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C3×ΓL(1,𝔽8)⋊#63121 #30655334

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

Aliases

direct productC3×ΓL(1,𝔽8)⋊#63121
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×#63121)

Splittings

Fitting, socleC3×C5×C7×C13 ⋊ C3×C4
derived, HallC5×C7×C13 ⋊ C32×C4
HallC3×C5×C13×ΓL(1,𝔽8) ⋊ C4
HallC7⋊#63121 ⋊ C32
HallC3×C13×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)
HallC3×C5×ΓL(1,𝔽8) ⋊ #261
HallC7×C13 ⋊ C32×AGL(1,𝔽5)
HallC5×C13 ⋊ C3×ΓL(1,𝔽8)⋊C4
HallC3×ΓL(1,𝔽8) ⋊ #63121
HallC5×C7 ⋊ C32×#261
SylowC13 ⋊ C3×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
SylowC7 ⋊ C32×#63121
SylowC5 ⋊ C3×ΓL(1,𝔽8)⋊#261

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)⋊#63121
commutator, hypocenterC5×C7×C13C32×C4
FrattiniC1G
Fitting, socleC3×C5×C7×C13C3×C4
prime radicalC3×ΓL(1,𝔽8)×D65C2
cosocleC7×D65C2×C32
LCSG ▷ C5×C7×C13
UCSC1 ◁ C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1365

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16380, 384);