Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#2069⋊C3) |
| unambiguous semidirect product | #68722⋊C3 |
| Fitting, derived | C3×#338 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C3⋊(#396⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C3×#338 | C2×C3 |
| Frattini | C22×C4 | D3×A4 |
| socle | C2×C3 | #396⋊C3 |
| hypercenter | C4 | C3⋊(#126⋊C3) |
| LCS | G ▷ C3×#338 | |
| UCS | C1 ◁ C2 ◁ C4 | |
139 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 19 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:
2,3,2^2,3,2^4|c^2=f,d^2=fghi,f^2=i,c^a=chi,d^a=dgh,e^a=e^2,f^a=fi,c^b=dg,d^b=cd,f^b=fgh,g^b=h,h^b=ghi,d^c=dfg,h^c=hi,f^d=fi,g^d=gi39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154164);