Finite Group Databasebeta

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C2×C3×ΓL(1,𝔽8)×D65 #30655580

order16380
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×ΓL(1,𝔽8)×D65
unambiguous semidirect product(C2×C5)⋊(C3×ΓL(1,𝔽8)×D13)

Splittings

Fitting, socleC2×C3×C5×C7×C13 ⋊ C2×C3
derived, HallC5×C7×C13 ⋊ C22×C32
HallC3×C5×C13×ΓL(1,𝔽8) ⋊ C22
HallC2×C7×D65 ⋊ C32
HallC3×C13×ΓL(1,𝔽8) ⋊ C2×D5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D13
HallC7×C13 ⋊ C2×C32×D5
HallC5×C13 ⋊ C22×C3×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C32×D13
SylowC13 ⋊ C2×C3×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C32×D65
SylowC5 ⋊ C2×C3×ΓL(1,𝔽8)×D13

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)×D65
commutator, cosocle, hypocenterC5×C7×C13C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C13C2×C3
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2×C3

240 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2730

Conjugacy classes & character table

1020 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16380, 630);