Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C9×#436 ⋊ C2 |
| Sylow | C9 ⋊ #1687 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63492 |
| commutator, cosocle | C42×C9 | C23 |
| Frattini | C3×C42 | C22×D3 |
| Fitting | C9×#436 | C2 |
| socle | C22×C3 | #930 |
| hypercenter | #436 | D9 |
| hypocenter | C9 | #1687 |
| LCS | G ▷ C42×C9 ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #436 | |
297 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^7,3^2|b^2=df,d^2=f,e^2=g,h^3=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2i^2,i^a=i^2,c^b=cf,e^c=eg135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 15713);