Finite Group Databasebeta

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C2×C3×C5×C17×AGL(1,𝔽7) #36339398

order21420
automorphism?
homologyAbelianization (H1) = C22×C32×C5×C17 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C5×C17×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C32×C5×C17)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C3×C5×C17)
unambiguous semidirect product(C2×D7)⋊(C32×C5×C17)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C3×C5×C17)
unambiguous semidirect product(C2×C3×D7)⋊(C3×C5×C17)
unambiguous semidirect product(C2×C5×D7)⋊(C32×C17)
unambiguous semidirect product(C2×C3×C5×D7)⋊(C3×C17)
unambiguous semidirect product(C2×D7×C17)⋊(C32×C5)
unambiguous semidirect product(C2×C3×D7×C17)⋊(C3×C5)
unambiguous semidirect product(C2×C5×D7×C17)⋊C32
unambiguous semidirect product(C2×C3×C5×D7×C17)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C17 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C32×C5×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C22
HallC2×C5×D7×C17 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7×C17 ⋊ C22×C32
HallC2×D7×C17 ⋊ C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ C22×C17
HallC2×C5×D7 ⋊ C32×C17
HallC7×C17 ⋊ C22×C32×C5
HallC3×ΓL(1,𝔽8) ⋊ C22×C5×C17
HallC5×C7 ⋊ C22×C32×C17
HallC2×D7 ⋊ C32×C5×C17

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5×C17AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C32×C5×C17
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C17C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5×C17

240 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 136 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

3570 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 230);