Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #7555⋊C3 |
| Fitting, Sylow | #7555 ⋊ C3 |
| derived | C22×#95 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1437⋊C3 |
| commutator, cosocle, hypocenter | C22×#95 | C2×C3 |
| Frattini, socle | C25 | C2×A4 |
| Fitting | #7555 | C3 |
| LCS | G ▷ C22×#95 | |
| UCS | C1 ◁ C2 | |
826 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 826 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=i,b^a=bcdefg,c^a=begi,d^a=df,e^a=ef,f^a=fg,g^a=f,h^a=efghi,i^a=h,c^b=ce,d^b=df,d^c=dg40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083481);