Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2401⋊C9 |
| Sylow | #2401 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | #126⋊C3 |
| commutator, hypocenter | #95 | C22×C9 |
| Frattini, socle | C23×C3 | C22×A4 |
| Fitting | C3×#2401 | C3 |
| prime radical | C24×C3 | C2×A4 |
| cosocle | C3×#95 | C22×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C3×C4 | |
193 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 37 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|b^2=gi,c^3=d,e^2=g,f^2=gh,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^c=fi,f^c=efgi,g^c=gh,h^c=i,i^c=hi,f^e=fgi108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153460);