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C2×C3×C5×AGL(1,𝔽31) #42124691

order27900
automorphism?
homologyAbelianization (H1) = C22×C32×C52 · Schur multiplier (H2) = C2×C3×C5

Aliases

BorelB(2,𝔽31)
direct productC2×C3×C5×AGL(1,𝔽31)
unambiguous semidirect product(D31⋊C3)⋊(C2×C3×C52)
unambiguous semidirect product(D31⋊C5)⋊(C2×C32×C5)
unambiguous semidirect product(C2×D31⋊C3)⋊(C3×C52)
unambiguous semidirect product((C31⋊C3)⋊C5)⋊(C22×C3×C5)
unambiguous semidirect product(C2×D31⋊C5)⋊(C32×C5)
unambiguous semidirect product(C2×(C31⋊C3)⋊C5)⋊(C2×C3×C5)
unambiguous semidirect product(C2×C3×D31⋊C3)⋊C52
unambiguous semidirect product(C2×C3×D31⋊C5)⋊(C3×C5)
unambiguous semidirect product(C2×C5×D31⋊C3)⋊(C3×C5)
unambiguous semidirect product(C2×C5×D31⋊C5)⋊C32
unambiguous semidirect product(C2×C3×C5×D31⋊C3)⋊C5
unambiguous semidirect product(C2×C3×C5×D31⋊C5)⋊C3

Splittings

Fitting, socleC2×C3×C5×C31 ⋊ C2×C3×C5
derived, SylowC31 ⋊ C22×C32×C52
HallC3×C5×(C31⋊C3)⋊C5 ⋊ C22
HallC2×C5×D31⋊C5 ⋊ C32
HallC2×C3×D31⋊C3 ⋊ C52
HallC5×ΓL(1,𝔽32) ⋊ C22×C32
HallC3×C31⋊C3 ⋊ C22×C52
HallC2×D31 ⋊ C32×C52

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5AGL(1,𝔽31)
commutator, cosocle, hypocenterC31C22×C32×C52
FrattiniC1G
Fitting, socleC2×C3×C5×C31C2×C3×C5
LCSG ▷ C31
UCSC1 ◁ C2×C3×C5

480 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Conjugacy classes & character table

930 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 438);