Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#67994 |
| Fitting | C3×#338 ⋊ D3 |
| derived | C3×#338⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C3⋊#4622 |
| commutator, cosocle, hypocenter | C3×#338⋊C3 | C2 |
| Frattini | C22×C4 | C3⋊S4 |
| Fitting | C3×#338 | D3 |
| socle | C2×C3 | #4622 |
| hypercenter | C4 | C3⋊#857 |
| LCS | G ▷ C3×#338⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
158 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 17 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:
2,3^2,2^6|d^2=f,e^2=fgh,f^2=i,b^a=b^2,c^a=c^2,d^a=deh,e^a=efghi,f^a=fh,g^a=ghi,d^b=eg,e^b=defg,f^b=fghi,g^b=hi,h^b=gh,e^d=efg,h^d=hi,f^e=fi,g^e=gi36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154141);