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C3⋊(C53⋊AGL(1,𝔽11)) #31775829

order17490
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C53⋊AGL(1,𝔽11))
unambiguous semidirect productC53⋊(C3⋊AGL(1,𝔽11))
unambiguous semidirect product(C11⋊C5)⋊D159
unambiguous semidirect productD1749⋊C5

Splittings

Fitting, socle, derived, HallC3×C11×C53 ⋊ C2×C5
HallC3×C53×C11⋊C5 ⋊ C2
HallD1749 ⋊ C5
HallC53×C11⋊C5 ⋊ D3
HallC11×C53 ⋊ C5×D3
HallC3×C11⋊C5 ⋊ D53
HallC3×C53 ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ D159
SylowC53 ⋊ C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C5×D53
SylowC11 ⋊ C5×D159
SylowC3 ⋊ C53⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C11×C53C2×C5
LCSG ▷ C3×C11×C53
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1749

Polycyclic presentation:

2,3,5,11,53|b^a=b^2,d^a=d^10,e^a=e^52,d^c=d^4

Conjugacy classes & character table

564 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD53×C3⋊AGL(1,𝔽11), AGL(1,𝔽11)×D159, D3×C53⋊AGL(1,𝔽11), …
maximal normal subgroup ofD53×C3⋊AGL(1,𝔽11), AGL(1,𝔽11)×D159, D3×C53⋊AGL(1,𝔽11), …
maximal quotient ofC3⋊((C11⋊C5)⋊Dic53), C2×C3⋊(C53⋊AGL(1,𝔽11))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17490, 24);