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C2×ΓL(1,𝔽8)×D127⋊C3 #44544802

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

Aliases

direct productC2×ΓL(1,𝔽8)×D127⋊C3
unambiguous semidirect product(C127⋊C3)⋊(C22×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C127⋊C3)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D127⋊C3)⋊C3
unambiguous semidirect product(C2×#19128624)⋊C3
unambiguous semidirect product(C2×#19128625)⋊C3
unambiguous semidirect product(C2×ΓL(1,𝔽8)×D127)⋊C3

Splittings

Fitting, socleC2×C7×C127 ⋊ C2×C32
derived, HallC7×C127 ⋊ C22×C32
HallΓL(1,𝔽8)×C127⋊C3 ⋊ C22
HallC2×C7×D127 ⋊ C32
HallC2×D127 ⋊ C3×ΓL(1,𝔽8)
SylowC127 ⋊ C22×C3×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×D127⋊C3

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D127⋊C3
commutator, cosocle, hypocenterC7×C127C22×C32
FrattiniC1G
Fitting, socleC2×C7×C127C2×C32
LCSG ▷ C7×C127
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1778

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 184);