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#127×AGL(1,ℤ/9) #13153281

order1728
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4

Aliases

direct product#127×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×#127)

Splittings

Fitting3−1+2×#127 ⋊ C2
Sylow3−1+2 ⋊ C2×#127

Subgroups

subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#127C2
hypercenter#127AGL(1,ℤ/9)
hypocenterC9C2×C3×#127
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #127

1548 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 1548 conjugacy classes of subgroups — too many to draw. Only its 288 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#127×C3⋊AGL(1,ℤ/9), AGL(1,ℤ/9)×#453, AGL(1,ℤ/9)×#535, …
maximal normal subgroup ofAGL(1,ℤ/9)×#453, AGL(1,ℤ/9)×#535, C2×#127×AGL(1,ℤ/9), …
maximal quotient of#127×C3⋊AGL(1,ℤ/9), C2×#127×AGL(1,ℤ/9), AGL(1,ℤ/9)×#493, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19344);