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D111⋊AGL(1,𝔽11) #37567531

order24420
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD111⋊AGL(1,𝔽11)
unambiguous semidirect product(C3⋊AGL(1,𝔽11))⋊D37
unambiguous semidirect product(C37⋊AGL(1,𝔽11))⋊D3
unambiguous semidirect product(D33⋊D37)⋊C5

Splittings

Fitting, socle, derived, HallC3×C11×C37 ⋊ C22×C5
HallC3×C37×C11⋊C5 ⋊ C22
HallD33⋊D37 ⋊ C5
HallC37×C11⋊C5 ⋊ C2×D3
HallC11×C37 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ C2×D37
HallC3×C37 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ D3×D37
SylowC37 ⋊ D3×AGL(1,𝔽11)
HallC3×C11 ⋊ C2×C5×D37
SylowC11 ⋊ C5×D3×D37
SylowC3 ⋊ D37×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C11×C37C22×C5
LCSG ▷ C3×C11×C37
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1221

Polycyclic presentation:

2^2,5,3,11,37|e^a=e^10,f^a=f^36,d^b=d^2,e^b=e^10,e^c=e^4

Conjugacy classes & character table

357 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24420, 52);