Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C3⋊(#2029⋊C3) |
| unambiguous semidirect product | #71897⋊C3 |
| Fitting, derived | C3×#543 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(#120⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C3×#543 | C2×C3 |
| Frattini | C24 | D3×A4 |
| socle | C22×C3 | #120⋊C3 |
| LCS | G ▷ C3×#543 | |
| UCS | C1 ◁ C22 | |
301 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2,3,2^4,3,2^2|c^2=h,d^2=h,c^a=ce,d^a=def,g^a=g^2,c^b=cd,d^b=c,e^b=fi,f^b=efi,d^c=dh,f^c=fi,e^d=ei,f^d=fi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154182);