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Cayley diagram

C2×C8×ΓL(1,𝔽8) #66582

order336
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×D4
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×C8×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7×C8)⋊C3

Splittings

Fitting, HallC2×C7×C8 ⋊ C3
derived, SylowC7 ⋊ C2×C3×C8

Subgroups

subgroupG/H
center, hypercenterC2×C8ΓL(1,𝔽8)
commutator, hypocenterC7C2×C3×C8
FrattiniC4C22×ΓL(1,𝔽8)
FittingC2×C7×C8C3
socleC22×C7C3×C4
prime radicalC22×ΓL(1,𝔽8)C4
cosocleC4×C7C22×C3
LCSG ▷ C7
UCSC1 ◁ C2×C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 2 generators on 17 points:

(4,5,6)(7,9,8)(10,11,12,14,13,15,16,17);(1,2)(3,4,5,7,6,8,9)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×ΓL(1,𝔽8)⋊C16, #6386⋊C3, C8×ΓL(1,𝔽8)⋊C4, …
maximal normal subgroup ofC2×ΓL(1,𝔽8)⋊C16, #6386⋊C3, C8×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC4×C8×ΓL(1,𝔽8), ΓL(1,𝔽8)×C22⋊C8, ΓL(1,𝔽8)×C4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 51);