Finite Group Databasebeta

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C23×ΓL(1,𝔽32)⋊C9 #44545542

order32085
automorphism?
homologyAbelianization (H1) = C5×C9×C23 · Schur multiplier (H2) = C1

Aliases

direct productC23×ΓL(1,𝔽32)⋊C9
unambiguous semidirect productΓL(1,𝔽32)⋊(C9×C23)
unambiguous semidirect product(C31⋊C9)⋊(C5×C23)
unambiguous semidirect product(C23×ΓL(1,𝔽32))⋊C9
unambiguous semidirect product(C23×C31⋊C9)⋊C5

Splittings

derived, SylowC31 ⋊ C5×C9×C23
HallC23×C31⋊C9 ⋊ C5
HallC23×ΓL(1,𝔽32) ⋊ C9
HallC23×C31 ⋊ C5×C9
HallC31⋊C9 ⋊ C5×C23
HallΓL(1,𝔽32) ⋊ C9×C23

Subgroups

subgroupG/H
center, hypercenterC3×C23(C31⋊C3)⋊C5
commutator, hypocenterC31C5×C9×C23
FrattiniC3C23×(C31⋊C3)⋊C5
Fitting, socleC3×C23×C31C3×C5
prime radicalC3×C23×ΓL(1,𝔽32)C3
cosocleC3×C31C3×C5×C23
LCSG ▷ C31
UCSC1 ◁ C3×C23

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 6417

Conjugacy classes & character table

1173 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32085, 4);