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D5×ΓL(1,𝔽8)×D51 #36339394

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

Aliases

direct productD5×ΓL(1,𝔽8)×D51
unambiguous semidirect product(C3×D5)⋊(ΓL(1,𝔽8)×D17)
unambiguous semidirect product(D5×C17)⋊(D3×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D5×D51)⋊C3

Splittings

Fitting, socle, derivedC3×C5×C7×C17 ⋊ C22×C3
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) ⋊ C2×D17
HallC7×C17 ⋊ C3×D3×D5
HallC5×C17 ⋊ C2×D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ D5×D17
HallC5×C7 ⋊ C2×C3×D51
SylowC17 ⋊ D3×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×D5×D51
SylowC5 ⋊ C2×ΓL(1,𝔽8)×D51

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C17C22×C3
LCSG ▷ C3×C5×C7×C17
UCSC1

176 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 48 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 226);