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AGL(1,𝔽5)×H(3,𝔽3) #10587373

order540
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C32

Aliases

direct productAGL(1,𝔽5)×H(3,𝔽3)
unambiguous semidirect product(C32×AGL(1,𝔽5))⋊C3

Splittings

Fitting, HallC5×H(3,𝔽3) ⋊ C4
SylowC5 ⋊ C4×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C32×AGL(1,𝔽5)
commutator, socleC3×C5C32×C4
FittingC5×H(3,𝔽3)C4
prime radicalD5×H(3,𝔽3)C2
cosocleC3×D5C2×C32
hypercenterH(3,𝔽3)AGL(1,𝔽5)
hypocenterC5C4×H(3,𝔽3)
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C3 ◁ H(3,𝔽3)

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 45

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×(C3⋊D3)⋊C3, AGL(1,𝔽5)×#272, C2×AGL(1,𝔽5)×H(3,𝔽3), …
maximal normal subgroup ofAGL(1,𝔽5)×(C3⋊D3)⋊C3, AGL(1,𝔽5)×#272, C2×AGL(1,𝔽5)×H(3,𝔽3), …
maximal quotient ofH(3,𝔽3)×#170, C2×AGL(1,𝔽5)×H(3,𝔽3), AGL(1,𝔽5)×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 33);