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C3×C4×AΓL(1,𝔽9) #13180384

order1728
isoclinismisoclinic class of AΓL(1,𝔽9)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×C4×AΓL(1,𝔽9)

Splittings

FittingC33×C4 ⋊ SD16
SylowC33 ⋊ C4×SD16

Subgroups

subgroupG/H
center, hypercenterC3×C4AΓL(1,𝔽9)
commutator#157C22×C3×C4
FrattiniC2C2×C3×AΓL(1,𝔽9)
FittingC33×C4SD16
socleC2×C33C2×SD16
prime radicalC2×C3×GO+(4,𝔽2)C22
cosocleC2×#157C23×C3
hypocenterC32C3×C4×SD16
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C3×C4

314 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 96

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽5)×AΓL(1,𝔽9), C3×C4×AGL(2,𝔽3), C2×C3×C4×AΓL(1,𝔽9), …
maximal normal subgroup ofC2×C3×C4×AΓL(1,𝔽9), C3×C8×AΓL(1,𝔽9), C3×C4×C5×AΓL(1,𝔽9)
maximal quotient ofC3×AGL(1,𝔽5)×AΓL(1,𝔽9), C2×C3×C4×AΓL(1,𝔽9), C3×C8×AΓL(1,𝔽9), …
automorphism group ofC13×#157

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46447);