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C3×C4×ΓL(1,𝔽8)×C127 #44544757

order32004
automorphism?
homologyAbelianization (H1) = C32×C4×C127 · Schur multiplier (H2) = C3

Aliases

direct productC3×C4×ΓL(1,𝔽8)×C127
unambiguous semidirect product(C4×C7)⋊(C32×C127)
unambiguous semidirect product(C3×C4×C7)⋊(C3×C127)

Splittings

FittingC3×C4×C7×C127 ⋊ C3
derived, SylowC7 ⋊ C32×C4×C127
HallC4×C7×C127 ⋊ C32
HallC7×C127 ⋊ C32×C4
HallC4×C7 ⋊ C32×C127

Subgroups

subgroupG/H
center, hypercenterC3×C4×C127ΓL(1,𝔽8)
commutator, hypocenterC7C32×C4×C127
FrattiniC2C2×C3×ΓL(1,𝔽8)×C127
FittingC3×C4×C7×C127C3
socleC2×C3×C7×C127C2×C3
prime radicalC2×C3×ΓL(1,𝔽8)×C127C2
cosocleC2×C7C2×C32×C127
LCSG ▷ C7
UCSC1 ◁ C3×C4×C127

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 10668

Conjugacy classes & character table

7620 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 139);