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Cayley diagram

C4×C5×D3 #1212

order120
isoclinismisoclinic class of D3
automorphismAut(G) = C22×C4×D3 (order 96) · Inn(G) = D3 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC4×C5×D3

Splittings

FittingC3×C4×C5 ⋊ C2
derived, SylowC3 ⋊ C2×C4×C5
HallC3×C5 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4×C5D3
commutator, hypocenterC3C2×C4×C5
FrattiniC2C2×C5×D3
FittingC3×C4×C5C2
socleC2×C3×C5C22
prime radicalC2×C5×D3C2
cosocleC2×C3C22×C5
LCSG ▷ C3
UCSC1 ◁ C4×C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#168, #6650, C5×D3×C8, …
maximal normal subgroup ofD3×#168, #6650, C5×D3×C8, …
maximal quotient ofC5×D3×C8, C5×ΓL(1,𝔽25), C4×C5×Dic3, …
automorphism group ofC22×C25, D3×C25
outer automorphism group ofC22×C25, Q8×C25

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 22);