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(ΓL(1,𝔽8)⋊D7)⋊AGL(1,𝔽11) #44568552

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊AGL(1,𝔽11)
unambiguous semidirect product(ΓL(1,𝔽8)⋊AGL(1,𝔽11))⋊D7
unambiguous semidirect product((ΓL(1,𝔽8)⋊D7)⋊D11)⋊C5

Splittings

Fitting, socle, derived, HallC72×C11 ⋊ C22×C3×C5
Hall(C7⋊D7)⋊AGL(1,𝔽11) ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
Hall(ΓL(1,𝔽8)⋊D7)⋊D11 ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
Hall(C7⋊D7)⋊D11 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC7×ΓL(1,𝔽8) ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ D7×AGL(1,𝔽7)
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
SylowC11 ⋊ C5×D7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C11C22×C3×C5
LCSG ▷ C72×C11
UCSC1

168 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 462

Polycyclic presentation:

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

Conjugacy classes & character table

197 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 254);