Finite Group Databasebeta

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H(3,𝔽3)×#627 #14518631

order1944
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C32

Aliases

direct productH(3,𝔽3)×#627
unambiguous semidirect product(C32×#627)⋊C3

Splittings

SylowC32×H(3,𝔽3) ⋊ Q8

Subgroups

subgroupG/H
center, FrattiniC2×C3C32×D32
commutator, socle, cosocleC2×C33C22×C32
Fitting, prime radicalC2×C32×H(3,𝔽3)C22
hypercenterC2×H(3,𝔽3)D32
hypocenterC32Q8×H(3,𝔽3)
LCSG ▷ C2×C33 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×H(3,𝔽3)

625 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 98 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

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 3025);