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C3×C5×ΓL(1,𝔽8)⋊#595 #36339338

order21420
automorphism?
homologyAbelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×ΓL(1,𝔽8)⋊#595
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C5×#595)
unambiguous semidirect product(C7⋊#595)⋊(C32×C5)
unambiguous semidirect product(C3×C7⋊#595)⋊(C3×C5)
unambiguous semidirect product(C5×C7⋊#595)⋊C32
unambiguous semidirect product(C3×C5×C7⋊#595)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C3×C4
derived, HallC7×C17 ⋊ C32×C4×C5
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC5×C7⋊#595 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7×C17 ⋊ C32×C4
HallC7⋊#595 ⋊ C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ #595
HallC5×C17 ⋊ C3×ΓL(1,𝔽8)⋊C4
HallC3×ΓL(1,𝔽8) ⋊ C5×#595
HallC5×C7 ⋊ C32×#595
SylowC17 ⋊ C3×C5×ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C32×C5×#595

Subgroups

subgroupG/H
center, hypercenterC3×C5ΓL(1,𝔽8)⋊#595
commutator, hypocenterC7×C17C32×C4×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C17C3×C4
prime radicalC3×C5×ΓL(1,𝔽8)×D17C2
cosocleC7×D17C2×C32×C5
LCSG ▷ C7×C17
UCSC1 ◁ C3×C5

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1785

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 170);