Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#3459 ⋊ C2 |
| Sylow | C3 ⋊ #23595 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #953 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#3459 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #3459 | D3 |
| hypocenter | C3 | #23595 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ #3459 | |
1616 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1616 conjugacy classes of subgroups — too many to draw. Only its 199 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|d^2=e,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,i^c=i^296 conjugacy classes — view the full character table.
| maximal subgroup of | #72271088, #72271105, D3×#23595, … |
| maximal normal subgroup of | #72271088, #72271105, D3×#23595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 198069);