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C2×C13⋊C3×ΓL(1,𝔽32) #27130636

order12090
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×C13⋊C3×ΓL(1,𝔽32)
unambiguous semidirect product(C2×C13)⋊(C3×ΓL(1,𝔽32))
unambiguous semidirect product(C2×C31×C13⋊C3)⋊C5

Splittings

Fitting, socle, HallC2×C13×C31 ⋊ C3×C5
derived, HallC13×C31 ⋊ C2×C3×C5
HallC2×C13×ΓL(1,𝔽32) ⋊ C3
HallC2×C31×C13⋊C3 ⋊ C5
HallC13×ΓL(1,𝔽32) ⋊ C2×C3
HallC31×C13⋊C3 ⋊ C2×C5
HallC2×C31 ⋊ C5×C13⋊C3
SylowC31 ⋊ C2×C5×C13⋊C3
HallC2×C13 ⋊ C3×ΓL(1,𝔽32)
SylowC13 ⋊ C2×C3×ΓL(1,𝔽32)

Subgroups

subgroupG/H
center, hypercenterC2C13⋊C3×ΓL(1,𝔽32)
commutator, cosocle, hypocenterC13×C31C2×C3×C5
FrattiniC1G
Fitting, socleC2×C13×C31C3×C5
LCSG ▷ C13×C31
UCSC1 ◁ C2

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×C13⋊C3×ΓL(1,𝔽32), C13⋊C3×Dic31⋊C5, ΓL(1,𝔽32)×Dic13⋊C3, …
maximal normal subgroup ofC4×C13⋊C3×ΓL(1,𝔽32), C13⋊C3×Dic31⋊C5, ΓL(1,𝔽32)×Dic13⋊C3, …
maximal quotient ofC4×C13⋊C3×ΓL(1,𝔽32), C22×C13⋊C3×ΓL(1,𝔽32), C2×C13⋊C9×ΓL(1,𝔽32), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12090, 8);