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C4×D4×AGL(1,ℤ/9) #13152647

order1728
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC4×D4×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×C4×D4)
unambiguous semidirect product(C4×D4×D9)⋊C3

Splittings

FittingC4×D4×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C2×C4×D4

Subgroups

subgroupG/H
centerC2×C4C22×AGL(1,ℤ/9)
commutatorC2×C9C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingC4×D4×3−1+2C2
prime radicalC2×D4×AGL(1,ℤ/9)C2
cosocleC22×C9C24×C3
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C2×C3×C4×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C4 ◁ C4×D4

1325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/9)×#442, AGL(1,ℤ/9)×#525, AGL(1,ℤ/9)×#532, …
maximal normal subgroup ofAGL(1,ℤ/9)×#442, AGL(1,ℤ/9)×#525, AGL(1,ℤ/9)×#532, …
maximal quotient ofAGL(1,ℤ/9)×#442, AGL(1,ℤ/9)×#444, D4×AGL(1,𝔽5)×AGL(1,ℤ/9), …
automorphism group ofC5×Dic18, C5×D36, C2×C5×Dic9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 18710);