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C5×#66322 #13108124

order1620
isoclinismisoclinic class of #4175
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×#66322

Splittings

derivedC33 ⋊ C3×C4×C5
HallC5×AΣL(1,𝔽27) ⋊ C4
SylowAΣL(1,𝔽27) ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5#4175
commutator, hypocenterC33C3×C4×C5
FrattiniC2×C32C3×C5×D3
Fitting, prime radicalC2×C5×AΣL(1,𝔽27)C2
socleC2×C3×C5(C3⋊D3)⋊C3
cosocleC2×C33C2×C3×C5
LCSG ▷ C33
UCSC1 ◁ C2×C5

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5)(6,7,8,9)(10,11,13)(12,15,17,16,14,18);(11,14,15)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 30);