Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #87297⋊C3 |
| Fitting | C3×#585 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | A4×#67 |
| commutator, cosocle | C3×#41 | C23×C3 |
| Frattini | C4 | C22×D3×A4 |
| Fitting | C3×#585 | C2×C3 |
| socle | C2×C3 | C2×D4×A4 |
| hypercenter | #41 | D3×A4 |
| hypocenter | C3×Q8 | C2×C3×D4 |
| LCS | G ▷ C3×#41 ▷ C3×Q8 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
704 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^3,3,2^3,3,2|b^2=i,e^2=i,f^2=i,g^2=i,c^a=ce,e^a=ei,h^a=h^2,e^c=ei,f^d=fg,g^d=f,g^f=gi90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157418);