Finite Group Databasebeta

← Prev (#13154017) Next (#13154019) →

#120×#272 #13154018

order1728
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct product#120×#272

Splittings

FittingH(3,𝔽3)×#120 ⋊ C2
SylowH(3,𝔽3) ⋊ C2×#120

Subgroups

subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#120C2
hypercenterC3×#120C3⋊D3
hypocenterH(3,𝔽3)C2×#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#120

3641 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 3641 conjugacy classes of subgroups — too many to draw. Only its 261 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.

Relationships

maximal subgroup of#120×((C3⋊D3)⋊C3)⋊C2, #272×B(4,𝔽2), #72253277
maximal normal subgroup of#120×((C3⋊D3)⋊C3)⋊C2, #272×B(4,𝔽2), #72253277
maximal quotient of#272×B(4,𝔽2)
automorphism group ofC22⋊C4×3−1+2, C4⋊C4×3−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20081);