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D187⋊AGL(1,𝔽7) #30628110

order15708
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD187⋊AGL(1,𝔽7)
unambiguous semidirect product(C11⋊AGL(1,𝔽7))⋊D17
unambiguous semidirect product(C17⋊AGL(1,𝔽7))⋊D11
unambiguous semidirect product(D77⋊D17)⋊C3

Splittings

Fitting, socle, derived, HallC7×C11×C17 ⋊ C22×C3
HallD77⋊D17 ⋊ C3
HallC11×C17×ΓL(1,𝔽8) ⋊ C22
HallC17×ΓL(1,𝔽8) ⋊ C2×D11
HallC11×ΓL(1,𝔽8) ⋊ C2×D17
HallC11×C17 ⋊ C2×AGL(1,𝔽7)
HallC7×C17 ⋊ C2×C3×D11
HallC7×C11 ⋊ C2×C3×D17
HallΓL(1,𝔽8) ⋊ D11×D17
SylowC17 ⋊ D11×AGL(1,𝔽7)
SylowC11 ⋊ D17×AGL(1,𝔽7)
SylowC7 ⋊ C3×D11×D17

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C17C22×C3
LCSG ▷ C7×C11×C17
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1309

Polycyclic presentation:

2^2,3,7,11,17|e^a=e^10,f^a=f^16,d^b=d^6,e^b=e^10,d^c=d^2

Conjugacy classes & character table

305 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15708, 31);