Finite Group Databasebeta

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C5×C3⋊((C31⋊C3)⋊AGL(1,𝔽5)) #42124802

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

Aliases

direct productC5×C3⋊((C31⋊C3)⋊AGL(1,𝔽5))
unambiguous semidirect productC3⋊(C5×(C31⋊C3)⋊AGL(1,𝔽5))
unambiguous semidirect product(C5×C3⋊(C31⋊AGL(1,𝔽5)))⋊C3

Splittings

Fitting, socleC3×C52×C31 ⋊ C3×C4
derivedC3×C5×C31 ⋊ C3×C4×C5
HallC3×C52×C31⋊C3 ⋊ C4
HallC52×C31 ⋊ C3×Dic3
HallC3×C31⋊C3 ⋊ C5×AGL(1,𝔽5)
SylowC31 ⋊ C3×C5×ΓL(1,𝔽16)
SylowC52 ⋊ C3⋊((C31⋊C3)⋊C4)

Subgroups

subgroupG/H
center, hypercenterC5C3⋊((C31⋊C3)⋊AGL(1,𝔽5))
commutator, hypocenterC3×C5×C31C3×C4×C5
FrattiniC1G
Fitting, socleC3×C52×C31C3×C4
prime radicalC3×C5×D5×C31⋊C3C2
cosocleC3×D5×C31C2×C3×C5
LCSG ▷ C3×C5×C31
UCSC1 ◁ C5

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

435 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 549);