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C3×AGL(1,𝔽7)×D85 #36339352

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

Aliases

direct productC3×AGL(1,𝔽7)×D85
unambiguous semidirect productD7⋊(C32×D85)
unambiguous semidirect product(C5×AGL(1,𝔽7))⋊(C3×D17)
unambiguous semidirect product(C17×AGL(1,𝔽7))⋊(C3×D5)
unambiguous semidirect product(D7×D85)⋊C32
unambiguous semidirect product(C3×D7×D85)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C22×C3
derived, HallC5×C7×C17 ⋊ C22×C32
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallD7×D85 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ C2×D5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D17
HallC7×C17 ⋊ C2×C32×D5
HallC5×C17 ⋊ C2×C3×AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×D85
HallC5×C7 ⋊ C2×C32×D17
SylowC17 ⋊ C3×D5×AGL(1,𝔽7)
SylowC7 ⋊ C2×C32×D85
SylowC5 ⋊ C3×D17×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽7)×D85
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 76 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 76 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 184);