Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C32×2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C22×C3⋊S4 |
| commutator, cosocle, hypocenter | Q8⋊H(3,𝔽3) | C23 |
| Fitting | C32×2−1+4 | D3 |
| hypercenter | C3×D4 | C3⋊S4 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
688 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 64 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:
2^3,3^3,2^3|b^2=i,g^2=i,h^2=i,c^a=ci,d^a=d^2,e^a=e^2,g^a=gh,h^a=hi,c^b=ci,e^d=ef,g^d=h,h^d=ghi,h^g=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30570);