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C3×C13×C17×ΓL(1,𝔽8) #28358199

order13923
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C13×C17 · Schur multiplier (H2) = C3

Aliases

direct productC3×C13×C17×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C32×C13×C17)
unambiguous semidirect product(C3×C7)⋊(C3×C13×C17)
unambiguous semidirect product(C7×C17)⋊(C32×C13)
unambiguous semidirect product(C3×C7×C17)⋊(C3×C13)

Splittings

Fitting, socleC3×C7×C13×C17 ⋊ C3
derived, SylowC7 ⋊ C32×C13×C17
HallC7×C13×C17 ⋊ C32
HallC7×C17 ⋊ C32×C13
HallC7×C13 ⋊ C32×C17

Subgroups

subgroupG/H
center, hypercenterC3×C13×C17ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C32×C13×C17
FrattiniC1G
Fitting, socleC3×C7×C13×C17C3
LCSG ▷ C7
UCSC1 ◁ C3×C13×C17

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 4641

Conjugacy classes & character table

3315 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C13×C17×AGL(1,𝔽7), C3×C13×C17⋊AGL(1,𝔽7), C3×C17×ΓL(1,𝔽8)⋊D13, …
maximal normal subgroup ofC3×C13×C17×AGL(1,𝔽7), C3×C13×C17⋊AGL(1,𝔽7), C3×C17×ΓL(1,𝔽8)⋊D13, …
maximal quotient ofC2×C3×C13×C17×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13923, 10);