Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3524⋊C9 |
| Sylow | #3524 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22×C3 | C23×A4 |
| commutator | C2×Q8 | C23×C9 |
| Fitting, prime radical | C3×#3524 | C3 |
| cosocle | C2×C3×Q8 | C23×C3 |
| hypercenter | C3×#126 | A4 |
| hypocenter | Q8 | C9×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#126 | |
460 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 91 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:
2^3,3,2,3,2^3|a^2=i,b^2=i,c^2=ei,d^3=f,g^2=i,h^2=i,b^a=be,c^a=ci,g^d=gh,h^d=g,h^g=hi156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153780);