Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×C4×M32 ⋊ C2 |
| Sylow | C3 ⋊ #33748 |
592 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|c^2=fg,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^2,c^b=ch,d^c=dh132 conjugacy classes — view the full character table.
| maximal subgroup of | #72225820, D3×#33748, #72225941, … |
| maximal normal subgroup of | #72225820, D3×#33748, #72225941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 325264);