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C2×C3×C11×ΓL(1,𝔽8) #13063465

order1386
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C32×C11 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×C11×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C32×C11)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C11)
unambiguous semidirect product(C2×C7×C11)⋊C32
unambiguous semidirect product(C2×C3×C7×C11)⋊C3

Splittings

Fitting, socleC2×C3×C7×C11 ⋊ C3
derived, SylowC7 ⋊ C2×C32×C11
HallC2×C7×C11 ⋊ C32
HallC7×C11 ⋊ C2×C32
HallC2×C7 ⋊ C32×C11

Subgroups

subgroupG/H
center, hypercenterC2×C3×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C32×C11
FrattiniC1G
Fitting, socleC2×C3×C7×C11C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C11

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 462

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)×Dic11, C3×C11×ΓL(1,𝔽8)⋊C4, C3×C11⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofC3×ΓL(1,𝔽8)×Dic11, C3×C11×ΓL(1,𝔽8)⋊C4, C3×C11⋊(ΓL(1,𝔽8)⋊C4), …
maximal quotient ofC3×C4×C11×ΓL(1,𝔽8), C22×C3×C11×ΓL(1,𝔽8), C2×C3×ΓL(1,𝔽8)×C121, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1386, 20);