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Cayley diagram

D8×ΓL(1,𝔽8) #66584

order336
isoclinismisoclinic class of D8×ΓL(1,𝔽8)
automorphismAut(G) = AGL(1,ℤ/8)×AGL(1,𝔽7) (order 1344) · Inn(G) = D4×ΓL(1,𝔽8) · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productD8×ΓL(1,𝔽8)
unambiguous semidirect product(C7×D8)⋊C3

Splittings

Fitting, HallC7×D8 ⋊ C3
SylowC7 ⋊ C3×D8

Subgroups

subgroupG/H
centerC2D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C22×C3
FrattiniC4C22×ΓL(1,𝔽8)
FittingC7×D8C3
socleC2×C7C3×D4
hypercenterD8ΓL(1,𝔽8)
hypocenterC7C3×D8
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4 ◁ D8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6400⋊C3, #6401⋊C3, ΓL(1,𝔽8)×D16, …
maximal normal subgroup of#6400⋊C3, #6401⋊C3, ΓL(1,𝔽8)×D16, …
maximal quotient ofΓL(1,𝔽8)×#102, ΓL(1,𝔽8)×#107, ΓL(1,𝔽8)×D16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 53);