Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
No aliases for this group.
| Fitting | C32×#585 ⋊ C3 |
| Sylow | #585 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×C3×A4 |
| commutator, cosocle | C3×Q8 | C23×C32 |
| Frattini, socle | C2×C3 | C23×C3×A4 |
| Fitting | C32×#585 | C3 |
| hypercenter | C32×#41 | A4 |
| hypocenter | Q8 | C23×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C3×C4 ◁ C32×#41 | |
1134 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,g^2=i,h^2=i,c^a=ci,e^d=ef,g^d=gh,h^d=g,h^g=hi182 conjugacy classes — view the full character table.
| maximal subgroup of | #72242713, #72252733 |
| maximal normal subgroup of | #72242713, #72252733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30928);