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C3×C5×(C31⋊C3)⋊AGL(1,𝔽5) #42124754

order27900
automorphism?
homologyAbelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×(C31⋊C3)⋊AGL(1,𝔽5)
unambiguous semidirect product(C5×C31⋊AGL(1,𝔽5))⋊C32
unambiguous semidirect product(C3×C5×C31⋊AGL(1,𝔽5))⋊C3

Splittings

Fitting, socleC3×C52×C31 ⋊ C3×C4
derivedC5×C31 ⋊ C32×C4×C5
HallC3×C52×C31⋊C3 ⋊ C4
HallC5×C31⋊AGL(1,𝔽5) ⋊ C32
HallC52×C31 ⋊ C32×C4
HallC3×C31⋊C3 ⋊ C5×AGL(1,𝔽5)
SylowC31 ⋊ C32×C5×AGL(1,𝔽5)
SylowC52 ⋊ C3×(C31⋊C3)⋊C4

Subgroups

subgroupG/H
center, hypercenterC3×C5(C31⋊C3)⋊AGL(1,𝔽5)
commutator, hypocenterC5×C31C32×C4×C5
FrattiniC1G
Fitting, socleC3×C52×C31C3×C4
prime radicalC3×C5×D5×C31⋊C3C2
cosocleD5×C31C2×C32×C5
LCSG ▷ C5×C31
UCSC1 ◁ C3×C5

156 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 60 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

525 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 501);