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Cayley diagram

C5×C8×A4 #91583

order480
isoclinismisoclinic class of A4
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = A4 · Out(G) = C23×C4
homologyAbelianization (H1) = C3×C5×C8 · Schur multiplier (H2) = C2

Aliases

direct productC5×C8×A4
unambiguous semidirect product(C22×C8)⋊(C3×C5)
unambiguous semidirect product(C22×C5×C8)⋊C3

Splittings

Fitting, HallC22×C5×C8 ⋊ C3
derivedC22 ⋊ C3×C5×C8
SylowC22×C8 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5×C8A4
commutator, hypocenterC22C3×C5×C8
FrattiniC4C2×C5×A4
FittingC22×C5×C8C3
socleC23×C5C3×C4
prime radicalC2×C5×A4C4
cosocleC22×C4C2×C3×C5
LCSG ▷ C22
UCSC1 ◁ C5×C8

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 17 points:

(2,3,4)(5,10,7,12,8,6,11,9)(13,14,15,16,17);(1,4)(2,3)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14511720, #14511721, #14511722, …
maximal subgroup ofC5×A4⋊C16, A4⋊#672, A4×#672, …
maximal normal subgroup ofC5×A4⋊C16, A4⋊#672, A4×#672, …
maximal quotient ofC5×C8×SL(2,𝔽3), C5×A4×C16, C5×#504⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 659);