Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#1783 ⋊ C2 |
| Sylow | C3 ⋊ #23907 |
1942 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 1942 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,i^a=i^2,c^b=cg,f^b=fh,e^c=eh,f^c=fh,i^c=i^2,f^d=fh78 conjugacy classes — view the full character table.
| maximal subgroup of | #72255076, #72255078, #72255084, … |
| maximal normal subgroup of | #72255076, #72255078, #72255084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 201212);