Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | #392⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#392⋊C3)⋊C3 |
| Fitting | C32×#392 ⋊ C3 |
| Sylow | #392 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C3×#127⋊C3 |
| commutator, cosocle | C3×#95 | C2×C32 |
| Frattini, socle | C23×C3 | C2×C3×A4 |
| Fitting | C32×#392 | C3 |
| hypercenter | C2×C32 | #127⋊C3 |
| hypocenter | #95 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×#95 ▷ #95 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 | |
394 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3^3,2^5|a^2=gh,e^2=g,f^2=gi,e^a=eh,f^a=fhi,c^b=cd,e^b=ef,f^b=ehi,g^b=ghi,h^b=i,i^b=hi,f^e=fgh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7300);