Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C3 |
No aliases for this group.
| Fitting | C32×Q82 ⋊ C2 |
| Sylow | C32 ⋊ #3483 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C25 |
| Fitting | C32×Q82 | C2 |
| hypercenter | Q82 | C3⋊D3 |
| hypocenter | C32 | #3483 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
3606 conjugacy classes of subgroups; only the 855 normal subgroups are shown.
This group has 3606 conjugacy classes of subgroups — too many to draw. Only its 855 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,b^a=bf,c^a=cg,h^a=h^2,i^a=i^2,d^b=df,h^c=h^2,i^c=i^2,h^e=h^2,i^e=i^2150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 138250);