Finite Group Databasebeta

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Cayley diagram

C5×A4⋊C8 #91179

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

Aliases

direct productC5×A4⋊C8
unambiguous semidirect productA4⋊(C5×C8)

Splittings

derivedA4 ⋊ C5×C8

Subgroups

subgroupG/H
center, hypercenterC4×C5S4
commutator, hypocenterA4C5×C8
FrattiniC4C5×S4
FittingC22×C4×C5D3
socleC23×C5Dic3
prime radicalC2×C5×A4C4
cosocleC4×A4C2×C5
LCSG ▷ A4
UCSC1 ◁ C4×C5

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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751570, #11751572, D5×A4⋊C8, …
maximal normal subgroup of#11751570, #11751572, D5×A4⋊C8, …
maximal quotient ofC5×SL(2,𝔽3)⋊C8, C5×A4⋊C16, C5×#4628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 255);