Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C3⋊#85666 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×C22⋊A4 |
| commutator, hypocenter | C3×H(5,𝔽2) | C3×C4 |
| Fitting, cosocle | C3×#585 | C2×C3 |
| socle | C2×C3 | C4×C22⋊A4 |
| prime radical | C3×#5950 | C2 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
576 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2,3,2^5,3,2|a^2=c,c^2=i,h^a=h^2,d^b=de,e^b=d,f^b=fg,g^b=f,g^d=gi,f^e=fi66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157446);