Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D3×#576⋊C3 |
| unambiguous semidirect product | (D3×#576)⋊C3 |
| Fitting | C3×#576 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D4×A4 |
| commutator, cosocle | C3×#41 | C23×C3 |
| Frattini | C4 | C22×D3×A4 |
| Fitting | C3×#576 | C2×C3 |
| socle | C2×C3 | C2×D4×A4 |
| hypercenter | Q16 | D3×A4 |
| hypocenter | C3×Q8 | C2×C3×D4 |
| LCS | G ▷ C3×#41 ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
711 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 711 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#226⋊A4, D3×#3585⋊C3, D3×#85468, … |
| maximal normal subgroup of | D3×#3585⋊C3, D3×#85468, D3×#85467 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157341);