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AGL(1,𝔽5)×3−1+2 #10587374

order540
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽5)×3−1+2
unambiguous semidirect product(C9×AGL(1,𝔽5))⋊C3

Splittings

Fitting, HallC5×3−1+2 ⋊ C4
SylowC5 ⋊ C4×3−1+2

Subgroups

subgroupG/H
center, FrattiniC3C32×AGL(1,𝔽5)
commutator, socleC3×C5C32×C4
FittingC5×3−1+2C4
prime radicalC32×D5C2×C3
cosocleC3×D5C2×C32
hypercenter3−1+2AGL(1,𝔽5)
hypocenterC5C4×3−1+2
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C3 ◁ 3−1+2

48 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)×AGL(1,ℤ/9), C2×AGL(1,𝔽5)×3−1+2, AGL(1,𝔽5)×AΣL(1,𝔽27), …
maximal normal subgroup ofAGL(1,𝔽5)×AGL(1,ℤ/9), C2×AGL(1,𝔽5)×3−1+2, AGL(1,𝔽5)×AΣL(1,𝔽27), …
maximal quotient of3−1+2×#170, C2×AGL(1,𝔽5)×3−1+2, AGL(1,𝔽5)×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 34);