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D9×C19⋊AGL(1,𝔽11) #49178337

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

Aliases

direct productD9×C19⋊AGL(1,𝔽11)
unambiguous semidirect product(D9×C19)⋊AGL(1,𝔽11)
unambiguous semidirect product(D9×C11⋊C5)⋊D19
unambiguous semidirect product(D9×D209)⋊C5

Splittings

Fitting, derived, HallC9×C11×C19 ⋊ C22×C5
HallC9×C19×C11⋊C5 ⋊ C22
HallD9×D209 ⋊ C5
HallC19×C11⋊C5 ⋊ C2×D9
HallC9×C11⋊C5 ⋊ C2×D19
HallC11×C19 ⋊ C2×C5×D9
HallC9×C19 ⋊ C2×AGL(1,𝔽11)
HallC9×C11 ⋊ C2×C5×D19
HallC11⋊C5 ⋊ D9×D19
SylowC19 ⋊ D9×AGL(1,𝔽11)
SylowC11 ⋊ C5×D9×D19
SylowC9 ⋊ C2×C19⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC9×C11×C19C22×C5
FrattiniC3D3×C19⋊AGL(1,𝔽11)
socleC3×C11×C19C2×C5×D3
LCSG ▷ C9×C11×C19
UCSC1

120 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 39
transitive degreeμt(G) = 1881

A minimal faithful permutation representation — 2 generators on 39 points:

(1,6,4,3,8,5,7,2,9)(11,15,16,14,19,12,13,20,17,18)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39);(2,4)(3,7)(5,8)(6,9)(10,11,14,19,13,16,20,15,18,17,12)(21,22,24,26,28,30,32,34,36,38,39,37,35,33,31,29,27,25,23)

Conjugacy classes & character table

444 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 46);