Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #448 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AΓL(1,𝔽9) |
| commutator, cosocle | C2×C3×#157 | C23 |
| Fitting, socle | C2×C33 | C2×SD16 |
| prime radical | #11724116 | C2 |
| hypocenter | C33 | #448 |
| LCS | G ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
386 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=e,d^2=f,b^a=bd,d^a=df,h^a=hi,i^a=hi^2,c^b=ce,d^b=df,h^b=h^2,g^c=g^2,h^d=i,i^d=h^2,h^f=h^2,i^f=i^242 conjugacy classes — view the full character table.
| maximal subgroup of | #72259769, #72259903 |
| maximal quotient of | #72259903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46203);