Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | D9⋊(C22⋊SL(2,𝔽3)) |
| Fitting, derived | C22×Q8×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×Q8×C9 | C2×C3 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C22×Q8×C9 | |
| UCS | C1 ◁ C2 | |
534 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 534 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:
2,3,2^4,3,2,3|e^2=h,f^2=h,g^3=i^2,g^a=g^2i,i^a=i^2,c^b=d,d^b=cd,e^b=f,f^b=ef,g^b=gi,f^e=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30711);