Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×C3×GL(2,ℤ/4) |
| Fitting | C26×C3 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3 | C2×S4 |
| commutator, cosocle | C2×A4 | C24×C3 |
| Frattini | C2 | C23×C3×S4 |
| Fitting | C26×C3 | D3 |
| socle | C25×C3 | C2×D3 |
| hypercenter | C24×C3 | S4 |
| hypocenter | A4 | C22×C3×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3 | |
3356 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 3356 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157723);