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C22×C3×C5×ΓL(1,𝔽32) #22604888

order9300
isoclinismisoclinic class of ΓL(1,𝔽32)
automorphism?
homologyAbelianization (H1) = C22×C3×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC22×C3×C5×ΓL(1,𝔽32)
unambiguous semidirect product(C22×C3×C31)⋊C52
unambiguous semidirect product(C22×C3×C5×C31)⋊C5

Splittings

Fitting, socleC22×C3×C5×C31 ⋊ C5
derived, SylowC31 ⋊ C22×C3×C52
HallC22×C3×C31 ⋊ C52
HallC22×C31 ⋊ C3×C52
HallC3×C31 ⋊ C22×C52

Subgroups

subgroupG/H
center, hypercenterC22×C3×C5ΓL(1,𝔽32)
commutator, hypocenterC31C22×C3×C52
FrattiniC1G
Fitting, socleC22×C3×C5×C31C5
LCSG ▷ C31
UCSC1 ◁ C22×C3×C5

160 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

660 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×C5×ΓL(1,𝔽32)⋊C9, C5×C22⋊(ΓL(1,𝔽32)⋊C9), C5×C22⋊C9×ΓL(1,𝔽32), …
maximal normal subgroup ofC22×C5×ΓL(1,𝔽32)⋊C9, C5×C22⋊(ΓL(1,𝔽32)⋊C9), C5×C22⋊C9×ΓL(1,𝔽32), …
maximal quotient ofC22×C5×C9×ΓL(1,𝔽32), C2×C3×C5×D3×ΓL(1,𝔽32), C22×C32×C5×ΓL(1,𝔽32)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9300, 162);