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Cayley diagram

A4×D20 #91961

order480
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

direct productA4×D20
unambiguous semidirect productC4⋊(D5×A4)
unambiguous semidirect product(C22×D20)⋊C3

Splittings

FittingC22×C4×C5 ⋊ C2×C3
HallC22×D20 ⋊ C3
SylowC5 ⋊ D4×A4

Subgroups

subgroupG/H
center, FrattiniC2C2×D5×A4
commutator, socle, cosocleC23×C5C22×C3
FittingC22×C4×C5C2×C3
hypercenterC4D5×A4
hypocenterC22×C5C3×D4
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C2 ◁ C4

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751579, #11751580, #11751581, …
maximal normal subgroup of#11751579, #11751580, #11751581, …
maximal quotient ofC4⋊(C5⋊(#41⋊C3)), SL(2,𝔽3)×D20, #65910⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1037);