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C5×A4×C17×ΓL(1,𝔽8) #36339397

order21420
automorphism?
homologyAbelianization (H1) = C32×C5×C17 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×A4×C17×ΓL(1,𝔽8)
unambiguous semidirect product(C22×ΓL(1,𝔽8))⋊(C3×C5×C17)
unambiguous semidirect product(C7×A4)⋊(C3×C5×C17)
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊(C3×C5×C17)
unambiguous semidirect product(C22×C5×ΓL(1,𝔽8))⋊(C3×C17)
unambiguous semidirect product(C5×C7×A4)⋊(C3×C17)
unambiguous semidirect product(C5×C22⋊ΓL(1,𝔽8))⋊(C3×C17)
unambiguous semidirect product(C22×C17×ΓL(1,𝔽8))⋊(C3×C5)
unambiguous semidirect product(C7×A4×C17)⋊(C3×C5)
unambiguous semidirect product(C17×C22⋊ΓL(1,𝔽8))⋊(C3×C5)
unambiguous semidirect product(C5×C17×C22⋊ΓL(1,𝔽8))⋊C3
unambiguous semidirect product(C5×C7×A4×C17)⋊C3
unambiguous semidirect product(C22×C5×C17×ΓL(1,𝔽8))⋊C3

Splittings

Fitting, socle, HallC22×C5×C7×C17 ⋊ C32
derived, HallC22×C7 ⋊ C32×C5×C17
HallC5×C7×C17 ⋊ C3×A4
HallC22×C7×C17 ⋊ C32×C5
HallC22×C5×C17 ⋊ C3×ΓL(1,𝔽8)
HallC22×C5×C7 ⋊ C32×C17
HallC7×C17 ⋊ C3×C5×A4
HallC22×C17 ⋊ C3×C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×A4×C17
HallC22×C5 ⋊ C3×C17×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C5×A4×C17
SylowC22 ⋊ C3×C5×C17×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC5×C17A4×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C32×C5×C17
FrattiniC1G
Fitting, socleC22×C5×C7×C17C32
LCSG ▷ C22×C7
UCSC1 ◁ C5×C17

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2380

Conjugacy classes & character table

1700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 229);