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C2×C3×C5⋊(C17⋊AGL(1,𝔽7)) #36339402

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

Aliases

direct productC2×C3×C5⋊(C17⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×C5)⋊(C3×C17⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×C17)⋊(C3×C5⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C3×D85)
unambiguous semidirect product(C2×D595)⋊C32
unambiguous semidirect product(C2×C3×D595)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C17 ⋊ C2×C3
derived, HallC5×C7×C17 ⋊ C22×C32
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallC2×D595 ⋊ 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 ⋊ C2×C3×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C32×D85
SylowC5 ⋊ C2×C3×C17⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊(C17⋊AGL(1,𝔽7))
commutator, cosocle, hypocenterC5×C7×C17C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C17C2×C3
LCSG ▷ C5×C7×C17
UCSC1 ◁ C2×C3

240 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 82 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

1302 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 234);