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ΓL(1,𝔽8)×D825 #46831428

order34650
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D825
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D275)
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D75)
unambiguous semidirect productC25⋊(ΓL(1,𝔽8)×D33)
unambiguous semidirect product(C7×D825)⋊C3

Splittings

Fitting, derivedC3×C7×C11×C25 ⋊ C2×C3
HallC3×C11×ΓL(1,𝔽8)×C25 ⋊ C2
HallC7×C11×C25 ⋊ C3×D3
HallC3×ΓL(1,𝔽8)×C25 ⋊ D11
HallC3×C11×ΓL(1,𝔽8) ⋊ D25
HallC11×C25 ⋊ D3×ΓL(1,𝔽8)
HallC7×C25 ⋊ C3×D33
HallC7×C11 ⋊ C3×D75
HallC3×ΓL(1,𝔽8) ⋊ D275
SylowC25 ⋊ ΓL(1,𝔽8)×D33
SylowC11 ⋊ ΓL(1,𝔽8)×D75
SylowC7 ⋊ C3×D825

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C7×C11×C25C2×C3
FrattiniC5ΓL(1,𝔽8)×D165
socleC3×C5×C7×C11C3×D5
LCSG ▷ C3×C7×C11×C25
UCSC1

108 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 108 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5775

Conjugacy classes & character table

2070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34650, 71);