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C2×(C2×AGL(1,ℤ/9))⋊C8 #13143144

order1728
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×(C2×AGL(1,ℤ/9))⋊C8
unambiguous semidirect product(C2×(C2×D9)⋊C8)⋊C3

Splittings

FittingC22×C8×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C2×C22⋊C8

Subgroups

subgroupG/H
centerC22×C4C2×AGL(1,ℤ/9)
commutatorC2×C9C22×C3×C8
FrattiniC2×C3×C4C22×C3×D3
FittingC22×C8×3−1+2C2
socleC23×C3C3×C4×D3
prime radicalC23×AGL(1,ℤ/9)C4
cosocleC2×C4×C9C23×C3
hypercenterC22×C8AGL(1,ℤ/9)
hypocenterC9C2×C3×C22⋊C8
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C4 ◁ C22×C8

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 288

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 9207);