Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #63419⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #10588303⋊C3 |
| derived | C42⋊C9 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #63419⋊C3 |
| commutator, hypocenter | C42⋊C9 | C3×C4 |
| Frattini, socle | C23×C3 | C3×S4 |
| Fitting | C2×C32×C42 | D3 |
| prime radical | C23×C32 | S4 |
| cosocle | C2×C42⋊C9 | C2×C3 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 ◁ C2 | |
196 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3,2,3^2,2^4|a^2=c,d^3=e,f^2=h,g^2=i,d^a=d^2e^2,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,d^b=de,f^d=gh,g^d=fg,h^d=i,i^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2817);