Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of C3⋊(#3567⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C3⋊(#3567⋊C3) |
| unambiguous semidirect product | #87141⋊C3 |
| Fitting, derived | C3×#564 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×#564 | C2×C3 |
| Frattini | C22 | D3×C22⋊A4 |
| socle | C22×C3 | C2×C22⋊A4 |
| LCS | G ▷ C3×#564 | |
| UCS | C1 | |
378 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3,2^4,3,2^2|c^2=i,d^2=hi,e^2=i,f^2=hi,e^a=ei,f^a=fhi,g^a=g^2,c^b=d,d^b=cdh,e^b=f,f^b=efh,h^b=i,i^b=hi,e^c=eh,f^c=fhi,e^d=ehi,f^d=fi27 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157441);