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C2×C3×C4×AGL(1,ℤ/9) #13048839

order1296
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C3×C4×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C2×C32×C4)
unambiguous semidirect product(C2×C4×D9)⋊C32
unambiguous semidirect product(C2×C3×C4×D9)⋊C3

Splittings

FittingC2×C3×C4×3−1+2 ⋊ C2
derivedC9 ⋊ C22×C32×C4
SylowC3×3−1+2 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4AGL(1,ℤ/9)
commutator, hypocenterC9C22×C32×C4
FrattiniC2×C3C22×C32×D3
FittingC2×C3×C4×3−1+2C2
socleC22×C32C2×C3×D3
prime radicalC22×C3×AGL(1,ℤ/9)C2
cosocleC2×C9C23×C32
LCSG ▷ C9
UCSC1 ◁ C2×C3×C4

566 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 226 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

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×D4×AGL(1,ℤ/9), C2×C3×C4×#4183, C2×C4×D3×AGL(1,ℤ/9), …
maximal normal subgroup ofC2×C3×D4×AGL(1,ℤ/9), C2×C4×D3×AGL(1,ℤ/9), C2×D12×AGL(1,ℤ/9)
maximal quotient ofC2×AGL(1,ℤ/9)×AGL(1,𝔽13)
automorphism group ofC13×Dic9, C2×C13×D9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2557);