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C3×AGL(1,𝔽5)×(C31⋊C3)⋊C5 #42124671

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

Aliases

direct productC3×AGL(1,𝔽5)×(C31⋊C3)⋊C5
unambiguous semidirect product(AGL(1,𝔽5)×C31⋊C3)⋊(C3×C5)
unambiguous semidirect product(AGL(1,𝔽5)×ΓL(1,𝔽32))⋊C32
unambiguous semidirect product(C3×AGL(1,𝔽5)×C31⋊C3)⋊C5
unambiguous semidirect product(C3×AGL(1,𝔽5)×ΓL(1,𝔽32))⋊C3

Splittings

Fitting, socleC3×C5×C31 ⋊ C3×C4×C5
derivedC5×C31 ⋊ C32×C4×C5
HallC3×C5×(C31⋊C3)⋊C5 ⋊ C4
HallAGL(1,𝔽5)×ΓL(1,𝔽32) ⋊ C32
HallC5×ΓL(1,𝔽32) ⋊ C32×C4
HallC3×C31⋊C3 ⋊ C5×AGL(1,𝔽5)
SylowC31 ⋊ C32×C5×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽5)×(C31⋊C3)⋊C5
commutator, hypocenterC5×C31C32×C4×C5
FrattiniC1G
Fitting, socleC3×C5×C31C3×C4×C5
prime radicalC3×D5×(C31⋊C3)⋊C5C2
cosocleD5×C31C2×C32×C5
LCSG ▷ C5×C31
UCSC1 ◁ C3

156 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 56 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

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 418);