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Cayley diagram

C3×D5×Dic3 #67128

order360
isoclinismisoclinic class of D3×D5
automorphismAut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = D3×D5 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×D5×Dic3

Splittings

derivedC3×C5 ⋊ C2×C3×C4
HallC32×C5 ⋊ C2×C4
SylowC32 ⋊ C4×D5
SylowC5 ⋊ C2×C3×Dic3

Subgroups

subgroupG/H
center, hypercenterC2×C3D3×D5
commutator, hypocenterC3×C5C2×C3×C4
FrattiniC2C3×D3×D5
Fitting, socleC2×C32×C5C22
prime radicalC2×C32×D5C2
cosocleC2×C3×C5C22×C3
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×Dic3×AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), Dic3×ΓL(1,𝔽16), …
maximal normal subgroup ofC3×Dic3×AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), Dic3×ΓL(1,𝔽16), …
maximal quotient ofC3×D5×C3⋊C8, C3×#6649, C3×Dic3×Dic5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 58);