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C3×ΓL(1,𝔽8)×C109 #20293545

order6867
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C109 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×C109
unambiguous semidirect productC7⋊(C32×C109)
unambiguous semidirect product(C3×C7)⋊(C3×C109)

Splittings

Fitting, socleC3×C7×C109 ⋊ C3
derived, SylowC7 ⋊ C32×C109
HallC7×C109 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3×C109ΓL(1,𝔽8)
commutator, hypocenterC7C32×C109
FrattiniC1G
Fitting, socleC3×C7×C109C3
LCSG ▷ C7
UCSC1 ◁ C3×C109

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1635 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×C109, C3×ΓL(1,𝔽8)⋊D109, C3×ΓL(1,𝔽8)×D109, …
maximal normal subgroup ofC3×AGL(1,𝔽7)×C109, C3×ΓL(1,𝔽8)⋊D109, C3×ΓL(1,𝔽8)×D109, …
maximal quotient ofC2×C3×ΓL(1,𝔽8)×C109, A4×ΓL(1,𝔽8)×C109, C3×C22⋊ΓL(1,𝔽8)×C109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6867, 13);