Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #2017⋊C9 |
| Sylow | #2017 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | #127⋊C3 |
| commutator, hypocenter | #338 | C2×C9 |
| Frattini | C22×C3×C4 | C2×A4 |
| Fitting, prime radical | C3×#2017 | C3 |
| socle | C2×C3 | #392⋊C3 |
| cosocle | C3×#338 | C2×C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C3×C4 | |
140 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3^2,2^6|a^2=fgi,b^3=c,d^2=f,e^2=h,f^2=i,h^2=i,b^a=bfgh,d^a=dgi,e^a=efgh,d^b=de,e^b=dfgh,f^b=ghi,g^b=fh,h^b=fi,e^d=efg,h^d=hi,f^e=fi,g^e=gi90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153372);