Finite Group Databasebeta

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#157×B(4,𝔽2) #72242742

order2304
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C3

Aliases

direct product#157×B(4,𝔽2)

Splittings

FittingC32×B(4,𝔽2) ⋊ C4
SylowC32 ⋊ C4×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2#120×#157
commutatorC23×C32C23×C4
FrattiniC23C23×#157
FittingC32×B(4,𝔽2)C4
socleC2×C32C4×#120
prime radicalC3⋊D3×B(4,𝔽2)C2
cosocleC23×C3⋊D3C24
hypercenterB(4,𝔽2)#157
hypocenterC32C4×B(4,𝔽2)
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ B(4,𝔽2)

2928 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 2928 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 48

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofB(4,𝔽2)×PSU(3,𝔽2), B(4,𝔽2)×GO+(4,𝔽2), B(4,𝔽2)×AGL(1,𝔽9)
maximal normal subgroup ofB(4,𝔽2)×PSU(3,𝔽2), B(4,𝔽2)×GO+(4,𝔽2), B(4,𝔽2)×AGL(1,𝔽9)

GAP

Past the SmallGroups library coverage — no SmallGroups id.