Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #21684⋊C3 |
| Fitting, Sylow | #21684 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #21684 | C3 |
| socle | C23 | C2×C4×A4 |
| prime radical | #2397⋊C3 | C2 |
| cosocle | #376 | C22×C3 |
| hypercenter | C8 | #127⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C4 ◁ C8 | |
239 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bcgi,c^a=dfgh,d^a=cdfi,f^a=gi,g^a=fgi,c^b=cf,d^b=dg,e^b=ei,d^c=di48 conjugacy classes — view the full character table.
| maximal subgroup of | #72340609, #72340711 |
| maximal normal subgroup of | #72340609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083869);