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#123×AGL(1,ℤ/9) #13153063

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

Aliases

direct product#123×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×#123)
unambiguous semidirect product(D9×#123)⋊C3

Splittings

Fitting3−1+2×#123 ⋊ C2
Sylow3−1+2 ⋊ C2×#123

Subgroups

subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#123C2
prime radicalC2×D4×AGL(1,ℤ/9)C2
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C2×C3×#123
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #123

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/9)×#525, AGL(1,ℤ/9)×#546, AGL(1,ℤ/9)×#534
maximal normal subgroup ofAGL(1,ℤ/9)×#525, AGL(1,ℤ/9)×#546, AGL(1,ℤ/9)×#534

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19126);