Finite Group Databasebeta

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D8×C22⋊H(3,𝔽3) #13152225

order1728
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C32

Aliases

direct productD8×C22⋊H(3,𝔽3)
unambiguous semidirect product(C22×D8)⋊H(3,𝔽3)
unambiguous semidirect product(C32×D8)⋊A4
unambiguous semidirect product(C3×A4×D8)⋊C3

Splittings

FittingC22×C32×D8 ⋊ C3
SylowC22×D8 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC2×C3C3×D4×A4
commutator, cosocleC22×C3×C4C22×C32
FrattiniC3×C4C22×C3×A4
FittingC22×C32×D8C3
socleC23×C3C32×D4
hypercenterC32×D8A4
hypocenterC22D8×H(3,𝔽3)
LCSG ▷ C22×C3×C4 ▷ C23 ▷ C22
UCSC1 ◁ C2×C3 ◁ C32×C4 ◁ C32×D8

536 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 70 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

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 18288);