Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #82601⋊C3 |
| Fitting | C2×C3×#120 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×#120⋊C3 |
| commutator, cosocle | C25×C3 | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C2×C3×#120 | C2×C3 |
| socle | C23×C3 | C22×A4 |
| hypocenter | C24×C3 | C3×D4 |
| LCS | G ▷ C25×C3 ▷ C24×C3 | |
| UCS | C1 ◁ C2 | |
527 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3,2^5,3|b^a=bd,e^a=eg,f^a=fgh,i^a=i^2,e^b=eg,f^b=fgh,e^c=ef,f^c=e,g^c=h,h^c=gh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155297);