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C5×C32⋊(C13⋊C3) #13182175

order1755
isoclinismisoclinic class of #66848
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C32

Aliases

direct productC5×C32⋊(C13⋊C3)
unambiguous semidirect productC32⋊(C5×C13⋊C3)
unambiguous semidirect productC13⋊(C5×H(3,𝔽3))
unambiguous semidirect product(C32×C5)⋊(C13⋊C3)
unambiguous semidirect product(C5×C13)⋊H(3,𝔽3)
unambiguous semidirect product(C3×C13⋊C3)⋊(C3×C5)
unambiguous semidirect product(C3×C5×C13⋊C3)⋊C3

Splittings

FittingC32×C5×C13 ⋊ C3
HallC5×C13 ⋊ H(3,𝔽3)
SylowC13 ⋊ C5×H(3,𝔽3)

Subgroups

subgroupG/H
centerC3×C5C3×C13⋊C3
commutator, cosocleC3×C13C32×C5
FrattiniC3C3×C5×C13⋊C3
FittingC32×C5×C13C3
socleC3×C5×C13C32
hypercenterC32×C5C13⋊C3
hypocenterC13C5×H(3,𝔽3)
LCSG ▷ C3×C13 ▷ C13
UCSC1 ◁ C3×C5 ◁ C32×C5

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 585

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,4,5)(6,7,9,8,11,15,12,16,18,10,13,17,14)(20,23,24)(22,26,27);(7,8,10)(9,12,15)(11,14,13)(16,18,17)(19,20,22)(21,24,26)(23,27,25)

Conjugacy classes & character table

235 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1755, 8);