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C3×SD16×ΓL(1,𝔽8) #11759235

order1008
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×SD16×ΓL(1,𝔽8)
unambiguous semidirect product(C7×SD16)⋊C32
unambiguous semidirect product(C3×C7×SD16)⋊C3

Splittings

FittingC3×C7×SD16 ⋊ C3
HallC7×SD16 ⋊ C32
SylowC7 ⋊ C32×SD16

Subgroups

subgroupG/H
centerC2×C3D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C22×C32
FrattiniC4C22×C3×ΓL(1,𝔽8)
FittingC3×C7×SD16C3
socleC2×C3×C7C3×D4
prime radicalC3×D4×ΓL(1,𝔽8)C2
hypercenterC3×SD16ΓL(1,𝔽8)
hypocenterC7C32×SD16
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×SD16

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)×AGL(1,ℤ/8), C3×SD16×AGL(1,𝔽7), C3×#6477⋊C3, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)×AGL(1,ℤ/8), C3×SD16×AGL(1,𝔽7), C3×#6477⋊C3, …
maximal quotient ofSD16×ΓL(1,𝔽8)2, ΓL(1,𝔽8)×#1142⋊C3, C3×ΓL(1,𝔽8)×#214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 406);