Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×#380 ⋊ C2 |
| Sylow | C3 ⋊ #14680 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting | C2×C3×#380 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C25×D3 | C22 |
| hypercenter | C2×#380 | D3 |
| hypocenter | C3 | #14680 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#380 | |
6494 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=h,b^2=e,b^a=be,c^a=cf,d^a=dg,i^d=i^2144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 100725);