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D51⋊(C3⋊AGL(1,𝔽11)) #46778009

order33660
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD51⋊(C3⋊AGL(1,𝔽11))
unambiguous semidirect product(D51⋊D33)⋊C5

Splittings

Fitting, socle, derived, HallC32×C11×C17 ⋊ C22×C5
HallC32×C17×C11⋊C5 ⋊ C22
HallD51⋊D33 ⋊ C5
HallC17×C11⋊C5 ⋊ D32
HallC32×C11⋊C5 ⋊ C2×D17
HallC11×C17 ⋊ C5×D32
HallC32×C17 ⋊ C2×AGL(1,𝔽11)
HallC32×C11 ⋊ C2×C5×D17
HallC11⋊C5 ⋊ (C3⋊D3)⋊D17
SylowC17 ⋊ D3×C3⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×(C3⋊D3)⋊D17
SylowC32 ⋊ D17×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C11×C17C22×C5
LCSG ▷ C32×C11×C17
UCSC1

176 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 34
transitive degreeμt(G) = 1122

A minimal faithful permutation representation — 2 generators on 34 points:

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

Polycyclic presentation:

2^2,5,3^2,11,17|e^a=e^2,g^a=g^16,d^b=d^2,e^b=e^2,f^b=f^10,f^c=f^4

Conjugacy classes & character table

366 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33660, 177);