Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #383 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C23 | D33 |
| commutator | C22×C33 | C22×C4 |
| Fitting, socle, prime radical, cosocle | C23×C33 | C23 |
| hypocenter | C33 | #383 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
834 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 834 conjugacy classes of subgroups — too many to draw. Only its 174 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=f,b^2=d,c^2=ef,b^a=be,c^a=cdef,i^a=i^2,c^b=cdf,g^b=g^2,h^c=h^2144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 22803);