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A4×C17×ΓL(1,𝔽8) #16862786

order4284
automorphism?
homologyAbelianization (H1) = C32×C17 · Schur multiplier (H2) = C2×C3

Aliases

direct productA4×C17×ΓL(1,𝔽8)
unambiguous semidirect product(C22×ΓL(1,𝔽8))⋊(C3×C17)
unambiguous semidirect product(C7×A4)⋊(C3×C17)
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊(C3×C17)
unambiguous semidirect product(C22×C17×ΓL(1,𝔽8))⋊C3
unambiguous semidirect product(C7×A4×C17)⋊C3
unambiguous semidirect product(C17×C22⋊ΓL(1,𝔽8))⋊C3

Splittings

Fitting, socle, HallC22×C7×C17 ⋊ C32
derived, HallC22×C7 ⋊ C32×C17
HallC7×C17 ⋊ C3×A4
HallC22×C17 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×A4×C17
SylowC22 ⋊ C3×C17×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC17A4×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C32×C17
FrattiniC1G
Fitting, socleC22×C7×C17C32
LCSG ▷ C22×C7
UCSC1 ◁ C17

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 476

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC5×A4×C17×ΓL(1,𝔽8), C17×ΓL(1,𝔽8)×A5, A4×C17×C49⋊C3, …
maximal normal subgroup ofC5×A4×C17×ΓL(1,𝔽8), C7×A4×C17×ΓL(1,𝔽8)
maximal quotient ofC5×A4×C17×ΓL(1,𝔽8), A4×C17×C49⋊C3, C7×A4×C17×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4284, 73);