Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10593110⋊C3 |
| derived | #63703 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11724050 |
| commutator, cosocle, hypocenter | #63703 | C2×C3 |
| Frattini, socle | C2×C3 | C3×#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| LCS | G ▷ #63703 | |
| UCS | C1 ◁ C2 | |
311 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 22 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:
2,3^3,2^5|c^3=d,g^2=i,h^2=i,c^a=c^2d^2,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,c^b=cd,e^c=f,f^c=ef,g^c=h,h^c=ghi,h^e=hi,g^f=gi,h^g=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219864, #72220106, C2×#10593110⋊C3, … |
| maximal normal subgroup of | C2×#10593110⋊C3, #72245676, #72245712, … |
| maximal quotient of | C2×#10593110⋊C3, #72252942 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30595);