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(C11⋊C5)⋊(D37⋊C3) #27131981

order12210
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(C11⋊C5)⋊(D37⋊C3)
unambiguous semidirect product(C37⋊C3)⋊AGL(1,𝔽11)
unambiguous semidirect product(C11⋊(D37⋊C3))⋊C5
unambiguous semidirect product(C37⋊AGL(1,𝔽11))⋊C3

Splittings

Fitting, socle, derived, HallC11×C37 ⋊ C2×C3×C5
HallC11⋊C5×C37⋊C3 ⋊ C2
HallC37⋊AGL(1,𝔽11) ⋊ C3
HallC11⋊(D37⋊C3) ⋊ C5
HallC37×C11⋊C5 ⋊ C2×C3
HallC11×C37⋊C3 ⋊ C2×C5
HallD407 ⋊ C3×C5
HallC37⋊C3 ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ D37⋊C3
SylowC37 ⋊ C3×AGL(1,𝔽11)
SylowC11 ⋊ C5×D37⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC11×C37C2×C3×C5
LCSG ▷ C11×C37
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,5,11,37|d^a=d^10,e^a=e^36,e^b=e^26,d^c=d^4

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×D37⋊C3, C2×(C11⋊C5)⋊(D37⋊C3), (C11⋊C5)⋊(#66526⋊C2), …
maximal normal subgroup ofAGL(1,𝔽11)×D37⋊C3, C2×(C11⋊C5)⋊(D37⋊C3), (C11⋊C5)⋊(#66526⋊C2), …
maximal quotient of(C11⋊C5)⋊(Dic37⋊C3), C2×(C11⋊C5)⋊(D37⋊C3), (C11⋊C5)⋊(#66524⋊C2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12210, 36);