Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3529⋊C9 |
| Sylow | #3529 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | C22⋊A4 |
| commutator, hypocenter | H(3,𝔽4) | C2×C9 |
| Frattini, socle | C22×C3 | C2×C22⋊A4 |
| Fitting, prime radical | C3×#3529 | C3 |
| cosocle | C3×H(3,𝔽4) | C2×C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C2×C3×C4 | |
439 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 439 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) = 288 |
Polycyclic presentation:
2,3^2,2^6|a^2=h,b^3=c,d^2=h,e^2=h,d^b=e,e^b=deh,f^b=fg,g^b=f,e^d=eh,f^d=fhi,g^d=gi,f^e=fi,g^e=gh102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153852);