Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4×#857 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×#857 |
| unambiguous semidirect product | (C22×#857)⋊C3 |
| Fitting | C22×C42 ⋊ C3×D3 |
| derived | C22×C42⋊C3 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42⋊C3 | C2×C3 |
| Frattini | C22 | A4×S4 |
| Fitting | C22×C42 | C3×D3 |
| socle | C24 | C3×S4 |
| LCS | G ▷ C22×C42⋊C3 | |
| UCS | C1 | |
222 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | C42⋊C3×#857, #72225929, A4×#857⋊C2, … |
| maximal normal subgroup of | A4×#857⋊C2, S4×#857, #72226115 |
| maximal quotient of | C42⋊C3×#857, #72225929 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155452);