Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | #125×C22⋊C9 |
| unambiguous semidirect product | (C22×#125)⋊C9 |
| Sylow | C22×#125 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C3 | C23×A4 |
| commutator | C24 | C23×C9 |
| Fitting | C22×C3×#125 | C3 |
| socle, prime radical, cosocle | C24×C3 | C23×C3 |
| hypercenter | C3×#125 | A4 |
| hypocenter | C22 | C9×#125 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#125 | |
528 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153714);