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Cayley diagram

C2×C4×C5×D3 #6805

order240
isoclinismisoclinic class of D3
automorphismAut(G) = C4×D3×#120 (order 768) · Inn(G) = D3 · Out(G) = C4×#120
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×C5×D3

Splittings

FittingC2×C3×C4×C5 ⋊ C2
derived, SylowC3 ⋊ C22×C4×C5
HallC3×C5 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4×C5D3
commutator, hypocenterC3C22×C4×C5
FrattiniC2C22×C5×D3
FittingC2×C3×C4×C5C2
socleC22×C3×C5C22
prime radicalC22×C5×D3C2
cosocleC2×C3C23×C5
LCSG ▷ C3
UCSC1 ◁ C2×C4×C5

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D3)⋊#168, C5×(C2×D3)⋊C8, C2×D3×#168, …
maximal normal subgroup of(C2×D3)⋊#168, C5×(C2×D3)⋊C8, C2×D3×#168, …
maximal quotient ofC4×C5×Dic6, C42×C5×D3, C5×#872, …
automorphism group ofC22×C5×C11, Dic3×C25, C2×D3×C25, …
outer automorphism group ofC22×C5×C11, C22×C3×C25, #41×C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 166);