Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C2×D4×C3⋊S4 |
| Fitting | C23×C3×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C3⋊S4 |
| commutator, cosocle | C2×C3×A4 | C24 |
| Frattini | C2 | C23×C3⋊S4 |
| Fitting | C23×C3×D4 | D3 |
| socle | C24×C3 | C22×D3 |
| hypercenter | C2×D4 | C3⋊S4 |
| hypocenter | C3×A4 | C22×D4 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
3062 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 3062 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×A4⋊S4, C2×S4×C3⋊S4, D3×D4×C3⋊S4, … |
| maximal normal subgroup of | #72252920, #72252952, C2×D3×D4×S4, … |
| maximal quotient of | C2×D4×A4⋊S4, D3×D4×C3⋊S4, #72242452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157728);