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C2×C3×C11×ΓL(1,𝔽32) #23779403

order10230
isoclinismisoclinic class of ΓL(1,𝔽32)
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C11 · Schur multiplier (H2) = C1

Aliases

direct productC2×C3×C11×ΓL(1,𝔽32)
unambiguous semidirect product(C2×C3×C31)⋊(C5×C11)

Splittings

Fitting, socle, HallC2×C3×C11×C31 ⋊ C5
derived, SylowC31 ⋊ C2×C3×C5×C11
HallC3×C11×C31 ⋊ C2×C5
HallC2×C11×C31 ⋊ C3×C5
HallC11×C31 ⋊ C2×C3×C5
HallC2×C3×C31 ⋊ C5×C11
HallC3×C31 ⋊ C2×C5×C11
HallC2×C31 ⋊ C3×C5×C11

Subgroups

subgroupG/H
center, hypercenterC2×C3×C11ΓL(1,𝔽32)
commutator, cosocle, hypocenterC31C2×C3×C5×C11
FrattiniC1G
Fitting, socleC2×C3×C11×C31C5
LCSG ▷ C31
UCSC1 ◁ C2×C3×C11

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

726 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C4×C11×ΓL(1,𝔽32), C3×C11×Dic31⋊C5, C3×Dic11×ΓL(1,𝔽32), …
maximal normal subgroup ofC3×C4×C11×ΓL(1,𝔽32), C3×C11×Dic31⋊C5, C3×Dic11×ΓL(1,𝔽32), …
maximal quotient ofC3×C4×C11×ΓL(1,𝔽32), C22×C3×C11×ΓL(1,𝔽32), C2×C9×C11×ΓL(1,𝔽32), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10230, 2);