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C5×AGL(1,𝔽7)×AGL(1,𝔽11) #37528409

order23100
automorphism?
homologyAbelianization (H1) = C22×C3×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC5×AGL(1,𝔽7)×AGL(1,𝔽11)
unambiguous semidirect productD7⋊(C3×C5×AGL(1,𝔽11))
unambiguous semidirect productD11⋊(C52×AGL(1,𝔽7))
unambiguous semidirect product(D11×AGL(1,𝔽7))⋊C52
unambiguous semidirect product(D7×AGL(1,𝔽11))⋊(C3×C5)
unambiguous semidirect product(ΓL(1,𝔽8)×AGL(1,𝔽11))⋊(C2×C5)
unambiguous semidirect product(AGL(1,𝔽7)×C11⋊C5)⋊(C2×C5)
unambiguous semidirect product(ΓL(1,𝔽8)⋊AGL(1,𝔽11))⋊(C2×C5)
unambiguous semidirect product(C5×D11×AGL(1,𝔽7))⋊C5
unambiguous semidirect product(C5×D7×AGL(1,𝔽11))⋊C3

Splittings

Fitting, socleC5×C7×C11 ⋊ C22×C3×C5
derived, HallC7×C11 ⋊ C22×C3×C52
HallC5×D7×AGL(1,𝔽11) ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC5×C7×C11⋊C5 ⋊ C22×C3
HallD11×AGL(1,𝔽7) ⋊ C52
HallD7×D11 ⋊ C3×C52
HallC5×C11⋊C5 ⋊ C2×AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C22×C52
HallΓL(1,𝔽8) ⋊ C2×C5×AGL(1,𝔽11)
SylowC11 ⋊ C2×C52×AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×C5×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽7)×AGL(1,𝔽11)
commutator, cosocle, hypocenterC7×C11C22×C3×C52
FrattiniC1G
Fitting, socleC5×C7×C11C22×C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C5

320 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 385

Conjugacy classes & character table

385 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 220);