Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #87490⋊C3 |
| Fitting | C22×#41 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×A4×S4 |
| commutator, cosocle, hypocenter | Q8×A4 | C22×C3 |
| Fitting | C22×#41 | C3×D3 |
| socle | C23 | C2×D3×A4 |
| hypercenter | C4 | A4×S4 |
| LCS | G ▷ Q8×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
419 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3,2^2,3,2^3|b^2=g,d^2=g,e^2=g,b^a=bg,f^a=f^2,h^a=hi,d^c=de,e^c=d,e^d=eg,h^f=i,i^f=hi52 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157497);