Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #500 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #88953 |
| commutator, Fitting, cosocle | C2×C33×C4 | C23 |
| Frattini, hypercenter | C2×C4 | D33 |
| socle, prime radical | C22×C33 | C2×D4 |
| hypocenter | C33 | #500 |
| LCS | G ▷ C2×C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
530 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 530 conjugacy classes of subgroups — too many to draw. Only its 116 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=df,d^2=f,b^a=be,c^a=cd,d^a=df,i^a=i^2,c^b=cd,d^b=df,g^b=g^2,h^c=h^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 22730);