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C3⋊(C3⋊(C19⋊AGL(1,𝔽11))) #34037872

order18810
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C19⋊AGL(1,𝔽11)))
unambiguous semidirect productC19⋊(C3⋊(C3⋊AGL(1,𝔽11)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊D57)
unambiguous semidirect product(C3⋊D627)⋊C5

Splittings

Fitting, socle, derived, HallC32×C11×C19 ⋊ C2×C5
HallC32×C19×C11⋊C5 ⋊ C2
HallC3⋊D627 ⋊ C5
HallC19×C11⋊C5 ⋊ C3⋊D3
HallC32×C11⋊C5 ⋊ D19
HallC11×C19 ⋊ C5×C3⋊D3
HallC32×C19 ⋊ AGL(1,𝔽11)
HallC32×C11 ⋊ C5×D19
HallC11⋊C5 ⋊ C3⋊D57
SylowC19 ⋊ C3⋊(C3⋊AGL(1,𝔽11))
SylowC11 ⋊ C5×C3⋊D57
SylowC32 ⋊ C19⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C11×C19C2×C5
LCSG ▷ C32×C11×C19
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 1881

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

606 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C3⋊(C19⋊AGL(1,𝔽11)), D57×C3⋊AGL(1,𝔽11), D19×C3⋊(C3⋊AGL(1,𝔽11)), …
maximal normal subgroup ofD3×C3⋊(C19⋊AGL(1,𝔽11)), D57×C3⋊AGL(1,𝔽11), D19×C3⋊(C3⋊AGL(1,𝔽11)), …
maximal quotient ofC3⋊(C3⋊(C19⋊(Dic11⋊C5))), C2×C3⋊(C3⋊(C19⋊AGL(1,𝔽11)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18810, 65);