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C2×C3×D5×C17×ΓL(1,𝔽8) #36339403

order21420
automorphism?
homologyAbelianization (H1) = C22×C32×C17 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×D5×C17×ΓL(1,𝔽8)
unambiguous semidirect productC5⋊(C22×C3×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C5)⋊(C2×C3×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D5)⋊(C32×C17)
unambiguous semidirect product(C2×C3×C7×D5)⋊(C3×C17)
unambiguous semidirect product(C2×C7×D5×C17)⋊C32
unambiguous semidirect product(C2×C3×C7×D5×C17)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C17 ⋊ C2×C3
derived, HallC5×C7 ⋊ C22×C32×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallC2×C7×D5×C17 ⋊ C32
HallC5×C7×C17 ⋊ C22×C32
HallC3×C5×ΓL(1,𝔽8) ⋊ C22×C17
HallC2×C7×D5 ⋊ C32×C17
HallC7×C17 ⋊ C2×C32×D5
HallC5×C17 ⋊ C22×C3×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C32×D5×C17
SylowC5 ⋊ C22×C3×C17×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2×C3×C17D5×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC5×C7C22×C32×C17
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C17C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C2×C3×C17

240 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 112 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

2040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 235);