Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #3556⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #3556⋊C9 |
| derived | #564 ⋊ C2×C9 |
| Sylow | #3556 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #3556⋊C3 |
| commutator, hypocenter | #564 | C2×C9 |
| Frattini, socle | C22×C3 | C2×C22⋊A4 |
| Fitting | C3×#3556 | C3 |
| prime radical | C23×C3 | C22⋊A4 |
| cosocle | C3×#564 | C2×C3 |
| LCS | G ▷ #564 | |
| UCS | C1 ◁ C3 | |
334 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 32 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:
2,3^2,2^6|b^3=c,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bi,d^a=di,e^a=ehi,d^b=de,e^b=di,f^b=fg,g^b=fhi,h^b=hi,i^b=h,f^d=fi,g^d=gh,f^e=fh,g^e=ghi54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153860);