Finite Group Databasebeta

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#13048005

order1296
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C32

Aliases

No aliases for this group.

Splittings

FittingC3×C8×H(3,𝔽3) ⋊ C2
SylowC3×H(3,𝔽3) ⋊ D8

Subgroups

subgroupG/H
centerC2×C3C4⋊(C3⋊(C3⋊D3))
commutator, cosocleC3×C4×H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊(C3⋊D3)
FittingC3×C8×H(3,𝔽3)C2
socleC2×C32C4⋊(C3⋊D3)
hypercenterC3×C8C3⋊(C3⋊D3)
hypocenterC3×H(3,𝔽3)D8
LCSG ▷ C3×C4×H(3,𝔽3) ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Polycyclic presentation:

2^4,3^4|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^2,g^a=g^2,g^f=gh

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1723);