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D19⋊C3×C3⋊AGL(1,𝔽11) #49178508

order37620
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productD19⋊C3×C3⋊AGL(1,𝔽11)
unambiguous semidirect productD19⋊(C3×C3⋊AGL(1,𝔽11))
unambiguous semidirect product(C3×D19⋊C3)⋊AGL(1,𝔽11)
unambiguous semidirect product(C11⋊C5×D19⋊C3)⋊D3
unambiguous semidirect product(D33×D19⋊C3)⋊C5
unambiguous semidirect product(D19×C3⋊AGL(1,𝔽11))⋊C3

Splittings

Fitting, socle, derivedC3×C11×C19 ⋊ C22×C3×C5
HallC3×C11⋊C5×C19⋊C3 ⋊ C22
HallD33×D19⋊C3 ⋊ C5
HallC3×C11×C19⋊C3 ⋊ C22×C5
HallC19×C11⋊C5 ⋊ C2×C3×D3
HallC11×C19 ⋊ C2×C3×C5×D3
HallC3×C19⋊C3 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ D3×D19⋊C3
SylowC19 ⋊ C2×C3×C3⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×D3×D19⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C11×C19C22×C3×C5
LCSG ▷ C3×C11×C19
UCSC1

176 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 627

A minimal faithful permutation representation — 2 generators on 33 points:

(2,3)(5,9,10,8,13,6,7,14,11,12)(15,16,22)(17,26,30)(18,20,27)(19,29,21)(23,32,28)(24,31,33);(1,2,3)(4,5,8,13,7,10,14,9,12,11,6)(16,17)(18,20)(19,23)(21,22)(24,28)(25,30)(26,32)(27,33)(29,31)

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 217);