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C2×C5×ΓL(1,𝔽8)×D51 #36339408

order21420
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×ΓL(1,𝔽8)×D51
unambiguous semidirect product(C2×C3)⋊(C5×ΓL(1,𝔽8)×D17)
unambiguous semidirect product(C2×C17)⋊(C5×D3×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D51)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C7×D51)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C17 ⋊ C2×C3
derivedC3×C7×C17 ⋊ C22×C3×C5
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallC3×C17×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7×C17 ⋊ C2×C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D17
HallC7×C17 ⋊ C2×C3×C5×D3
HallC5×C17 ⋊ C2×D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ C2×C5×D17
HallC5×C7 ⋊ C2×C3×D51
SylowC17 ⋊ C2×C5×D3×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×C5×D51

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×D51
commutator, cosocle, hypocenterC3×C7×C17C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C17C2×C3
LCSG ▷ C3×C7×C17
UCSC1 ◁ C2×C5

176 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 3570

Conjugacy classes & character table

1350 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 240);