Finite Group Databasebeta

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#137×AGL(1,ℤ/9) #13153519

order1728
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct product#137×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×#137)
unambiguous semidirect product(D9×#137)⋊C3

Splittings

Fitting3−1+2×#137 ⋊ C2
Sylow3−1+2 ⋊ C2×#137

Subgroups

subgroupG/H
centerC2C2×D4×AGL(1,ℤ/9)
commutator, cosocleC4×C9C24×C3
FrattiniC3×C4C23×C3×D3
Fitting3−1+2×#137C2
socleC2×C3C2×C3×D3×D4
prime radical#41×AGL(1,ℤ/9)C2
hypercenter#137AGL(1,ℤ/9)
hypocenterC9C2×C3×#137
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19582);