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#120×AGL(1,ℤ/9) #13152780

order1728
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct product#120×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×#120)

Splittings

Fitting3−1+2×#120 ⋊ C2
Sylow3−1+2 ⋊ C2×#120

Subgroups

subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#120C2
hypercenter#120AGL(1,ℤ/9)
hypocenterC9C2×C3×#120
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #120

2051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72220089, AGL(1,ℤ/9)×B(4,𝔽2), #120×C3⋊AGL(1,ℤ/9), …
maximal normal subgroup ofAGL(1,ℤ/9)×B(4,𝔽2), AGL(1,ℤ/9)×#535, AGL(1,ℤ/9)×#525, …
maximal quotient ofAGL(1,ℤ/9)×#453, #72220089, AGL(1,ℤ/9)×B(4,𝔽2), …
automorphism group ofC4×Dic9, C4⋊Dic9, #3676, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 18843);