Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #78128⋊C3 |
| Fitting | C3×#399 ⋊ C2×C3 |
| derived | C3×#95 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5720⋊C3 |
| commutator, cosocle, hypocenter | C3×#95 | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C3×#399 | C2×C3 |
| socle | C23×C3 | C22×A4 |
| LCS | G ▷ C3×#95 | |
| UCS | C1 ◁ C2 | |
318 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3,2^2,3,2^3|d^2=g,e^2=ghi,d^a=dhi,e^a=eh,f^a=f^2,d^b=dh,e^b=ei,d^c=de,e^c=di,g^c=gi,h^c=hi,i^c=h,e^d=egh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155206);