Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | H(3,𝔽3)×#126 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22×C3 | C23×C3⋊D3 |
| commutator, cosocle | C22×H(3,𝔽3) | C24 |
| Fitting | H(3,𝔽3)×#126 | C2 |
| hypercenter | C3×#126 | C3⋊D3 |
| hypocenter | H(3,𝔽3) | #553 |
| LCS | G ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22×C3 ◁ C3×#126 | |
1309 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|c^2=e,d^2=f,b^a=bf,c^a=cf,d^a=df,g^a=g^2,h^a=h^2,c^b=cf,d^b=de,d^c=df,h^g=hi131 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20319);