Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊(#1780⋊C3) |
| unambiguous semidirect product | #72136⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#120⋊C3) |
| commutator, hypocenter | C22×C3×Q8 | C3×C4 |
| Frattini | C22×C4 | D3×A4 |
| Fitting, cosocle | C22×C3×#41 | C2×C3 |
| socle | C23×C3 | C4×A4 |
| prime radical | C3×#5948 | C2 |
| LCS | G ▷ C22×C3×Q8 | |
| UCS | C1 ◁ C4 | |
270 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 23 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:
2,3,2^5,3,2|a^2=c,c^2=i,d^2=i,e^2=i,d^a=df,e^a=efg,h^a=h^2,d^b=de,e^b=d,f^b=g,g^b=fg,e^d=ei60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154189);