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) ⋊ #560 |
| 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) | #560 |
| LCS | G ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22×C3 ◁ C3×#126 | |
1386 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|c^2=f,d^2=e,c^a=cf,d^a=def,g^a=g^2,h^a=h^2,c^b=ce,d^b=df,d^c=df,h^g=hi113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20320);