Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of C3⋊(#126⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊(#1839⋊C3) |
| unambiguous semidirect product | #69393⋊C3 |
| derived | C3×C42 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#126⋊C3) |
| commutator, hypocenter | C3×C42 | C3×C8 |
| Frattini | C22×C4 | D3×A4 |
| Fitting, cosocle | C3×C43 | C2×C3 |
| socle | C23×C3 | C4×A4 |
| prime radical | C2×C3×C42⋊C3 | C4 |
| LCS | G ▷ C3×C42 | |
| UCS | C1 ◁ C4 | |
129 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 27 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:
2,3,2^4,3,2^2|a^2=c,c^2=d,e^2=h,f^2=i,e^a=ehi,f^a=fh,g^a=g^2,e^b=ef,f^b=ei,h^b=hi,i^b=h72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154168);