Finite Group Databasebeta

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C2×C3×AGL(1,𝔽7)×C127 #44544855

order32004
automorphism?
homologyAbelianization (H1) = C22×C32×C127 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×AGL(1,𝔽7)×C127
unambiguous semidirect productD7⋊(C2×C32×C127)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C3×C127)
unambiguous semidirect product(C2×D7)⋊(C32×C127)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C3×C127)
unambiguous semidirect product(C2×C3×D7)⋊(C3×C127)

Splittings

Fitting, socleC2×C3×C7×C127 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C32×C127
HallC3×ΓL(1,𝔽8)×C127 ⋊ C22
HallC2×D7×C127 ⋊ C32
HallC7×C127 ⋊ C22×C32
HallC3×ΓL(1,𝔽8) ⋊ C22×C127
HallC2×D7 ⋊ C32×C127

Subgroups

subgroupG/H
center, hypercenterC2×C3×C127AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C32×C127
FrattiniC1G
Fitting, socleC2×C3×C7×C127C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C127

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5334

Conjugacy classes & character table

5334 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 237);