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C112×A4×ΓL(1,𝔽8) #44479140

order30492
automorphism?
homologyAbelianization (H1) = C32×C112 · Schur multiplier (H2) = C2×C3×C11

Aliases

direct productC112×A4×ΓL(1,𝔽8)
unambiguous semidirect product(C22×ΓL(1,𝔽8))⋊(C3×C112)
unambiguous semidirect product(C7×A4)⋊(C3×C112)
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊(C3×C112)
unambiguous semidirect product(C22×C11×ΓL(1,𝔽8))⋊(C3×C11)
unambiguous semidirect product(C7×C11×A4)⋊(C3×C11)
unambiguous semidirect product(C11×C22⋊ΓL(1,𝔽8))⋊(C3×C11)

Splittings

Fitting, socle, HallC22×C7×C112 ⋊ C32
derived, HallC22×C7 ⋊ C32×C112
HallC7×C112 ⋊ C3×A4
HallC22×C112 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C112×A4
SylowC22 ⋊ C3×C112×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC112A4×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C32×C112
FrattiniC1G
Fitting, socleC22×C7×C112C32
LCSG ▷ C22×C7
UCSC1 ◁ C112

392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3388

Conjugacy classes & character table

2420 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 363);