Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #2273⋊C9 |
| derived, Sylow | #2273 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C22⋊(C42⋊C3) |
| commutator, hypocenter | #2273 | C9 |
| Frattini, socle | C23×C3 | C22⋊A4 |
| Fitting, cosocle | C3×#2273 | C3 |
| prime radical | C25×C3 | A4 |
| LCS | G ▷ #2273 | |
| UCS | C1 ◁ C2×C3 | |
353 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 29 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:
3^2,2^7|a^3=b,c^2=h,d^2=g,c^a=cd,d^a=cg,e^a=ef,f^a=e,g^a=h,h^a=gh,i^a=ghi,f^c=fgi,e^d=egi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153652);