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Cayley diagram

C3×AΓL(1,𝔽8) #92732

order504
isoclinismisoclinic class of AΓL(1,𝔽8)
automorphismAut(G) = D3×AΓL(1,𝔽8) (order 1008) · Inn(G) = AΓL(1,𝔽8) · Out(G) = D3
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×AΓL(1,𝔽8)
unambiguous semidirect productAGL(1,𝔽8)⋊C32
unambiguous semidirect product(C3×AGL(1,𝔽8))⋊C3

Splittings

Fitting, socleC23×C3 ⋊ ΓL(1,𝔽8)
derived, HallAGL(1,𝔽8) ⋊ C32
SylowC23 ⋊ C3×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC3AΓL(1,𝔽8)
commutator, cosocle, hypocenterAGL(1,𝔽8)C32
FrattiniC1G
Fitting, socleC23×C3ΓL(1,𝔽8)
LCSG ▷ AGL(1,𝔽8)
UCSC1 ◁ C3

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

(1,6,5,8,4,3)(2,7);(2,3,4,5,8,6,7)(9,10,11)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AΓL(1,𝔽8), C2×C3×AΓL(1,𝔽8), C9×AΓL(1,𝔽8), …
maximal normal subgroup ofD3×AΓL(1,𝔽8), C2×C3×AΓL(1,𝔽8), C9×AΓL(1,𝔽8), …
maximal quotient ofC2×C3×AΓL(1,𝔽8), C9×AΓL(1,𝔽8), #13094542, …
automorphism group ofAGL(1,𝔽8)⋊C9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 158);