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H(3,𝔽3)×ΓL(1,𝔽16) #13108366

order1620
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C32

Aliases

direct productH(3,𝔽3)×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(AGL(1,𝔽5)×H(3,𝔽3))
unambiguous semidirect product(C32×ΓL(1,𝔽16))⋊C3

Splittings

Fitting, HallC3×C5×H(3,𝔽3) ⋊ C4
SylowC3×H(3,𝔽3) ⋊ AGL(1,𝔽5)
SylowC5 ⋊ Dic3×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C32×ΓL(1,𝔽16)
commutator, socleC32×C5C32×C4
FittingC3×C5×H(3,𝔽3)C4
prime radicalC3×D5×H(3,𝔽3)C2
cosocleC32×D5C2×C32
hypercenterH(3,𝔽3)ΓL(1,𝔽16)
hypocenterC3×C5C4×H(3,𝔽3)
LCSG ▷ C32×C5 ▷ C3×C5
UCSC1 ◁ C3 ◁ H(3,𝔽3)

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 90

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽5)×H(3,𝔽3), #272×ΓL(1,𝔽16), #72254483, …
maximal normal subgroup ofD3×AGL(1,𝔽5)×H(3,𝔽3), #272×ΓL(1,𝔽16), #72254483
maximal quotient of#72254495

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 272);