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C2×ΓL(1,𝔽8)×(C31⋊C3)⋊C5 #34086172

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

Aliases

direct productC2×ΓL(1,𝔽8)×(C31⋊C3)⋊C5
unambiguous semidirect product(C2×C31⋊C3)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×ΓL(1,𝔽8)×C31⋊C3)⋊C5
unambiguous semidirect product(C2×C7×(C31⋊C3)⋊C5)⋊C3
unambiguous semidirect product(C2×ΓL(1,𝔽8)×ΓL(1,𝔽32))⋊C3
unambiguous semidirect product(C2×#10620253⋊C5)⋊C3
unambiguous semidirect product(C2×#10620252⋊C5)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×(C31⋊C3)⋊C5
commutator, cosocle, hypocenterC7×C31C2×C32×C5
FrattiniC1G
Fitting, socleC2×C7×C31C32×C5
LCSG ▷ C7×C31
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 434

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 77);