Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | Q82×C9 ⋊ C2 |
| Sylow | C9 ⋊ #3483 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D9 |
| commutator, cosocle | C22×C9 | C25 |
| Frattini, socle | C22×C3 | C24×D3 |
| Fitting | Q82×C9 | C2 |
| hypercenter | Q82 | D9 |
| hypocenter | C9 | #3483 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1803 conjugacy classes of subgroups; only the 582 normal subgroups are shown.
This group has 1803 conjugacy classes of subgroups — too many to draw. Only its 582 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=f,c^2=f,d^2=f,e^2=fg,h^3=i,b^a=bf,c^a=cg,h^a=h^2i^2,i^a=i^2,d^b=df,h^c=h^2i^2,i^c=i^2,h^e=h^2i^2,i^e=i^2150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 138249);