Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×A4×D9 |
| commutator, cosocle | C2×Q8×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C22×Q8×C9 | C2×C3 |
| hypercenter | C23 | A4×D9 |
| hypocenter | Q8×C9 | C3×D4 |
| LCS | G ▷ C2×Q8×C9 ▷ Q8×C9 | |
| UCS | C1 ◁ C22 ◁ C23 | |
312 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3,2^3,3,2,3|a^2=h,e^2=h,f^2=h,g^3=i,b^a=bd,g^a=g^2i^2,i^a=i^2,e^c=ef,f^c=e,f^e=fh147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13231);