Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #1779⋊C9 |
| derived | C24 ⋊ C8×C9 |
| Sylow | #1779 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | #120⋊C3 |
| commutator, hypocenter | C24 | C8×C9 |
| Frattini | C22×C3×C4 | C2×A4 |
| Fitting | C3×#1779 | C3 |
| socle | C23×C3 | C4×A4 |
| prime radical | C25×C3 | C3×C4 |
| cosocle | C24×C3×C4 | C2×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C3×C4 | |
326 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2,3,2^5|a^2=c,b^3=d,c^2=e,f^a=fh,g^a=gi,f^b=fg,g^b=f,h^b=hi,i^b=h120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153356);