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C2×C17⋊(ΓL(1,𝔽8)⋊D23) #45659275

order32844
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C17⋊(ΓL(1,𝔽8)⋊D23)
unambiguous semidirect product(C2×C17)⋊(ΓL(1,𝔽8)⋊D23)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D391
unambiguous semidirect product(C2×C23)⋊(C17⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×D2737)⋊C3

Splittings

Fitting, socleC2×C7×C17×C23 ⋊ C2×C3
derived, HallC7×C17×C23 ⋊ C22×C3
HallC2×D2737 ⋊ C3
HallC17×ΓL(1,𝔽8)×C23 ⋊ C22
HallΓL(1,𝔽8)×C23 ⋊ C2×D17
HallC17×C23 ⋊ C2×AGL(1,𝔽7)
HallC17×ΓL(1,𝔽8) ⋊ C2×D23
HallC7×C23 ⋊ C2×C3×D17
HallC7×C17 ⋊ C2×C3×D23
SylowC23 ⋊ C2×C17⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×D391
SylowC17 ⋊ C2×ΓL(1,𝔽8)⋊D23
SylowC7 ⋊ C2×C3×D391

Subgroups

subgroupG/H
center, hypercenterC2C17⋊(ΓL(1,𝔽8)⋊D23)
commutator, cosocle, hypocenterC7×C17×C23C22×C3
FrattiniC1G
Fitting, socleC2×C7×C17×C23C2×C3
LCSG ▷ C7×C17×C23
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5474

Conjugacy classes & character table

1964 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32844, 40);