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C3×D5×H(3,𝔽3) #11717657

order810
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C34

Aliases

direct productC3×D5×H(3,𝔽3)
unambiguous semidirect productC5⋊(C2×C3×H(3,𝔽3))
unambiguous semidirect product(C32×D5)⋊C32

Splittings

Fitting, HallC3×C5×H(3,𝔽3) ⋊ C2
SylowC5 ⋊ C2×C3×H(3,𝔽3)

Subgroups

subgroupG/H
centerC32C32×D5
commutator, cosocleC3×C5C2×C33
FrattiniC3C33×D5
FittingC3×C5×H(3,𝔽3)C2
socleC32×C5C2×C32
hypercenterC3×H(3,𝔽3)D5
hypocenterC5C2×C3×H(3,𝔽3)
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C32 ◁ C3×H(3,𝔽3)

224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 135

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽5)×H(3,𝔽3), C3×(C3⋊ΓL(1,𝔽16))⋊C3, H(3,𝔽3)×ΓL(1,𝔽16), …
maximal normal subgroup ofC3×AGL(1,𝔽5)×H(3,𝔽3), C3×(C3⋊ΓL(1,𝔽16))⋊C3, H(3,𝔽3)×ΓL(1,𝔽16), …
maximal quotient ofC3×Dic5×H(3,𝔽3), C2×C3×D5×H(3,𝔽3), C3×D5×AΣL(1,𝔽27), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(810, 64);