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(C3⋊D3)⋊(C19⋊AGL(1,𝔽11)) #49178552

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

Aliases

unambiguous semidirect product(C3⋊D3)⋊(C19⋊AGL(1,𝔽11))
unambiguous semidirect product((C3⋊D3)⋊D209)⋊C5

Splittings

Fitting, socle, derived, HallC32×C11×C19 ⋊ C22×C5
HallC32×C19×C11⋊C5 ⋊ C22
Hall(C3⋊D3)⋊D209 ⋊ C5
HallC19×C11⋊C5 ⋊ D32
HallC32×C11⋊C5 ⋊ C2×D19
HallC11×C19 ⋊ C5×D32
HallC32×C19 ⋊ C2×AGL(1,𝔽11)
HallC32×C11 ⋊ C2×C5×D19
HallC11⋊C5 ⋊ (C3⋊D3)⋊D19
SylowC19 ⋊ (C3⋊D3)⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×(C3⋊D3)⋊D19
SylowC32 ⋊ C2×C19⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C11×C19C22×C5
LCSG ▷ C32×C11×C19
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) = 36
transitive degreeμt(G) = 1254

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

429 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 261);