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C4×ΓL(1,𝔽8)×C127⋊C3 #44544725

order32004
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC4×ΓL(1,𝔽8)×C127⋊C3
unambiguous semidirect product(C4×C7×C127⋊C3)⋊C3
unambiguous semidirect product(C4×ΓL(1,𝔽8)×C127)⋊C3
unambiguous semidirect product(C4×#14581888)⋊C3
unambiguous semidirect product(C4×#14581887)⋊C3

Splittings

Fitting, HallC4×C7×C127 ⋊ C32
derived, HallC7×C127 ⋊ C32×C4
HallC4×C127 ⋊ C3×ΓL(1,𝔽8)
SylowC127 ⋊ C3×C4×ΓL(1,𝔽8)
HallC4×C7 ⋊ C3×C127⋊C3
SylowC7 ⋊ C3×C4×C127⋊C3

Subgroups

subgroupG/H
center, hypercenterC4ΓL(1,𝔽8)×C127⋊C3
commutator, hypocenterC7×C127C32×C4
FrattiniC2C2×ΓL(1,𝔽8)×C127⋊C3
FittingC4×C7×C127C32
socle, cosocleC2×C7×C127C2×C32
prime radicalC2×ΓL(1,𝔽8)×C127⋊C3C2
LCSG ▷ C7×C127
UCSC1 ◁ C4

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3556

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 107);