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ΓL(1,𝔽8)×#63120⋊C3 #30655293

order16380
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productΓL(1,𝔽8)×#63120⋊C3
unambiguous semidirect product(C7×#63120⋊C3)⋊C3
unambiguous semidirect product#19129741⋊C3
unambiguous semidirect product#19129742⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)×#63120)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C13 ⋊ C32×C4
HallC5×ΓL(1,𝔽8)×C13⋊C3 ⋊ C4
HallC7×#63120 ⋊ C32
HallΓL(1,𝔽8)×C13⋊C3 ⋊ AGL(1,𝔽5)
Hall#63120 ⋊ C3×ΓL(1,𝔽8)
HallC7×C13 ⋊ C32×AGL(1,𝔽5)
HallC5×C13 ⋊ C3×C4×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×Dic13⋊C3
SylowC13 ⋊ C3×AGL(1,𝔽5)×ΓL(1,𝔽8)
SylowC7 ⋊ C3×#63120⋊C3
SylowC5 ⋊ ΓL(1,𝔽8)×Dic13⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C13C32×C4
prime radicalD5×ΓL(1,𝔽8)×C13⋊C3C2
cosocleC7×D5×C13C2×C32
LCSG ▷ C5×C7×C13
UCSC1

144 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 455

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16380, 343);