Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, derived | C9×H(5,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11720760 |
| commutator, Fitting, cosocle, hypocenter | C9×H(5,𝔽2) | C2×C3 |
| Frattini | C23×C3 | D3×A4 |
| socle | C2×C3 | C3⋊(#120⋊C3) |
| LCS | G ▷ C9×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
185 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2^4,3,2,3|g^3=i,c^a=ce,d^a=def,g^a=g^2i^2,i^a=i^2,c^b=cd,d^b=c,e^b=fh,f^b=efh,g^b=gi^2,f^c=fh,e^d=eh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2904);