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C3×C7×C11×ΓL(1,𝔽8) #18012929

order4851
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C7×C11 · Schur multiplier (H2) = C3

Aliases

direct productC3×C7×C11×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C32×C7×C11)
unambiguous semidirect product(C3×C7)⋊(C3×C7×C11)
unambiguous semidirect product(C7×C11)⋊(C32×C7)
unambiguous semidirect product(C3×C7×C11)⋊(C3×C7)

Splittings

Fitting, socleC3×C72×C11 ⋊ C3
derivedC7 ⋊ C32×C7×C11
HallC72×C11 ⋊ C32
SylowC72 ⋊ C32×C11

Subgroups

subgroupG/H
center, hypercenterC3×C7×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C32×C7×C11
FrattiniC1G
Fitting, socleC3×C72×C11C3
LCSG ▷ C7
UCSC1 ◁ C3×C7×C11

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 693

Conjugacy classes & character table

1155 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×C11×A4×ΓL(1,𝔽8), C3×C7×C11×C22⋊ΓL(1,𝔽8), C3×C7×ΓL(1,𝔽8)×C11⋊C5, …
maximal normal subgroup ofC3×C7×ΓL(1,𝔽8)×C11⋊C5, C3×C5×C7×C11×ΓL(1,𝔽8)
maximal quotient ofC7×C11×A4×ΓL(1,𝔽8), C3×C7×C11×C22⋊ΓL(1,𝔽8), C3×C5×C7×C11×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4851, 12);