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C2×C3×ΓL(1,𝔽8)×ΓL(1,𝔽32) #34086211

order19530
automorphism?
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×ΓL(1,𝔽8)×ΓL(1,𝔽32)
unambiguous semidirect product(C2×C7)⋊(C32×ΓL(1,𝔽32))
unambiguous semidirect product(C2×C3×C7)⋊(C3×ΓL(1,𝔽32))
unambiguous semidirect product(C2×C3×ΓL(1,𝔽8)×C31)⋊C5

Splittings

Fitting, socleC2×C3×C7×C31 ⋊ C3×C5
derived, HallC7×C31 ⋊ C2×C32×C5
HallC2×C3×ΓL(1,𝔽8)×C31 ⋊ C5
HallC2×C7×ΓL(1,𝔽32) ⋊ C32
HallC3×ΓL(1,𝔽8)×C31 ⋊ C2×C5
HallC7×ΓL(1,𝔽32) ⋊ C2×C32
HallC2×C7×C31 ⋊ C32×C5
HallC2×C31 ⋊ C3×C5×ΓL(1,𝔽8)
SylowC31 ⋊ C2×C3×C5×ΓL(1,𝔽8)
HallC2×C7 ⋊ C32×ΓL(1,𝔽32)
SylowC7 ⋊ C2×C32×ΓL(1,𝔽32)

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)×ΓL(1,𝔽32)
commutator, cosocle, hypocenterC7×C31C2×C32×C5
FrattiniC1G
Fitting, socleC2×C3×C7×C31C3×C5
LCSG ▷ C7×C31
UCSC1 ◁ C2×C3

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1302

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 116);