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C7×C11⋊C5×C22⋊ΓL(1,𝔽8) #44568612

order32340
automorphism?
homologyAbelianization (H1) = C3×C5×C7 · Schur multiplier (H2) = C2

Aliases

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

Splittings

Fitting, socle, HallC22×C72×C11 ⋊ C3×C5
derivedC22×C7×C11 ⋊ C3×C5×C7
HallC22×C72×C11⋊C5 ⋊ C3
HallC7×C11×C22⋊ΓL(1,𝔽8) ⋊ C5
HallC72×C11⋊C5 ⋊ A4
HallC72×C11 ⋊ C5×A4
HallC22×C11⋊C5 ⋊ C7×ΓL(1,𝔽8)
HallC22×C72 ⋊ C3×C11⋊C5
SylowC72 ⋊ A4×C11⋊C5
HallC22×C11 ⋊ C5×C7×ΓL(1,𝔽8)
SylowC11 ⋊ C5×C7×C22⋊ΓL(1,𝔽8)
SylowC22 ⋊ C7×ΓL(1,𝔽8)×C11⋊C5

Subgroups

subgroupG/H
center, hypercenterC7C11⋊C5×C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7×C11C3×C5×C7
FrattiniC1G
Fitting, socleC22×C72×C11C3×C5
LCSG ▷ C22×C7×C11
UCSC1 ◁ C7

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 462

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 314);