Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×C4×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #32926 |
1922 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 1922 conjugacy classes of subgroups — too many to draw. Only its 247 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|b^2=e,b^a=be,c^a=cf,d^a=dg,i^a=i^2,g^c=gh,f^d=fh117 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#5244, D12×B(4,𝔽2), D3×#32926, … |
| maximal normal subgroup of | D4×#5244, D12×B(4,𝔽2), D3×#32926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 318596);