Finite Group Databasebeta

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#272×B(4,𝔽2) #72253275

order3456
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct product#272×B(4,𝔽2)

Splittings

FittingH(3,𝔽3)×B(4,𝔽2) ⋊ C2
SylowH(3,𝔽3) ⋊ C2×B(4,𝔽2)

Subgroups

subgroupG/H
center, socleC2×C3C3⋊D3×#120
commutator, cosocleC23×H(3,𝔽3)C24
FrattiniC23×C3C23×C3⋊D3
FittingH(3,𝔽3)×B(4,𝔽2)C2
hypercenterC3×B(4,𝔽2)C3⋊D3
hypocenterH(3,𝔽3)C2×B(4,𝔽2)
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×B(4,𝔽2)

5599 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 5599 conjugacy classes of subgroups — too many to draw. Only its 270 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

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofB(4,𝔽2)×((C3⋊D3)⋊C3)⋊C2
maximal normal subgroup ofB(4,𝔽2)×((C3⋊D3)⋊C3)⋊C2

GAP

Past the SmallGroups library coverage — no SmallGroups id.