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C2×C11×ΓL(1,𝔽8)×D15 #28357646

order13860
automorphism?
homologyAbelianization (H1) = C22×C3×C11 · Schur multiplier (H2) = C2

Aliases

direct productC2×C11×ΓL(1,𝔽8)×D15
unambiguous semidirect product(C2×C3)⋊(D5×C11×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C5)⋊(D3×C11×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D15)⋊(C3×C11)
unambiguous semidirect product(C2×C7×C11×D15)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)×D15
commutator, cosocle, hypocenterC3×C5×C7C22×C3×C11
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C11C2×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C2×C11

176 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 66 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13860, 313);