Finite Group Databasebeta

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Cayley diagram

C3×#676 #6673

order240
isoclinismisoclinic class of Dic10
automorphismAut(G) = C24×AGL(1,𝔽5) (order 320) · Inn(G) = C2×D5 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#676

Splittings

FittingC3×C5×C8 ⋊ C2
HallC3×C5 ⋊ M16
SylowC5 ⋊ C3×M16

Subgroups

subgroupG/H
centerC3×C4C2×D5
commutatorC2×C5C2×C3×C4
FrattiniC4C2×C3×D5
FittingC3×C5×C8C2
socleC2×C3×C5C2×C4
prime radicalC2×C3×D5C4
cosocleC4×C5C22×C3
hypercenterC3×C8D5
hypocenterC5C3×M16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C3×C8

40 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)(6,11,12,7,9,13,8,10)(14,15,16);(2,3)(4,5)(7,10)(11,13)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#676, #91246, #91249, …
maximal normal subgroup ofD3×#676, #91246, #91249, …
maximal quotient ofC3×#3946, C3×#3947, C3×#3952, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 34);