Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C3×C4 |
No aliases for this group.
| Sylow | C33 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×PSU(3,𝔽2) |
| commutator | C2×C3×C3⋊D3 | C22×C4 |
| Fitting, socle | C22×C33 | C2×Q8 |
| prime radical, cosocle | C22×C3×C3⋊D3 | C23 |
| hypocenter | C33 | #387 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C22 | |
378 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=d,b^2=d,c^2=e,b^a=bdf,h^a=i,i^a=h^2,c^b=cf,h^b=hi^2,i^b=h^2i^2,g^c=g^2,h^d=h^2,i^d=i^260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31865);