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C3×D5×ΓL(1,𝔽8)×D17 #36339356

order21420
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×D5×ΓL(1,𝔽8)×D17
unambiguous semidirect product(D5×C17)⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C5×D17)⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect productD85⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D5×D17)⋊C32
unambiguous semidirect product(C3×C7×D5×D17)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C22×C3
derived, HallC5×C7×C17 ⋊ C22×C32
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallC7×D5×D17 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ C2×D5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D17
HallC7×C17 ⋊ C2×C32×D5
HallC5×C17 ⋊ C22×C3×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C32×D17
SylowC17 ⋊ C2×C3×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C32×D5×D17
SylowC5 ⋊ C2×C3×ΓL(1,𝔽8)×D17

Subgroups

subgroupG/H
center, hypercenterC3D5×ΓL(1,𝔽8)×D17
commutator, cosocle, hypocenterC5×C7×C17C22×C32
FrattiniC1G
Fitting, socleC3×C5×C7×C17C22×C3
LCSG ▷ C5×C7×C17
UCSC1 ◁ C3

240 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 1785

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 188);