Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | #73210⋊C3 |
| unambiguous semidirect product | #73211⋊C3 |
| unambiguous semidirect product | #73292⋊C3 |
| unambiguous semidirect product | #73293⋊C3 |
| Fitting, derived, Sylow | #2840 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450⋊C3 |
| commutator, Fitting, cosocle, hypocenter | #2840 | C32 |
| Frattini, socle | C23 | A42 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C2 | |
147 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3^2,2^7|e^2=g,f^2=g,e^a=f,f^a=ef,h^a=i,i^a=hi,c^b=d,d^b=cd,h^b=i,i^b=hi,e^c=ei,f^c=fhi,e^d=ehi,f^d=fh,f^e=fg34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154768);