Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | #2276 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical | C23×C3 | C22×A4 |
| commutator, hypocenter | Q8 | C42×C9 |
| Fitting | C3×#2276 | C3 |
| cosocle | C22×C3×Q8 | C22×C3 |
| hypercenter | C3×#95 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#95 | |
333 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1152 |
Polycyclic presentation:
2^2,3,2^2,3,2^3|a^2=e,b^2=d,c^3=f,g^2=i,h^2=i,b^a=bi,g^c=h,h^c=ghi,h^g=hi228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153416);