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ΓL(1,𝔽8)⋊(D101⋊C5) #35234609

order21210
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊(D101⋊C5)
unambiguous semidirect product(C101⋊C5)⋊AGL(1,𝔽7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D101)⋊C5
unambiguous semidirect product(C7⋊(D101⋊C5))⋊C3

Splittings

Fitting, socle, derived, HallC7×C101 ⋊ C2×C3×C5
HallΓL(1,𝔽8)×C101⋊C5 ⋊ C2
HallC7⋊(D101⋊C5) ⋊ C3
HallΓL(1,𝔽8)⋊D101 ⋊ C5
HallC7×C101⋊C5 ⋊ C2×C3
HallΓL(1,𝔽8)×C101 ⋊ C2×C5
HallD707 ⋊ C3×C5
HallC101⋊C5 ⋊ AGL(1,𝔽7)
SylowC101 ⋊ C5×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D101⋊C5
SylowC7 ⋊ C3×D101⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C101C2×C3×C5
LCSG ▷ C7×C101
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 707

Polycyclic presentation:

2,3,5,7,101|d^a=d^6,e^a=e^100,d^b=d^2,e^c=e^95

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21210, 36);