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G = GL2(𝔽5)  order 480 = 25·3·5

General linear group on 𝔽52

non-abelian, not soluble

Aliases: GL2(𝔽5), C4.5S5, SL2(𝔽5)⋊1C4, C4.A5.2C2, C2.2(A5⋊C4), Aut(C52), SmallGroup(480,218)

Series: ChiefDerived Lower central Upper central

C1C2C4C4.A5 — GL2(𝔽5)
SL2(𝔽5) — GL2(𝔽5)
SL2(𝔽5) — GL2(𝔽5)
C1C4

30C2
10C3
6C5
15C22
15C4
30C4
30C4
10C6
20S3
6C10
6D5
6D5
5Q8
10C8
15D4
15C2×C4
30C2×C4
10C12
10D6
10Dic3
6F5
6F5
6C20
6F5
6F5
6D10
6Dic5
5C4○D4
15C42
15M4(2)
5SL2(𝔽3)
10C3⋊C8
10C4×S3
10C24
6C2×F5
6C2×F5
6C4×D5
15C4≀C2
5C4.A4
10C8⋊S3
6C4×F5
5U2(𝔽3)

Character table of GL2(𝔽5)

 class 12A2B34A4B4C4D4E4F4G568A8B1012A12B20A20B24A24B24C24D
 size 11302011303030303024202020242020242420202020
ρ1111111111111111111111111    trivial
ρ2111111-1-1-1-1111-1-111111-1-1-1-1    linear of order 2
ρ311-11-1-1i-ii-i111i-i1-1-1-1-1-ii-ii    linear of order 4
ρ411-11-1-1-ii-ii111-ii1-1-1-1-1i-ii-i    linear of order 4
ρ544014400000-1122-111-1-1-1-1-1-1    orthogonal lifted from S5
ρ644014400000-11-2-2-111-1-11111    orthogonal lifted from S5
ρ74401-4-400000-112i-2i-1-1-111i-ii-i    complex lifted from A5⋊C4
ρ84401-4-400000-11-2i2i-1-1-111-ii-ii    complex lifted from A5⋊C4
ρ94-40-24i-4i00000-120012i-2i-ii0000    complex faithful
ρ104-40-2-4i4i00000-12001-2i2ii-i0000    complex faithful
ρ114-4014i-4i00000-1-1001-ii-ii87ζ3878ζ3883ζ38385ζ385    complex faithful
ρ124-401-4i4i00000-1-1001i-ii-i85ζ38583ζ3838ζ3887ζ387    complex faithful
ρ134-4014i-4i00000-1-1001-ii-ii83ζ38385ζ38587ζ3878ζ38    complex faithful
ρ144-401-4i4i00000-1-1001i-ii-i8ζ3887ζ38785ζ38583ζ383    complex faithful
ρ15551-155111110-1-1-10-1-100-1-1-1-1    orthogonal lifted from S5
ρ16551-155-1-1-1-110-1110-1-1001111    orthogonal lifted from S5
ρ1755-1-1-5-5i-ii-i10-1-ii01100i-ii-i    complex lifted from A5⋊C4
ρ1855-1-1-5-5-ii-ii10-1i-i01100-ii-ii    complex lifted from A5⋊C4
ρ1966-20660000-21000100110000    orthogonal lifted from S5
ρ206620-6-60000-21000100-1-10000    orthogonal lifted from A5⋊C4
ρ216-600-6i6i-1-i-1+i1+i1-i01000-100-ii0000    complex faithful
ρ226-6006i-6i-1+i-1-i1-i1+i01000-100i-i0000    complex faithful
ρ236-6006i-6i1-i1+i-1+i-1-i01000-100i-i0000    complex faithful
ρ246-600-6i6i1+i1-i-1-i-1+i01000-100-ii0000    complex faithful

Permutation representations of GL2(𝔽5)
On 24 points - transitive group 24T1353
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
(1 5 2 12)(3 9 15 21)(6 19 23 20)(7 11 8 18)(13 17 14 24)

G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,5,2,12)(3,9,15,21)(6,19,23,20)(7,11,8,18)(13,17,14,24)>;

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), (1,5,2,12)(3,9,15,21)(6,19,23,20)(7,11,8,18)(13,17,14,24) );

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)], [(1,5,2,12),(3,9,15,21),(6,19,23,20),(7,11,8,18),(13,17,14,24)])

G:=TransitiveGroup(24,1353);

Matrix representation of GL2(𝔽5) in GL2(𝔽5) generated by

14
21
,
03
41
G:=sub<GL(2,GF(5))| [1,2,4,1],[0,4,3,1] >;

GL2(𝔽5) in GAP, Magma, Sage, TeX

{\rm GL}_2({\mathbb F}_5)
% in TeX

G:=Group("GL(2,5)");
// GroupNames label

G:=SmallGroup(480,218);
// by ID

G=gap.SmallGroup(480,218);
# by ID

Export

Subgroup lattice of GL2(𝔽5) in TeX
Character table of GL2(𝔽5) in TeX

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