Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2383⋊C9 |
| unambiguous semidirect product | (#126⋊C9)⋊C4 |
| derived | C42 ⋊ C2×C4×C9 |
| Sylow | #2383 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #560⋊C3 |
| commutator, hypocenter | C42 | C2×C4×C9 |
| Frattini, socle | C23×C3 | C22×A4 |
| Fitting | C3×#2383 | C3 |
| prime radical | C24×C3 | C2×A4 |
| cosocle | C2×C3×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2×C3 | |
208 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 44 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:
2^2,3,2,3,2^4|a^2=d,c^3=e,f^2=h,g^2=i,c^a=chi,f^a=fhi,g^a=gh,c^b=ch,f^b=fi,g^b=ghi,f^c=gh,g^c=fg,h^c=i,i^c=hi102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153442);