Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#1790 ⋊ C2 |
| Sylow | C3 ⋊ C4×B(4,𝔽2) |
1910 conjugacy classes of subgroups; only the 257 normal subgroups are shown.
This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 257 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=g,d^2=g,b^a=be,c^a=cf,f^b=fh,e^c=eh,i^d=i^296 conjugacy classes — view the full character table.
| maximal subgroup of | #72230802, C4×D3×B(4,𝔽2), #72260293, … |
| maximal normal subgroup of | C4×D3×B(4,𝔽2), #72270940, #72271008, … |
| maximal quotient of | #72230802, #72260293 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 175547);