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C2×C5×ΓL(1,𝔽8)×AGL(1,𝔽11) #37528422

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

Aliases

direct productC2×C5×ΓL(1,𝔽8)×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C2×C52×ΓL(1,𝔽8))
unambiguous semidirect product(C2×D11)⋊(C52×ΓL(1,𝔽8))
unambiguous semidirect product(C11⋊C5)⋊(C22×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C11⋊C5)⋊(C2×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C5×D11)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×ΓL(1,𝔽8)×D11)⋊C52
unambiguous semidirect product(C2×C7×AGL(1,𝔽11))⋊(C3×C5)
unambiguous semidirect product(C2×C5×ΓL(1,𝔽8)×D11)⋊C5
unambiguous semidirect product(C2×C5×C7×AGL(1,𝔽11))⋊C3

Splittings

Fitting, socleC2×C5×C7×C11 ⋊ C2×C3×C5
derived, HallC7×C11 ⋊ C22×C3×C52
HallC2×C5×C7×AGL(1,𝔽11) ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC5×C7×C11⋊C5 ⋊ C22×C3
HallC2×ΓL(1,𝔽8)×D11 ⋊ C52
HallC2×C7×D11 ⋊ C3×C52
HallC5×C11⋊C5 ⋊ C22×ΓL(1,𝔽8)
HallC11×ΓL(1,𝔽8) ⋊ C22×C52
HallC2×D11 ⋊ C52×ΓL(1,𝔽8)
SylowC11 ⋊ C22×C52×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×C5×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×AGL(1,𝔽11)
commutator, cosocle, hypocenterC7×C11C22×C3×C52
FrattiniC1G
Fitting, socleC2×C5×C7×C11C2×C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C2×C5

320 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 770

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 233);