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Cayley diagram

GL(2,𝔽5) #91142

order480
isoclinismisoclinic class of #6728
automorphismAut(G) = C22×S5 (order 480) · Inn(G) = S5 · Out(G) = C22
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

general linearGL(2,𝔽5)

Splittings

derivedSL(2,𝔽5) ⋊ C4

Subgroups

subgroupG/H
center, Frattini, Fitting, hypercenterC4S5
commutator, hypocenterSL(2,𝔽5)C4
socleC2A5⋊C4
prime radical, cosocle#6732C2
LCSG ▷ SL(2,𝔽5)
UCSC1 ◁ C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751490, #11751492, C2×GL(2,𝔽5), …
maximal normal subgroup of#11751490, #11751492, C2×GL(2,𝔽5), …
maximal quotient ofSL(2,𝔽5)⋊C8, C2×GL(2,𝔽5), C3×GL(2,𝔽5), …
automorphism group ofC52, C2×C52
outer automorphism group ofC52, C2×C52, H(3,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 218);