Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C3⋊#87548 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C22⋊A4 |
| commutator, cosocle, hypocenter | C3×H(5,𝔽2) | C22×C3 |
| Fitting | C3×#585 | C2×C3 |
| socle | C2×C3 | C22×C22⋊A4 |
| hypercenter | C4 | D3×C22⋊A4 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 | |
2080 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 2080 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^2,3,2^4,3,2|a^2=i,b^2=i,b^a=bi,h^a=h^2,d^c=de,e^c=d,f^c=fg,g^c=f,g^d=gi,f^e=fi57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157844);