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C5×C7×C11×C22⋊ΓL(1,𝔽8) #44568926

order32340
automorphism?
homologyAbelianization (H1) = C3×C5×C7×C11 · Schur multiplier (H2) = C2

Aliases

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

Splittings

Fitting, socle, HallC22×C5×C72×C11 ⋊ C3
derivedC22×C7 ⋊ C3×C5×C7×C11
HallC5×C72×C11 ⋊ A4
HallC22×C72×C11 ⋊ C3×C5
HallC22×C5×C72 ⋊ C3×C11
HallC72×C11 ⋊ C5×A4
HallC5×C72 ⋊ C11×A4
HallC22×C5×C11 ⋊ C7×ΓL(1,𝔽8)
HallC22×C72 ⋊ C3×C5×C11
SylowC72 ⋊ C5×C11×A4
HallC22×C11 ⋊ C5×C7×ΓL(1,𝔽8)
HallC22×C5 ⋊ C7×C11×ΓL(1,𝔽8)
SylowC22 ⋊ C5×C7×C11×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC5×C7×C11C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C3×C5×C7×C11
FrattiniC1G
Fitting, socleC22×C5×C72×C11C3
LCSG ▷ C22×C7
UCSC1 ◁ C5×C7×C11

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2310

Conjugacy classes & character table

4620 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 628);