Finite Group Databasebeta

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C2×C3⋊(C3⋊(C5⋊(D31⋊C5))) #42124831

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

Aliases

direct productC2×C3⋊(C3⋊(C5⋊(D31⋊C5)))
unambiguous semidirect product(C2×C3)⋊(C3⋊(C5⋊(D31⋊C5)))
unambiguous semidirect product(C2×C5)⋊(C3⋊(C3⋊(D31⋊C5)))
unambiguous semidirect product(C2×ΓL(1,𝔽32))⋊(C3⋊D15)
unambiguous semidirect product(C2×C3⋊D465)⋊C5

Splittings

Fitting, socleC2×C32×C5×C31 ⋊ C2×C5
derivedC32×C5×C31 ⋊ C22×C5
HallC32×C5×ΓL(1,𝔽32) ⋊ C22
HallC5×ΓL(1,𝔽32) ⋊ C2×C3⋊D3
HallC32×C31 ⋊ C2×C5×D5
SylowC31 ⋊ C2×C5×C3⋊D15
SylowC32 ⋊ C2×C5⋊(D31⋊C5)

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C3⋊(C5⋊(D31⋊C5)))
commutator, cosocle, hypocenterC32×C5×C31C22×C5
FrattiniC1G
Fitting, socleC2×C32×C5×C31C2×C5
LCSG ▷ C32×C5×C31
UCSC1 ◁ C2

288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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

510 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 578);