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D3×D5×C17×ΓL(1,𝔽8) #36339393

order21420
automorphism?
homologyAbelianization (H1) = C22×C3×C17 · Schur multiplier (H2) = C2

Aliases

direct productD3×D5×C17×ΓL(1,𝔽8)
unambiguous semidirect product(C5×D3)⋊(C2×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C3×D5)⋊(C2×C17×ΓL(1,𝔽8))
unambiguous semidirect productD15⋊(C2×C17×ΓL(1,𝔽8))
unambiguous semidirect product(D3×C7×D5)⋊(C3×C17)
unambiguous semidirect product(D3×C7×D5×C17)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C22×C3
derivedC3×C5×C7 ⋊ C22×C3×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallC3×C17×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7×C17 ⋊ C2×C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C22×C17
HallC7×C17 ⋊ C3×D3×D5
HallC5×C17 ⋊ C2×D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ C2×D5×C17
HallC5×C7 ⋊ C2×C3×D3×C17
SylowC7 ⋊ C3×D3×D5×C17
SylowC5 ⋊ C2×D3×C17×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC17D3×D5×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC3×C5×C7C22×C3×C17
FrattiniC1G
Fitting, socleC3×C5×C7×C17C22×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C17

176 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 60 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

1020 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 225);