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C2×C3×ΓL(1,𝔽8)×D71 #31822079

order17892
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×ΓL(1,𝔽8)×D71
unambiguous semidirect productC71⋊(C22×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C71)⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D71)⋊C32
unambiguous semidirect product(C2×C3×C7×D71)⋊C3

Splittings

Fitting, socleC2×C3×C7×C71 ⋊ C2×C3
derived, HallC7×C71 ⋊ C22×C32
HallC3×ΓL(1,𝔽8)×C71 ⋊ C22
HallC2×C7×D71 ⋊ C32
SylowC71 ⋊ C22×C3×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C32×D71

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)×D71
commutator, cosocle, hypocenterC7×C71C22×C32
FrattiniC1G
Fitting, socleC2×C3×C7×C71C2×C3
LCSG ▷ C7×C71
UCSC1 ◁ C2×C3

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2982

Conjugacy classes & character table

1110 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17892, 96);