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C3×ΓL(1,𝔽8)×C71⋊C7 #44531154

order31311
automorphism?
homologyAbelianization (H1) = C32×C7 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×C71⋊C7
unambiguous semidirect productC7⋊(C32×C71⋊C7)
unambiguous semidirect product(C3×C7)⋊(C3×C71⋊C7)
unambiguous semidirect productC71⋊(C3×C7×ΓL(1,𝔽8))
unambiguous semidirect product(C3×C71)⋊(C7×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×C71)⋊(C3×C7)
unambiguous semidirect product(C7×C71⋊C7)⋊C32
unambiguous semidirect product(C3×ΓL(1,𝔽8)×C71)⋊C7
unambiguous semidirect product(C3×C7×C71⋊C7)⋊C3

Splittings

Fitting, socleC3×C7×C71 ⋊ C3×C7
derivedC7×C71 ⋊ C32×C7
HallC7×C71⋊C7 ⋊ C32
SylowC71 ⋊ C3×C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×C71⋊C7
commutator, cosocle, hypocenterC7×C71C32×C7
FrattiniC1G
Fitting, socleC3×C7×C71C3×C7
LCSG ▷ C7×C71
UCSC1 ◁ C3

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1491

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31311, 14);