Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C28 |
| unambiguous semidirect product | C22⋊#63703 |
| derived, Sylow | C22×H(5,𝔽2) ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C22⋊(C22⋊A4) |
| commutator, hypocenter | C22×H(5,𝔽2) | C9 |
| Fitting, prime radical, cosocle | C22×C3×H(5,𝔽2) | C3 |
| socle | C23×C3 | C22⋊A4 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
2329 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 2329 conjugacy classes of subgroups — too many to draw. Only its 93 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:
3^2,2^7|a^3=b,c^a=cd,d^a=c,e^a=ef,f^a=e,g^a=gh,h^a=g,f^c=fi,e^d=ei84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153922);