Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #83192⋊C3 |
| Fitting | C2×C3×#127 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×#127⋊C3 |
| commutator, cosocle | C2×C3×C42 | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C2×C3×#127 | C2×C3 |
| socle | C23×C3 | C22×A4 |
| hypocenter | C3×C42 | C3×D4 |
| LCS | G ▷ C2×C3×C42 ▷ C3×C42 | |
| UCS | C1 ◁ C2 | |
403 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 32 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:
2^2,3,2^3,3,2^2|e^2=h,f^2=hi,b^a=bd,g^a=g^2,e^b=eh,f^b=fhi,e^c=ef,f^c=ehi,h^c=i,i^c=hi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155307);