Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | A4×#882 |
| unambiguous semidirect product | (C22×#882)⋊C3 |
| Fitting | C22×C3×C22⋊C4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×A4 |
| commutator, socle, cosocle | C24×C3 | C23×C3 |
| Fitting | C22×C3×C22⋊C4 | C2×C3 |
| hypercenter | C22⋊C4 | D3×A4 |
| hypocenter | C22×C3 | C3×#120 |
| LCS | G ▷ C24×C3 ▷ C22×C3 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
1982 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 1982 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#64210, D3×A4×#120, #72252962, … |
| maximal normal subgroup of | D3×A4×#120, #72252962, #72253012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157167);