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C3×M16×ΓL(1,𝔽8) #11759233

order1008
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×M16×ΓL(1,𝔽8)
unambiguous semidirect product(C7×M16)⋊C32
unambiguous semidirect product(C3×C7×M16)⋊C3

Splittings

FittingC3×C7×M16 ⋊ C3
HallC7×M16 ⋊ C32
SylowC7 ⋊ C32×M16

Subgroups

subgroupG/H
centerC3×C4C22×ΓL(1,𝔽8)
commutatorC2×C7C2×C32×C4
FrattiniC4C22×C3×ΓL(1,𝔽8)
FittingC3×C7×M16C3
socleC2×C3×C7C2×C3×C4
prime radicalC22×C3×ΓL(1,𝔽8)C4
cosocleC4×C7C22×C32
hypercenterC3×M16ΓL(1,𝔽8)
hypocenterC7C32×M16
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C3×M16

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)×#100, C3×M16×AGL(1,𝔽7), C3×#6397⋊C3, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)×#100, C3×M16×AGL(1,𝔽7), C3×#6397⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#1136⋊C3, M16×ΓL(1,𝔽8)2, C3×ΓL(1,𝔽8)×#208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 404);