Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting, Sylow | #63093 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D4×A4 |
| commutator, cosocle | #41 | C24×C3 |
| Frattini | C4 | C24×A4 |
| Fitting | #63093 | C3 |
| hypercenter | #576 | A4 |
| hypocenter | Q8 | C22×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #576 | |
2203 conjugacy classes of subgroups; only the 235 normal subgroups are shown.
This group has 2203 conjugacy classes of subgroups — too many to draw. Only its 235 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=i,h^2=i,b^a=bfgh,c^a=cfgh,d^a=fgi,e^a=defgi,g^a=dgh,c^b=ch,h^b=hi,h^c=hi,g^d=gi,f^e=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085282);