Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #41⋊(Q8⋊C9) |
| Sylow | Q8×#41 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | C22⋊A4 |
| commutator, hypocenter | Q82 | C2×C9 |
| Frattini, socle | C22×C3 | C2×C22⋊A4 |
| Fitting | C3×Q8×#41 | C3 |
| prime radical | C2×C3×#41 | A4 |
| cosocle | C3×Q82 | C2×C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C3×C4 | |
419 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2,3^2,2^6|a^2=hi,b^3=c,d^2=h,e^2=h,f^2=h,g^2=hi,d^b=de,e^b=d,f^b=dgh,g^b=efgi,e^d=eh,f^d=fh,g^d=ghi,f^e=fhi,g^e=ghi114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153844);