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Cayley diagram

C5×D3×C8 #6688

order240
isoclinismisoclinic class of D3
automorphismAut(G) = C23×C4×D3 (order 192) · Inn(G) = D3 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2

Aliases

direct productC5×D3×C8

Splittings

FittingC3×C5×C8 ⋊ C2
derived, SylowC3 ⋊ C2×C5×C8
HallC3×C5 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC5×C8D3
commutator, hypocenterC3C2×C5×C8
FrattiniC4C2×C5×D3
FittingC3×C5×C8C2
socleC2×C3×C5C2×C4
prime radicalC2×C5×D3C4
cosocleC3×C4C22×C5
LCSG ▷ C3
UCSC1 ◁ C5×C8

44 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:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#672, #90935, C5×D3×C16, …
maximal normal subgroup ofD3×#672, #90935, C5×D3×C16, …
maximal quotient ofC5×D3×C16, C5×#798, C5×C8×Dic3, …
automorphism group ofC22×C41, D3×C41
outer automorphism group ofC22×C41, Q8×C41

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 49);