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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

Subgroups: 466 in 48 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C4, S5, A5:C4, GL2(F5)
30C2
10C3
6C5
15C22
15C4
30C4
30C4
10C6
20S3
6C10
6D5
6D5
5Q8
10C8
15D4
15C2xC4
30C2xC4
10C12
10D6
10Dic3
6F5
6F5
6C20
6F5
6F5
6D10
6Dic5
5C4oD4
15C42
15M4(2)
5SL2(F3)
10C3:C8
10C4xS3
10C24
6C2xF5
6C2xF5
6C4xD5
15C4wrC2
5C4.A4
10C8:S3
6C4xF5
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 7 13 19)(3 24 10 23)(4 17 21 18)(5 9 6 16)(11 15 12 22)

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), (1,7,13,19)(3,24,10,23)(4,17,21,18)(5,9,6,16)(11,15,12,22)>;

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), (1,7,13,19)(3,24,10,23)(4,17,21,18)(5,9,6,16)(11,15,12,22) );

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)], [(1,7,13,19),(3,24,10,23),(4,17,21,18),(5,9,6,16),(11,15,12,22)]])

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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