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H(5,𝔽2)×#1181 #16836473

order3712
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productH(5,𝔽2)×#1181

Splittings

FittingC29×H(5,𝔽2) ⋊ C4
SylowC29 ⋊ C4×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC2C24×#1181
commutator, socleC2×C29C24×C4
FittingC29×H(5,𝔽2)C4
prime radicalH(5,𝔽2)×D29C2
cosocleC2×D29C25
hypercenterH(5,𝔽2)#1181
hypocenterC29C4×H(5,𝔽2)
LCSG ▷ C2×C29 ▷ C29
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) = 232

Conjugacy classes & character table

187 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 21491);