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C3⋊(ΓL(1,𝔽8)⋊D139) #31776136

order17514
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(ΓL(1,𝔽8)⋊D139)
unambiguous semidirect productΓL(1,𝔽8)⋊D417

Splittings

Fitting, socle, derivedC3×C7×C139 ⋊ C2×C3
HallC3×ΓL(1,𝔽8)×C139 ⋊ C2
HallC7×C139 ⋊ C3×D3
SylowC139 ⋊ C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ D139
SylowC7 ⋊ C3×D417

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C139C2×C3
LCSG ▷ C3×C7×C139
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2919

Polycyclic presentation:

2,3^2,7,139|c^a=c^2,d^a=d^6,e^a=e^138,d^b=d^2

Conjugacy classes & character table

1047 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3⋊AGL(1,𝔽7)×D139, AGL(1,𝔽7)×D417, D3×ΓL(1,𝔽8)⋊D139, …
maximal normal subgroup ofC3⋊AGL(1,𝔽7)×D139, AGL(1,𝔽7)×D417, D3×ΓL(1,𝔽8)⋊D139, …
maximal quotient ofC3⋊(ΓL(1,𝔽8)⋊Dic139), C2×C3⋊(ΓL(1,𝔽8)⋊D139)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17514, 49);