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Cayley diagram

SD16×ΓL(1,𝔽8) #66585

order336
isoclinismisoclinic class of D8×ΓL(1,𝔽8)
automorphismAut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = D4×ΓL(1,𝔽8) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productSD16×ΓL(1,𝔽8)
unambiguous semidirect product(C7×SD16)⋊C3

Splittings

Fitting, HallC7×SD16 ⋊ C3
SylowC7 ⋊ C3×SD16

Subgroups

subgroupG/H
centerC2D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C22×C3
FrattiniC4C22×ΓL(1,𝔽8)
FittingC7×SD16C3
socleC2×C7C3×D4
prime radicalD4×ΓL(1,𝔽8)C2
hypercenterSD16ΓL(1,𝔽8)
hypocenterC7C3×SD16
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4 ◁ SD16

40 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)(8,9,11,14)(10,12,15,13)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×AGL(1,𝔽7), #6477⋊C3, #6478⋊C3, …
maximal normal subgroup ofSD16×AGL(1,𝔽7), #6477⋊C3, #6478⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#102, ΓL(1,𝔽8)×#103, ΓL(1,𝔽8)×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 54);