Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C3⋊#10593109 |
| derived | C3×#63703 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#64188 |
| commutator, cosocle, hypocenter | C3×#63703 | C2 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | #1020 |
| LCS | G ▷ C3×#63703 | |
| UCS | C1 ◁ C2 | |
542 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|b^3=d,b^a=b^2d^2,c^a=c^2,d^a=d^2,e^a=ef,g^a=gh,e^b=f,f^b=ef,g^b=h,h^b=gh,h^e=hi,g^f=gi57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30616);