Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C23×C33 ⋊ C23 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D33 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C23×C33 | C23 |
| hypercenter | C23 | D33 |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
3490 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 3490 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|d^a=df,h^a=h^2,d^b=de,g^b=g^2,d^c=def,i^c=i^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #72220314, D3×#10595278, #72222978 |
| maximal normal subgroup of | D3×#10595278, #72222978 |
| maximal quotient of | #72220314, #72220386, #72220923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39001);