Finite Group Databasebeta

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C5×#13049954⋊C5 #45654445

order32550
automorphism?
homologyAbelianization (H1) = C2×C3×C52 · Schur multiplier (H2) = C5

Aliases

direct productC5×#13049954⋊C5
unambiguous semidirect product#13049954⋊C52
unambiguous semidirect product(C5×#13049954)⋊C5

Splittings

Fitting, socleC5×C7×C31 ⋊ C2×C3×C5
derived, HallC7×C31 ⋊ C2×C3×C52
HallC5×#10620252⋊C5 ⋊ C2
HallC5×C7×D31⋊C5 ⋊ C3
HallC5×C7×ΓL(1,𝔽32) ⋊ C2×C3
HallC5×D31⋊C5 ⋊ ΓL(1,𝔽8)
Hall#13049954 ⋊ C52
HallC5×ΓL(1,𝔽32) ⋊ C2×ΓL(1,𝔽8)
Hall#10620252 ⋊ C2×C52
HallC7×D31 ⋊ C3×C52
HallD31 ⋊ C52×ΓL(1,𝔽8)
SylowC31 ⋊ C2×C52×ΓL(1,𝔽8)
SylowC7 ⋊ C5×AGL(1,𝔽31)

Subgroups

subgroupG/H
center, hypercenterC5#13049954⋊C5
commutator, cosocle, hypocenterC7×C31C2×C3×C52
FrattiniC1G
Fitting, socleC5×C7×C31C2×C3×C5
LCSG ▷ C7×C31
UCSC1 ◁ C5

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1085

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32550, 81);