Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2801⋊C9 |
| Sylow | #2801 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #560⋊C3 |
| commutator, hypocenter | #95 | C22×C9 |
| Frattini, socle | C23×C3 | C22×A4 |
| Fitting | C3×#2801 | C3 |
| prime radical | C22×C3×D4 | A4 |
| cosocle | C3×#95 | C22×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2×C3 | |
189 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 31 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:
2^2,3^2,2^5|a^2=g,c^3=d,e^2=i,f^2=h,b^a=bg,c^a=cgh,e^a=ehi,f^a=fgi,c^b=cgh,e^b=egh,f^b=fhi,e^c=fgi,f^c=efg,h^c=ghi,i^c=h,f^e=fg72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153486);