Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24×C3 |
| unambiguous semidirect product | C4⋊(C3⋊#1020) |
| Fitting | C24×C3×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊#1020 |
| commutator, cosocle | C2×C3×C22⋊A4 | C22 |
| Fitting | C24×C3×C4 | D3 |
| socle | C25×C3 | C2×D3 |
| hypercenter | C4 | C3⋊#1020 |
| hypocenter | C3×C22⋊A4 | D4 |
| LCS | G ▷ C2×C3×C22⋊A4 ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
956 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 956 conjugacy classes of subgroups — too many to draw. Only its 45 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:
2^3,3^2,2^4|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,f^d=g,g^d=fg,h^d=i,i^d=hi60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157780);