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C2×ΓL(1,𝔽8)×D165 #28357647

order13860
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)×D165
unambiguous semidirect product(C2×C3)⋊(ΓL(1,𝔽8)×D55)
unambiguous semidirect product(C2×C5)⋊(ΓL(1,𝔽8)×D33)
unambiguous semidirect product(C2×C11)⋊(ΓL(1,𝔽8)×D15)
unambiguous semidirect product(C2×C7×D165)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C11 ⋊ C2×C3
derivedC3×C5×C7×C11 ⋊ C22×C3
HallC3×C5×C11×ΓL(1,𝔽8) ⋊ C22
HallC3×C11×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7×C11 ⋊ C2×C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D11
HallC7×C11 ⋊ C2×C3×D15
HallC3×ΓL(1,𝔽8) ⋊ C2×D55
HallC5×C11 ⋊ C2×D3×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C3×D33
SylowC11 ⋊ C2×ΓL(1,𝔽8)×D15
SylowC7 ⋊ C2×C3×D165
SylowC5 ⋊ C2×ΓL(1,𝔽8)×D33

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D165
commutator, cosocle, hypocenterC3×C5×C7×C11C22×C3
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C11C2×C3
LCSG ▷ C3×C5×C7×C11
UCSC1 ◁ C2

176 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 2310

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13860, 314);