Finite Group Databasebeta

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C22⋊C4×C22⋊H(3,𝔽3) #13152165

order1728
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C23×C32

Aliases

direct productC22⋊C4×C22⋊H(3,𝔽3)
unambiguous semidirect product(C22×C22⋊C4)⋊H(3,𝔽3)
unambiguous semidirect product(C32×C22⋊C4)⋊A4
unambiguous semidirect product(C3×A4×C22⋊C4)⋊C3

Splittings

FittingC22×C32×C22⋊C4 ⋊ C3
SylowC22×C22⋊C4 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC22×C3C22×C3×A4
commutatorC23×C3C2×C32×C4
FittingC22×C32×C22⋊C4C3
socle, cosocleC24×C3C22×C32
prime radicalC23×C22⋊H(3,𝔽3)C2
hypercenterC32×C22⋊C4A4
hypocenterC22C22⋊C4×H(3,𝔽3)
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C22×C3 ◁ C32×C22⋊C4

980 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 72

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 18228);