Finite Group Databasebeta

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C22×D5×H(3,𝔽3) #11762000

order1080
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32

Aliases

direct productC22×D5×H(3,𝔽3)
unambiguous semidirect productC5⋊(C23×H(3,𝔽3))
unambiguous semidirect product(C2×C5)⋊(C22×H(3,𝔽3))

Splittings

FittingC22×C5×H(3,𝔽3) ⋊ C2
HallC5×H(3,𝔽3) ⋊ C23
SylowC5 ⋊ C23×H(3,𝔽3)

Subgroups

subgroupG/H
centerC22×C3C32×D5
commutator, cosocleC3×C5C23×C32
FrattiniC3C22×C32×D5
FittingC22×C5×H(3,𝔽3)C2
socleC22×C3×C5C2×C32
hypercenterC22×H(3,𝔽3)D5
hypocenterC5C23×H(3,𝔽3)
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×H(3,𝔽3)

352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 180

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 346);