Finite Group Databasebeta

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C7×A4×C17×ΓL(1,𝔽8) #44472859

order29988
automorphism?
homologyAbelianization (H1) = C32×C7×C17 · Schur multiplier (H2) = C2×C3

Aliases

direct productC7×A4×C17×ΓL(1,𝔽8)
unambiguous semidirect product(C22×ΓL(1,𝔽8))⋊(C3×C7×C17)
unambiguous semidirect product(C7×A4)⋊(C3×C7×C17)
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊(C3×C7×C17)
unambiguous semidirect product(C22×C17×ΓL(1,𝔽8))⋊(C3×C7)
unambiguous semidirect product(C7×A4×C17)⋊(C3×C7)
unambiguous semidirect product(C17×C22⋊ΓL(1,𝔽8))⋊(C3×C7)

Splittings

Fitting, socle, HallC22×C72×C17 ⋊ C32
derivedC22×C7 ⋊ C32×C7×C17
HallC72×C17 ⋊ C3×A4
HallC22×C72 ⋊ C32×C17
HallC22×C17 ⋊ C3×C7×ΓL(1,𝔽8)
SylowC72 ⋊ C3×A4×C17
SylowC22 ⋊ C3×C7×C17×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC7×C17A4×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C32×C7×C17
FrattiniC1G
Fitting, socleC22×C72×C17C32
LCSG ▷ C22×C7
UCSC1 ◁ C7×C17

132 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 44 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

2380 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 500);