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D171⋊AGL(1,𝔽11) #49178338

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

Aliases

unambiguous semidirect productD171⋊AGL(1,𝔽11)
unambiguous semidirect product(C9⋊AGL(1,𝔽11))⋊D19
unambiguous semidirect product(C19⋊AGL(1,𝔽11))⋊D9
unambiguous semidirect product(D99⋊D19)⋊C5

Splittings

Fitting, derived, HallC9×C11×C19 ⋊ C22×C5
HallC9×C19×C11⋊C5 ⋊ C22
HallD99⋊D19 ⋊ 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 ⋊ D19×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC9×C11×C19C22×C5
FrattiniC3D57⋊AGL(1,𝔽11)
socleC3×C11×C19C2×C5×D3
LCSG ▷ C9×C11×C19
UCSC1

120 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 33 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

417 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 47);