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C5×C11⋊C5×C22⋊ΓL(1,𝔽8) #37528471

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

Aliases

direct productC5×C11⋊C5×C22⋊ΓL(1,𝔽8)
unambiguous semidirect productC22⋊(C5×ΓL(1,𝔽8)×C11⋊C5)
unambiguous semidirect productC7⋊(C5×A4×C11⋊C5)
unambiguous semidirect productC11⋊(C52×C22⋊ΓL(1,𝔽8))
unambiguous semidirect product(C22×C5)⋊(ΓL(1,𝔽8)×C11⋊C5)
unambiguous semidirect product(C5×C7)⋊(A4×C11⋊C5)
unambiguous semidirect product(C5×C11)⋊(C5×C22⋊ΓL(1,𝔽8))
unambiguous semidirect product(C22×C11⋊C5)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×C11⋊C5)⋊(C5×A4)
unambiguous semidirect product(C11×C22⋊ΓL(1,𝔽8))⋊C52
unambiguous semidirect product(C22×C5×C11⋊C5)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C5×C7×C11⋊C5)⋊A4
unambiguous semidirect product(C5×C11×C22⋊ΓL(1,𝔽8))⋊C5

Splittings

Fitting, socleC22×C5×C7×C11 ⋊ C3×C5
derived, HallC22×C7×C11 ⋊ C3×C52
HallC22×C5×C7×C11⋊C5 ⋊ C3
HallC5×C7×C11⋊C5 ⋊ A4
HallC22×C5×C11⋊C5 ⋊ ΓL(1,𝔽8)
HallC11×C22⋊ΓL(1,𝔽8) ⋊ C52
HallC7×C11 ⋊ C52×A4
HallC22×C11 ⋊ C52×ΓL(1,𝔽8)
HallC22×C7 ⋊ C3×C5×C11⋊C5
SylowC11 ⋊ C52×C22⋊ΓL(1,𝔽8)
SylowC7 ⋊ C5×A4×C11⋊C5
SylowC22 ⋊ C5×ΓL(1,𝔽8)×C11⋊C5

Subgroups

subgroupG/H
center, hypercenterC5C11⋊C5×C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7×C11C3×C52
FrattiniC1G
Fitting, socleC22×C5×C7×C11C3×C5
LCSG ▷ C22×C7×C11
UCSC1 ◁ C5

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1540

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 282);