Finite Group Databasebeta

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Cayley diagram

C3×#685 #6682

order240
isoclinismisoclinic class of Dic10
automorphismAut(G) = C25×AGL(1,𝔽5) (order 640) · Inn(G) = C2×D5 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#685

Splittings

FittingC2×C3×C4×C5 ⋊ C2
HallC3×C5 ⋊ C22⋊C4
SylowC5 ⋊ C3×C22⋊C4

Subgroups

subgroupG/H
centerC22×C3C2×D5
commutatorC2×C5C2×C3×C4
FrattiniC22C2×C3×D5
FittingC2×C3×C4×C5C2
socleC22×C3×C5C22
prime radicalC22×C3×D5C2
cosocleC22×C5C22×C3
hypercenterC2×C3×C4D5
hypocenterC5C3×C22⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#91353, #91354, #91356, …
maximal normal subgroup of#91353, #91354, #91356, …
maximal quotient ofC3×#3937, C3×#3938, C3×#3941, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 43);