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Cayley diagram

C4×C5×A4 #6791

order240
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C4×S4 (order 192) · Inn(G) = A4 · Out(G) = C22×C4
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC4×C5×A4
unambiguous semidirect product(C22×C4)⋊(C3×C5)
unambiguous semidirect product(C22×C4×C5)⋊C3

Splittings

Fitting, HallC22×C4×C5 ⋊ C3
derivedC22 ⋊ C3×C4×C5
SylowC22×C4 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC4×C5A4
commutator, hypocenterC22C3×C4×C5
FrattiniC2C2×C5×A4
FittingC22×C4×C5C3
socleC23×C5C2×C3
prime radicalC2×C5×A4C2
cosocleC23C2×C3×C5
LCSG ▷ C22
UCSC1 ◁ C4×C5

38 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:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11751573, #11751574, #11751575, …
maximal subgroup ofC5×A4⋊C8, A4⋊#168, A4×#168, …
maximal normal subgroup ofC5×A4⋊C8, A4⋊#168, A4×#168, …
maximal quotient ofC4×C5×SL(2,𝔽3), C5×C8×A4, C5×#131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 152);