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C3⋊(C5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11))) #46831546

order34650
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11)))
unambiguous semidirect productC5⋊(C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11)))
unambiguous semidirect productΓL(1,𝔽8)⋊(C3⋊(C5⋊AGL(1,𝔽11)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊(C5⋊AGL(1,𝔽7)))
unambiguous semidirect product(C3⋊(C5⋊(C11⋊AGL(1,𝔽7))))⋊C5
unambiguous semidirect product(C3⋊(C5⋊(C7⋊AGL(1,𝔽11))))⋊C3

Splittings

Fitting, socle, derivedC3×C5×C7×C11 ⋊ C2×C3×C5
HallC3×C5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC5×C7×C11⋊C5 ⋊ C3×D3
HallC3×C11×ΓL(1,𝔽8) ⋊ C5×D5
HallC5×C11⋊C5 ⋊ C3⋊AGL(1,𝔽7)
HallC7×C11 ⋊ C3×C5×D15
HallC3×ΓL(1,𝔽8) ⋊ C5⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×C3⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C3×C3⋊(C5⋊AGL(1,𝔽11))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C11C2×C3×C5
LCSG ▷ C3×C5×C7×C11
UCSC1

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

normal subgroups only

This group has 192 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

transitive degreeμt(G) = 1155

Polycyclic presentation:

2,3,5,3,5,7,11|d^a=d^2,e^a=e^4,f^a=f^6,g^a=g^10,f^b=f^2,g^c=g^4

Conjugacy classes & character table

285 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34650, 189);