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C2×C5×D3×(C31⋊C3)⋊C5 #42124695

order27900
automorphism?
homologyAbelianization (H1) = C22×C3×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC2×C5×D3×(C31⋊C3)⋊C5
unambiguous semidirect productC3⋊(C22×C5×(C31⋊C3)⋊C5)
unambiguous semidirect product(C2×C3)⋊(C2×C5×(C31⋊C3)⋊C5)
unambiguous semidirect product(C2×D3×C31⋊C3)⋊C52
unambiguous semidirect product(C2×D3×ΓL(1,𝔽32))⋊(C3×C5)
unambiguous semidirect product(C2×C5×D3×C31⋊C3)⋊C5
unambiguous semidirect product(C2×C5×D3×ΓL(1,𝔽32))⋊C3

Splittings

Fitting, socleC2×C3×C5×C31 ⋊ C2×C3×C5
derivedC3×C31 ⋊ C22×C3×C52
HallC3×C5×(C31⋊C3)⋊C5 ⋊ C22
HallC2×D3×C31⋊C3 ⋊ C52
HallC5×ΓL(1,𝔽32) ⋊ C2×C3×D3
HallC3×C31⋊C3 ⋊ C22×C52
SylowC31 ⋊ C2×C3×C52×D3

Subgroups

subgroupG/H
center, hypercenterC2×C5D3×(C31⋊C3)⋊C5
commutator, cosocle, hypocenterC3×C31C22×C3×C52
FrattiniC1G
Fitting, socleC2×C3×C5×C31C2×C3×C5
LCSG ▷ C3×C31
UCSC1 ◁ C2×C5

352 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 126 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, 442);