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#13154383

order1728
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingH(3,𝔽3)×AGL(1,ℤ/8) ⋊ C2
SylowH(3,𝔽3) ⋊ #576

Subgroups

subgroupG/H
center, socleC2×C3C2×D4×C3⋊D3
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C23×C3⋊D3
FittingH(3,𝔽3)×AGL(1,ℤ/8)C2
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterH(3,𝔽3)#576
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Polycyclic presentation:

2^6,3^3|e^2=f,b^a=bf,g^a=g^2,h^a=h^2,d^b=df,d^c=de,e^c=ef,e^d=ef,h^g=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20446);