Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #8265 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C42 | C22×D3 |
| commutator, socle, prime radical | C23×C3 | C2×C42 |
| Fitting | C3×#1273 | C2 |
| cosocle | C2×C3×C42 | C23 |
| hypercenter | #1273 | D3 |
| hypocenter | C3 | #8265 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C42 ◁ #1273 | |
438 conjugacy classes of subgroups.
| transitive degree | μt(G) = 768 |
Polycyclic presentation:
2^8,3|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,b^a=bd,c^a=ch,c^b=cg,i^b=i^2,i^c=i^2168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 60720);