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C2×C17×ΓL(1,𝔽8)×AGL(1,𝔽7) #44472724

order29988
automorphism?
homologyAbelianization (H1) = C22×C32×C17 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C17×ΓL(1,𝔽8)×AGL(1,𝔽7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×AGL(1,𝔽7))⋊(C3×C17)
unambiguous semidirect product(C2×D7×ΓL(1,𝔽8))⋊(C3×C17)
unambiguous semidirect product(C2×#64423)⋊(C3×C17)
unambiguous semidirect product(C2×#64422)⋊(C3×C17)
unambiguous semidirect product(C2×C7×C17×AGL(1,𝔽7))⋊C3
unambiguous semidirect product(C2×C17×#64423)⋊C3
unambiguous semidirect product(C2×C17×#64422)⋊C3
unambiguous semidirect product(C2×D7×C17×ΓL(1,𝔽8))⋊C3

Splittings

Fitting, socleC2×C72×C17 ⋊ C2×C32
derived, SylowC72 ⋊ C22×C32×C17
HallC17×ΓL(1,𝔽8)2 ⋊ C22
HallC2×C7×D7×C17 ⋊ C32
HallC72×C17 ⋊ C22×C32
HallΓL(1,𝔽8)2 ⋊ C22×C17
HallC2×C7×D7 ⋊ C32×C17

Subgroups

subgroupG/H
center, hypercenterC2×C17ΓL(1,𝔽8)×AGL(1,𝔽7)
commutator, cosocle, hypocenterC72C22×C32×C17
FrattiniC1G
Fitting, socleC2×C72×C17C2×C32
LCSG ▷ C72
UCSC1 ◁ C2×C17

248 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1428

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 365);