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C3×C3⋊(ΓL(1,𝔽32)⋊AGL(1,𝔽5)) #42124708

order27900
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

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

Splittings

Fitting, socleC32×C5×C31 ⋊ C4×C5
derivedC3×C5×C31 ⋊ C3×C4×C5
HallC32×C5×ΓL(1,𝔽32) ⋊ C4
HallC5×ΓL(1,𝔽32) ⋊ C3×Dic3
HallC32×C31 ⋊ C5×AGL(1,𝔽5)
SylowC31 ⋊ C3×C5×ΓL(1,𝔽16)
SylowC32 ⋊ ΓL(1,𝔽32)⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC3C3⋊(ΓL(1,𝔽32)⋊AGL(1,𝔽5))
commutator, hypocenterC3×C5×C31C3×C4×C5
FrattiniC1G
Fitting, socleC32×C5×C31C4×C5
prime radicalC32×D5×ΓL(1,𝔽32)C2
cosocleC3×D5×C31C2×C3×C5
LCSG ▷ C3×C5×C31
UCSC1 ◁ C3

124 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 40 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

243 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 455);