Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C33 ⋊ C2×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D33 |
| commutator | C2×C33 | C22×C8 |
| Fitting, cosocle | C2×C33×C4 | C23 |
| socle | C22×C33 | C22×C4 |
| prime radical | C22×#1113 | C4 |
| hypocenter | C33 | C2×C22⋊C8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
1006 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 154 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=e,b^2=e,c^2=def,e^2=f,c^a=cd,i^a=i^2,c^b=cd,g^b=g^2,h^c=h^2168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 21370);