Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #87210⋊C3 |
| Fitting | C3×#578 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D4×A4 |
| commutator, cosocle | C3×#41 | C23×C3 |
| Frattini | C4 | C22×D3×A4 |
| Fitting | C3×#578 | C2×C3 |
| socle | C2×C3 | C2×D4×A4 |
| hypercenter | D8 | D3×A4 |
| hypocenter | C3×Q8 | C2×C3×D4 |
| LCS | G ▷ C3×#41 ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
687 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 687 conjugacy classes of subgroups — too many to draw. Only its 77 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|e^2=i,f^2=i,g^2=i,b^a=bi,c^a=ci,h^a=h^2,c^b=ce,e^b=ei,e^c=ei,f^d=fg,g^d=f,g^f=gi78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157350);