Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#2404 ⋊ C2 |
| Sylow | C3 ⋊ #32899 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#937 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#2404 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #2404 | D3 |
| hypocenter | C3 | #32899 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2404 | |
3866 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 3866 conjugacy classes of subgroups — too many to draw. Only its 251 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=h,b^a=be,c^a=cf,d^a=dg,f^b=fh,e^c=eh,i^c=i^287 conjugacy classes — view the full character table.
| maximal subgroup of | #72271216, D3×#32899, #72271251 |
| maximal normal subgroup of | #72271216, D3×#32899, #72271251 |
| maximal quotient of | #72271910, #72271924 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 318343);