65115-1-1-101100    orthogonal lifted from S5
ρ751-15-1-1101-100    orthogonal lifted from S5
ρ86-2060001-2011    orthogonal lifted from S5
ρ96-20-3000110-1--15/2-1+-15/2    complex faithful
ρ106-20-3000110-1+-15/2-1--15/2    complex faithful
ρ11800-42-10-20011    orthogonal faithful
ρ121020-5-2100-1000    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

1001
1110
1101
0010
,
1000
1001
1111
1101
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

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