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C2×ΓL(1,𝔽8)×D407 #46803245

order34188
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)×D407
unambiguous semidirect product(C2×C11)⋊(ΓL(1,𝔽8)×D37)

Splittings

Fitting, socleC2×C7×C11×C37 ⋊ C2×C3
derived, HallC7×C11×C37 ⋊ C22×C3
HallC2×C7×D407 ⋊ C3
HallC11×ΓL(1,𝔽8)×C37 ⋊ C22
HallΓL(1,𝔽8)×C37 ⋊ C2×D11
HallC11×C37 ⋊ C22×ΓL(1,𝔽8)
HallC7×C37 ⋊ C2×C3×D11
HallC11×ΓL(1,𝔽8) ⋊ C2×D37
HallC7×C11 ⋊ C2×C3×D37
SylowC37 ⋊ C2×ΓL(1,𝔽8)×D11
SylowC11 ⋊ C2×ΓL(1,𝔽8)×D37
SylowC7 ⋊ C2×C3×D407

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D407
commutator, cosocle, hypocenterC7×C11×C37C22×C3
FrattiniC1G
Fitting, socleC2×C7×C11×C37C2×C3
LCSG ▷ C7×C11×C37
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5698

Conjugacy classes & character table

2050 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34188, 117);