Finite Group Databasebeta

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H(5,𝔽2)×#10624382 #37550702

order23168
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productH(5,𝔽2)×#10624382

Splittings

FittingH(5,𝔽2)×C181 ⋊ C4
SylowC181 ⋊ C4×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC2C24×#10624382
commutator, socleC2×C181C24×C4
FittingH(5,𝔽2)×C181C4
prime radicalH(5,𝔽2)×D181C2
cosocleC2×D181C25
hypercenterH(5,𝔽2)#10624382
hypocenterC181C4×H(5,𝔽2)
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ H(5,𝔽2)

1660 conjugacy classes of subgroups; only the 754 normal subgroups are shown.

normal subgroups only

This group has 1660 conjugacy classes of subgroups — too many to draw. Only its 754 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) = 1448

Conjugacy classes & character table

833 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23168, 21491);