Finite Group Databasebeta

← Prev (#11761864) Next (#11761866) →

C4⋊(C5⋊((C3⋊D3)⋊C3)) #11761865

order1080
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

unambiguous semidirect productC4⋊(C5⋊((C3⋊D3)⋊C3))
unambiguous semidirect product(C4⋊(C3⋊D15))⋊C3

Splittings

FittingC4×C5×H(3,𝔽3) ⋊ C2
HallC5×H(3,𝔽3) ⋊ D4
SylowH(3,𝔽3) ⋊ D20
SylowC5 ⋊ C4⋊((C3⋊D3)⋊C3)

Subgroups

subgroupG/H
centerC2C2×C5⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C5C22×C3
FrattiniC2×C3C2×C3×D15
FittingC4×C5×H(3,𝔽3)C2
socleC2×C3×C5C2×C3×D3
hypercenterC4C5⋊((C3⋊D3)⋊C3)
hypocenterC32×C5C3×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 180

Polycyclic presentation:

2^2,3,2,3^2,5|b^2=d,b^a=bd,e^a=e^2,f^a=f^2,g^a=g^4,e^c=ef

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 211);