Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊(#1791⋊C3) |
| unambiguous semidirect product | #72145⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#120⋊C3) |
| commutator, hypocenter | C3×H(5,𝔽2) | C3×C4 |
| Frattini | C22×C4 | D3×A4 |
| Fitting, cosocle | C3×#585 | C2×C3 |
| socle | C2×C3 | #379⋊C3 |
| prime radical | C3×#5950 | C2 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
254 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 21 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,d^a=df,e^a=efg,h^a=h^2,d^b=de,e^b=d,f^b=gi,g^b=fgi,g^d=gi,f^e=fi,g^e=gi54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154191);