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Cayley diagram

C2×AΓL(1,𝔽8) #66741

order336
isoclinismisoclinic class of AΓL(1,𝔽8)
automorphismAut(G) = AΓL(1,𝔽8) (order 168) · Inn(G) = AΓL(1,𝔽8) · Out(G) = C1
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×AΓL(1,𝔽8)
unambiguous semidirect productAGL(1,𝔽8)⋊(C2×C3)
unambiguous semidirect product(C2×AGL(1,𝔽8))⋊C3

Splittings

Fitting, socle, SylowC24 ⋊ ΓL(1,𝔽8)
derivedAGL(1,𝔽8) ⋊ C2×C3
HallC2×AGL(1,𝔽8) ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2AΓL(1,𝔽8)
commutator, cosocle, hypocenterAGL(1,𝔽8)C2×C3
FrattiniC1G
Fitting, socleC24ΓL(1,𝔽8)
LCSG ▷ AGL(1,𝔽8)
UCSC1 ◁ C2

29 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 14

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AΓL(1,𝔽8), C22×AΓL(1,𝔽8), D3×AΓL(1,𝔽8), …
maximal normal subgroup ofC4×AΓL(1,𝔽8), C22×AΓL(1,𝔽8), C2×C3×AΓL(1,𝔽8), …
maximal quotient ofC4×AΓL(1,𝔽8), C22×AΓL(1,𝔽8), C2×AGL(1,𝔽8)⋊C9, …
automorphism group ofC3×AGL(1,𝔽8), C4×AGL(1,𝔽8), C2×C3×AGL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 210);