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Cayley diagram

C23×ΓL(1,𝔽16) #92130

order480
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC23×ΓL(1,𝔽16)
unambiguous semidirect productD5⋊(C22×Dic3)

Splittings

Fitting, socleC23×C3×C5 ⋊ C4
derived, HallC3×C5 ⋊ C23×C4
SylowC5 ⋊ C23×Dic3
SylowC3 ⋊ C23×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC23ΓL(1,𝔽16)
commutator, hypocenterC3×C5C23×C4
FrattiniC1G
Fitting, socleC23×C3×C5C4
prime radicalC23×C3×D5C2
cosocleC3×D5C24
LCSG ▷ C3×C5
UCSC1 ◁ C23

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 4 generators on 14 points:

(8,9)(11,12,13,14);(5,6)(7,8,9)(10,11,14,12,13);(3,4);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×(C2×AGL(1,𝔽5))⋊Dic3, #11752021, #11752026, …
maximal normal subgroup ofC2×(C2×AGL(1,𝔽5))⋊Dic3, #11752021, #11752026, …
maximal quotient ofC22×D5⋊(C3⋊C8), C22×#6758, C2×D5⋊#208, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1206);