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C3⋊(C11⋊((C19⋊C3)⋊AGL(1,𝔽5))) #49178591

order37620
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C11⋊((C19⋊C3)⋊AGL(1,𝔽5)))
unambiguous semidirect productC11⋊(C3⋊((C19⋊C3)⋊AGL(1,𝔽5)))
unambiguous semidirect product(C19⋊C3)⋊(C3⋊(C11⋊AGL(1,𝔽5)))
unambiguous semidirect product(C3⋊(C11⋊(C19⋊AGL(1,𝔽5))))⋊C3

Splittings

Fitting, socle, derivedC3×C5×C11×C19 ⋊ C3×C4
HallC3×C5×C11×C19⋊C3 ⋊ C4
HallC3×C11×C19⋊C3 ⋊ AGL(1,𝔽5)
HallC5×C11×C19 ⋊ C3×Dic3
HallC3×C5×C19⋊C3 ⋊ Dic11
HallC11×C19 ⋊ C3×ΓL(1,𝔽16)
HallC3×C19⋊C3 ⋊ C11⋊AGL(1,𝔽5)
HallC5×C19 ⋊ C3×Dic33
HallC5×C11 ⋊ C3⋊((C19⋊C3)⋊C4)
SylowC19 ⋊ C3×C3⋊(C11⋊AGL(1,𝔽5))
SylowC11 ⋊ C3⋊((C19⋊C3)⋊AGL(1,𝔽5))
SylowC5 ⋊ C3⋊(C11⋊((C19⋊C3)⋊C4))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C11×C19C3×C4
prime radicalC3×D5×C11×C19⋊C3C2
cosocleC3×D5×C11×C19C2×C3
LCSG ▷ C3×C5×C11×C19
UCSC1

112 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3135

Polycyclic presentation:

2,3,2,3,5,11,19|a^2=c,d^a=d^2,e^a=e^2,f^a=f^10,g^a=g^18,g^b=g^7,e^c=e^4

Conjugacy classes & character table

603 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 300);